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Restricted uniform rational approximations Borwein, Peter B. 1979

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RESTRICTED UNIFORM RATIONAL APPROXIMATIONS PETER B. BORWEIN B. S c , U n i v e r s i t y of Western Ontar i o , 1974 M.Sc, U n i v e r s i t y of B r i t i s h Columbia, 1976 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY ' i n THE FACULTY OF GRADUATE STUDIES i n the Department of MATHEMATICS We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA-NS) Peter B. Borwein , 1979 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d tha t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l 1 not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date Fj. l» WW - i i -Thesis Supervisor: Professor D. Boyd ABSTRACT We consider the approximation of continuous functions from various r e s t r i c t e d classes of r a t i o n a l functions. More p r e c i s e l y , for a function f continuous on [0,1] we consider e = min ||f - r ||r , n r £ A n " [ 0 , l ] n n where I I R^ •, T i s the uniform norm on [0,1] and A i s some 1 1 LO,1J n r e s t r i c t e d subset of the r e a l r a t i o n a l functions of degree n . We are p r i m a r i l y concerned with the following three r e s t r i c t i o n s : monotone denominators, denominators having p o s i t i v e c o e f f i c i e n t s and both numerators and denominators having p o s i t i v e c o e f f i c i e n t s . We compare the order of approximation (the asymptotic behaviour of e ) from r e s t r i c t e d r a t i o n a l s to the orders of approximation by poly-nomials and unr e s t r i c t e d r a t i o n a l s . We show, for functions that are exactly k-times d i f f e r e n t i a b l e on [0,1], that r e s t r i c t e d r a t i o n a l approximation cannot be spectacularly more e f f i c i e n t than polynomial approximation. For example, i f A^ i s the cla s s of r a t i o n a l functions of degree n with monotone denominators and for some f the corresponding = 0( > then f i s k-times n continuously d i f f e r e n t i a b l e on (0,1] . The s i t u a t i o n i s more complicated f o r a n a l y t i c functions. In t h i s case r e s t r i c t e d r a t i o n a l approximations vary from being no more e f f i c i e n t than polynomials to being a r b i t r a r i l y f a s t e r . For functions of the form - i i i -/.a^ x , where 4- 0, we show that the order of approximation by ra t i o n a l s with denominators having p o s i t i v e c o e f f i c i e n t s i s the same as the polynomial order. F i n a l l y , we develop approximation theorems for r a t i o n a l functions with p o s i t i v e c o e f f i c i e n t s . (This class does not contain the poly-nomials so we cannot a p r i o r i assert denseness.) We show that i t i s possible to approximate p o s i t i v e functions f a i r l y e f f i c i e n t l y from t h i s c l a s s . For instance, i f f i s a n a l y t i c and p o s i t i v e on [0,1] then the order of approximation by r a t i o n a l functions of degree n with p o s i t i v e c o e f f i c i e n t s i s at l e a s t as rapid as ^ , where p > 1 . rn - • P - iv -T A B L E O F C O N T E N T S S E C T I O N 0 I N T R O D U C T I O N 1 S E C T I O N 1 N O T A T I O N • :i_ . 3 S E C T I O N 2 B E R N S T E I N - M A R K O V I N E Q U A L I T I E S 5 - F O R R F . . . 6 n - F O R R + 7 n - F O R R A T I O N A L S W I T H R E S T R I C T E D P O L E S . . . . 12 S E C T I O N 3 I N V E R S E T H E O R E M S F O R C k[a,b] . . . . . . 18 - E X A M P L E S 22 S E C T I O N 4 I N V E R S E T H E O R E M S A N D A N A L Y T I C F U N C T I O N S . . . 25 - F O R R A T I O N A L S W I T H R E S T R I C T E D P O L E S . . . . 26 - F O R A N A L Y T I C F U N C T I O N S W I T H P O S I T I V E C O E F F I C I E N T S 28 - T H E E F F E C T O F A S I N G U L A R I T Y 36 S E C T I O N 5 A P P R O X I M A T I N G x* 1 40 - F R O M n + A N D R + 41 n n OO - A P P R O X I M A T I N G Y c x™ F R O M R + . . . . 44 m n + 1 - F R O M R 45 n S E C T I O N 6 P O L Y N O M I A L S A N D T H E C L A S S R 4 " * 49 n - W R I T I N G P O L Y N O M I A L S A S R A T I O N A L S I N . . 49 n - R"1"4" A N D P O L Y N O M I A L S W I T H R E S T R I C T E D R O O T S . . 55 n - n A N D R A T I O N A L S W I T H R E S T R I C T E D P O L E S . . 60 n - v -II SECTION 7 APPROXIMATIONS FROM THE CLASS R . . . . 64 n - FOR C k[a,b] 65 - FOR ANALYTIC FUNCTIONS 67 SECTION 8 INVERSE THEOREMS FOR RATIONAL FUNCTIONS WITH NO POLES IN {re z > 0} 72 1/2 SECTION 9 APPROXIMATING x"^ 2 and e X FROM R 4^ . . 75 n SECTION 10 SOME COMMENTS ON INTERVAL DEPENDENCE . . . . .80 REFERENCES 85 BIBLIOGRAPHY 87 - v i -. • ACKNOWLEDGEMENTS I t i s my pleasure to thank Dr. D. Boyd f o r h i s a s s i s t a n c e and encouragement during the pre p a r a t i o n of t h i s t h e s i s . I would a l s o l i k e to thank Dr. Fournier and Dr. Adams f o r t h e i r comments and c o r r e c t i o n s , Cathy Agnew f o r her e x c e l l e n t typing and J e n n i f e r Moore f o r her p a t i e n t proofreading. - 1 -INTRODUCTION. A ce n t r a l problem i n approximation theory has been to r e l a t e s t r u c t u r a l properties of a function to the rate of convergence of a sequence of approximants. The so l u t i o n to t h i s problem ex i s t s i n a most complete and elegant form for functions continuous on the unit i n t e r v a l (or the c i r c l e ) being approximated uniformly by algebraic polynomials (or trigonometric polynomials) of degree n . Though many people have contributed to the century long development of t h i s theory, much of the c r e d i t must go to S.N. Bernstein. Informally, the order of approximation i s the asymptotic rate of convergence of a sequence of approximants. T y p i c a l of the r e s u l t s Bernstein and Jackson obtained i s the following: a function f i s t t l k-times continuously d i f f e r e n t i a b l e on the c i r c l e with k d e r i v a t i v e i n the cla s s l i p a for some 0 < a < 1 i f and only i f f may be uniformly approximated by trigonometric polynomials of degree n with order ^ ([11] p. 62). No such r e s u l t s e x i s t f o r uniform approxi-n mation on [0,1] by r a t i o n a l functions of degree n . (A r a t i o n a l function of degree n i s the quotient of two algebraic polynomials each of degree at most n .) We may, for instance, construct functions that are exactly once d i f f e r e n t i a b l e whose respective orders of approxi-mation, by r a t i o n a l functions, are as rapid as desired or as slow as 1 n It i s easy to construct functions that may be approximated more e f f i c i e n t l y by r a t i o n a l s than by polynomials ([11] p. 91) and for some classes, notably r e s t r i c t i o n s of meromorphic functions, t h i s i s always the case. However, i t was not u n t i l 1964 when D.J. Newman [16] constructed r a t i o n a l functions r of degree n so that sup |r (x) - |x|| <_ 3e ^ X 6 [ - l , l ] 1 1 - 2 -(compare to an asymptotic order of ^ by polynomials of degree n on [-1,1]), that much attention was paid to t h i s phenomenon. In the l i g h t of the above comments i t i s hard to see what type of conditions on f would y i e l d an exact r e s u l t of the Jackson-Bernstein type f o r r a t i o n a l approximations. In t h i s thesis we consider the question of approximation by r e s t r i c t e d classes of r a t i o n a l functions. The p a r t i c u l a r classes we consider include r a t i o n a l functions with monotone denominators, r a t i o n a l functions with p o s i t i v e c o e f f i c i e n t s and r a t i o n a l functions with r e s t r i c t e d poles. In a l l the above cases we recover many of the c h a r a c t e r i s t i c s of polynomial approximations. For example, i f a function f can be approximated uniformly on [0,1] by r a t i o n a l functions of degree n with monotone denominators with an order of approximation , then f i s k-times continuously d i f f e r e n t i a b l e on (0,1] . n As mentioned previously, no such r e s u l t e x i s t s f or u n r e s t r i c t e d r a t i o n a l functions although A.A. Goncar has shown that i f f has order of approximation ^ by r a t i o n a l functions on [0,1] then f has a. n k1"'1 d e r i v a t i v e and f ^ ) i s an element of l i p a except on a set of a r b i t r a r i l y small measure [ 9 ] . The classes of r a t i o n a l functions we consider l i e , i n general, between the polynomials and the r a t i o n a l s . Thus, i t i s appropriate to examine how approximations from these classes compare to both of these more f a m i l i a r cases. As would be expected these intermediary classes demonstrate some of the p e c u l i a r i t i e s of each type of approximation. - 3 -1 - NOTATION. Let II denote the algebraic polynomials with r e a l n c o e f f i c i e n t s of degree n . Let n + denote those p e II where b n r n n p has non-negative c o e f f i c i e n t s . Let R denote those r a t i o n a l n n,m functions p /q where p e II and q_ e II . We define three n m n n Tn m r e s t r i c t e d classes of r a t i o n a l s . Let R + , + be those p /q e R n m n m n,m where p e II and q_ e 31 . Let R be those p /q_ e R r n n Tn m n,m n TIT n,m where q e II . Let R [a,b] be those p /q_ e R where q_ T n m n,m n TU n,m TII i s non-decreasing on [a,b] . F i n a l l y , we define the class P.P.C. (polynomials with p o s i t i v e c o e f f i c i e n t s [ 2 ] ) to be the polynomials with p o s i t i v e c o e f f i c i e n t s i n x and (1-x) . That i s P(x) = Ia k £ x k ( l - x ) \ a k £ > 0 . + + t If a•> - 0 then n c p . P . c . c n c R c R c R and — n n n,m n,m n,m + + + + + I I G R C R c R C R . When n = m we often contract n n,m n,m n,m n,m i + R to R , R to R etc. n,m n n,m n We use the notation IIfII r . n for the supremum norm of f 1 1 1 1[a,b] on the i n t e r v a l [a,b] . We define the distance n ( f : [a,b]) = i n f ||f - p || . p eE n n We define II +(f: [a,b]), R"*"1" ( f : [a,b]), R + ( f : [a,b]), R * ( f : [a,b]) n n,m n,m ' n,m and R ( f : [a,b]) analogously; these f i v e numbers are r e s p e c t i v e l y , n,m + ++ + t the distances from f to * II , R , R , R [a,b] and R n n,m n,m n,m n,m When the i n t e r v a l of approximation i s understood then R ^ m ( f ; [a»b]) i s contracted to R ( f ) , etc. n,m - 4 -A l l the above infimums are attained. This observation guarantees the existence of best approximants from a l l the above classes. We use the notation C [a,b] f o r those functions which have k continuous de r i v a t i v e s on [a,b] . For f e C[a,b] we define the modulus of continuity by o)(f,h) = u(h) = max |f(x+t) - f(x) x,x+te[a,b] 111 <h F i n a l l y , we say that f e C[a,b] belongs to the class l i P M a i f , f o r a l l x,y e [a,b], f (x) - f (y) | <_ M|x - y | a - 5 -2 - BERNSTEIN-MARKOV INEQUALITIES. ' A Bernstein-Markov i n e q u a l i t y i s an i n e q u a l i t y that bounds the d e r i v a t i v e of a polynomial or r a t i o n a l function i n terms of i t s degree and i t s supremum norm on an i n t e r v a l . A model for such i n e q u a l i t i e s i s the following: INEQUALITY 1. (S.N. BERNSTEIN ([2] Vol. 1, p. 26)). Let t be a trigonometric polynomial of degree n on the c i r c l e [-IT,IT] . Then for a l l £ e [-IT,IT] , It'(?) I < n| It I I r , . n 1 — n [ —TT , TT ] This r e s u l t i s the key to the " i f " or inverse part of the Bernstein and Jackson r e s u l t mentioned i n the introduction. The above i n e q u a l i t y can be transformed into an inequ a l i t y about r e a l polynomials. INEQUALITY 2. (S.N. BERNSTEIN). If P n e II then for any x e (a,b) | P n ( x ) l ± ° ,,1/2 M p J I [ a , b ] • ((x-a)(b-x)) Of a s i m i l a r flavour i s the next i n e q u a l i t y . INEQUALITY 3. (A.A. MARKOV [13]). If P n e ^ then 2n 2 V I [a,b] - b-a I l p J I [a,b] No such i n e q u a l i t i e s e x i s t for un r e s t r i c t e d r a t i o n a l functions. -<52 Consider ([11] p. 83) r(x) = - 5 x , then |r(x)L_ , < 1 but x + 6 L ' J r ' ( 5 ) = In t h i s section we s h a l l show that Bernstein-Markov type ++ + + i n e q u a l i t i e s can be recovered for the classes R , R and R n n n n - 6 -INEQUALITY 4. Let r = p/q e R n[a,b] . Then, i f 0 < e < b - a, a ) H r ' | | [ a + £ j b ] r r and 1 1 1 [ a , b ] b) (m) ( ( m + l ) ( m + 2 ) 2 m  r m, 2 n n i i i i [a+e,b] — m 1 1 1 1 [ a , b j PROOF, a) Let 0 < e < b - a, l e t ? be a po i n t where where Ip(t) [a+e,b] , a + e <_ t, <_ b, and l e t t be a poin t [a,?] , a < t <_ £ . Then r U ; q(?) q ( 0 U ; From I n e q u a l i t y 3 and the monotonicity of q i t f o l l o w s that 2, P'(5) |q ( 0 < 2 n H p H [ a , U 2n Z|p(t ) l - (5-a) |q(?) | (C-a) |q(.C) 2n ( ? - a ) £(tl q(t ) 2n i , < — r ( t ) and that q.'(S>| | r ( r ) | < ' ' "La H O T 1 1 - e. |q(c) 2n •[a,C,3 r(c) 2n , , < — -— £ T h u s ' H r ' H [ a + £ , b ] = l r ' ( ? ) l H r l l [ a , b ] • b) Note that r ^ £ R ^ k a n <* that r ^ n a s a monotone ek denominator i f r does. Let 0 < e < b - a and l e t y = a H k m Then by a) - 7 -or ( k ) , . ; 4 n 2 2 2 k | | r ( k - l ) | l ' ' t a + \ - l + ( V Y k - l } ' b ] - Y k " Y k - l [a+f k_ rb] . 0 0 I I < m n 2 4 k + 1 . i ( k - l ) . . 1 1 [ a + Y k , b ] - e l | r ' 1 [ a + y ^ . b ] Thus, by i t e r a t i o n , I I _ (m) [ a + e , b ] -m 2.k+l n 5S_4 k=l e | r | 1[a,b] (m +l)(m +2) 2 m r m, "2 ,n i i I I = [m 4 ] — - r r -. . m 1 1 ' [a,bJ + t If 0 < a then R [a,b] c R [a,b] and the previous i n e q u a l i t y — n n applie s . However, for t h i s more r e s t r i c t e d c l a s s we can deduce a stronger i n e q u a l i t y . • INEQUALITY 5. Let r = p/q £ R^[a,b] . If 0 < L a < a < g < b then b - e 3/2 a n e [(a-a)(b-g)] T7T l l r l l r T , i + ~ l l r l l r ui a n d 1/2 1 1 1 1[a,b] a 1 1 1 1[a>bj b) r (m) [a,3] 3m 2 i i i i - i i M [ a , b ] where C depends only on a, a, 3 and b PROOF. Suppose 0 < x < y then, since q has non-negative c o e f f i c i e n t s , n q(x) = T I a x and n m m=0 n u x (l) \ q(y) =. I (|ajym) . m" n m=0 y n n-m v x i I m , . = I a x < q(x) . m=0 y n-m 1 m' Also, i f x > 0 then (2) xq'(x) = x £ m|a |x m ^ <_ nq(x) n=0 m' Let Z be a point where | r ' ( ? ) | = I | r ' I I r a g] • T h e n h ? ( 2) (3) [a,3] < 1?'^> - |q(?) q(c) r(s) '(c) q(?) + — I I r I I a ' ' ' ' [a,b] Set y = b - 8 then, by Inequality 2, (4) -^ V ( C ) q(?) 1IPII[a,C+y/n] - a ) ( , + Y / n - 0 ] 1 / 2 ^ ^ T | P | '[a^+y/n] [ ( ? - a ) ( ^ ) ] 1 / 2 ^ From (1) and (4) 1 P ' ( C ) (5) q(?) 3/2 [ ( a - a ) ( b - 3 ) ] 1/2 I [a,?+y/n] rg+y/n^n |q(?+y/n| ^ ? J 3/2 [(a-a)(b-3)] 1/2 I| r | 1[a.S+y/n] y/a Part a) now follows from (3) and (5) - 9 -Part b) i s deduced from part a) by a s i m i l a r i t e r a t i o n to that used i n the proof of I n e q u a l i t y 4 b ) . We i s o l a t e (2) of the proof of l a s t r e s u l t as a lemma. LEMMA 1. I f q £ II and q has non-negative c o e f f i c i e n t s then, n n n f o r any x > 0 q' (x) izz q(x) • •A. A fundamental d i f f e r e n c e between the c l a s s R and the c l a s s n + t + R i s that R i s closed under t r a n s l a t i o n w h i l e R i s not. Thus, n n n + n we expect and o b t a i n i n t e r v a l dependent r e s u l t s f o r the c l a s s R _^ 2 As the example r ( x ) = » „ showed we cannot expect any B e r n s t e i n -x +6 Markov type i n e q u a l i t y f o r R * [ - l , l ] or f o r that matter R^[a,b], where a < 0, nor can we hope to have a theorem that bounds r 1 ( 0 ) f o r r e R*[0,1] . However, on a s t r i c t l y p o s i t i v e i n t e r v a l we o b t a i n the f o l l o w i n g : .2 INEQUALITY 6. Let b - ~ > a > 0 and l e t r = ^  £ R* . Then a) l k ' M [ a > b ] < I2n 3 e 1 / a + f ] | i r l ! [ a ) b ] and W M ^ M l a . b ^ ^ l M l t a . b ] where C depends only on a and m PROOF, a) Let £ be a point where | r ' ( ? ) | = [a,b] Then r . m p ' ( Q q ' ( Q r >^ q(c) q(?) r ( ? ) - 10 -Suppose that £ < b - — . By Inequality 3 — n q ( 0 2 « " p l l ^ , ^ and by (1) of the proof of Inequality 5, (1) |p'(g) q(?) 2n I I P | I [ 5 > 5 4 ] - 1 n ? + n < 2n r • r e 1 1 1 1 [ a , b ] < 2n r e If ? > b we have, by Inequality 3, (2) Ip'(C) q(?) 2 n iipii[C4,g] < 2n r [a,b] By Lemma 1, (3) q'(5> < H < £ q(C) - a Combining the appropriate choice of (1) or (2) with (3) y i e l d s I r ' l L u l < [2n 3 e a + -] ||r | L 1 1 1[a,b] - a J 1 1 1 1[a,b] Part b) now follows by i t e r a t i o n . • The further r e s t r i c t e d c l a s s R s a t i s f i e s a s t i l l stronger n i n e q u a l i t y . - 11 -v> 1 i INEQUALITY 7. Let r = e Rn . If x > 0 then |r' (x) | <_ ^  r(x) . P R 0 0 F r , ( x ) = E ^ 2 0 _ £ L W . P O O PROOF. r (x) q ( x ) q ( x ) q ( x ) By Lemma 1 o < p'QQ < n R M -— q(x) — x q(x) and 0 < q'(x) p(x) < n p(x) - q(x) q(x) — x q(x) The r e s u l t now follows from the observation that r'(x) i s the diff e r e n c e of the above two p o s i t i v e q u a n t i t i e s . • + t While neither R nor R i s closed under d i f f e r e n t i a t i o n i t i s n n true that i f r e R (R ) then r ' e R„ (R„ ) . This i s not the case n n 2n 2n for R . Thus, we cannot i t e r a t e Inequality 7 to obtain an n in e q u a l i t y for higher d e r i v a t i v e s . An added complication of working 44- + + + + + + with R i s that while R - R <= R„ and R - R <= R„ , t h i s i s n n n zn n n 2n I | not the case for R . n It i s i n t e r e s t i n g and important to know how close the preceding r e s u l t s are to being best possible. Inequality 1 i s s a t i s f i e d exactly f o r f (9) = s i n n6 or T (0) = cos nO . Inequality 3 i s also s a t i s f i e d n n th exactly. Let T be the n Chebyshev polynomial on [-1,1] n -1 2 (T n(x) = cos n cos (x)), then T ^ ( l ) = n • Thus, i n Inequality 4 - 1 2 -2 the growth rate of n cannot be s i g n i f i c a n t l y improved. I f , once -e 2 1 again, we consider r(x) = —^ ^ » w e s e e that r'(e) = and x + e that the dependence on e ' at the. .right end-of the i n t e r v a l i s also e s s e n t i a l l y correct. Thus, i n some sense, up to a constant, Inequality 4 i s best possible. 2 Inequality 2 i l l u s t r a t e s that the n i n Inequality 3 i s mis-leading i n the sense that i f p e II and l i p [ I r .. ., , < 1 then n n 1 1 rn''[-1,1] — (0) <_n and that, i n f a c t , the d e r i v a t i v e can only grow l i k e a multiple of n provided p^ i s r e s t r i c t e d to a set bounded away from the endpoints of the i n t e r v a l . This leads to the following unanswered question: i f r e R^[0,2] and M r I I r r . n < 1 is n n 1 1 n 1 1[0,2] — | r ' ( l ) | < cn or does r ' ( l ) increase e s s e n t i a l l y f a s t e r than n ? ' n 1 — n We may ask, but also not answer, the same question for the c l a s s R* . 3/2 Thus, i n I n e q u a l i t y 5 we cannot guarantee that the n should not 3 2 be n, or that the n of Inequality 6 should not be n Inequality 7 i s exact. It i s s a t i s f i e d at every point by x 1 1 . n Consider S (x) = — , then I Is I I r „ 0 1 < 1 and S'(l) = ¥r . n i . n n M [ 0 , 2 ] — n 2 Thus, the bound of n i s also e s s e n t i a l l y correct on the i n t e r i o r of the i n t e r v a l . The remaining i n e q u a l i t i e s of t h i s section concern r a t i o n a l functions whose poles are bounded away from the i n t e r v a l under con-s i d e r a t i o n . Let D(x,6) be the closed disc i n the complex plane C with (real) center at x and radius 6 . - 13 -INEQUALITY 8. Let r = p/q e R n and suppose that q has no roots In D(x,e+6) where 6 <_e/(k+l) for some integer k . Then I 'II — 2n/k II || | r ' 1 [x-6,x+6.] - 6 6 l | r 1 1 [x-e-6,x+e+<5] We need the following lemma i n the proof of Inequality LEMMA 2. Suppose q e IT and suppose q has no roots i n D(x,(k+l)e) for some integer k . Then supremum -z,weD(x,E) q(z) q(w) < e 2n/k PROOF. Let z and w be points i n D(x,e) where o o |q(z )| = max |q(z)| and |q(w Q)| = min |q(z)| . Suppose ° zeD(x,e) zeD(x,e) q(z) = a. n (z+a ) . Then m=l m q(z o) q(w Q) n = n m=l z + a o . . m w + a o m = n | i + m=l z - w o o i w + a ' o m Since z ,w e D(x,e) and a i D(x,(k+l)e), o o m q(z o) q(w Q) /-. , 2e^ 2n/k < n (1 + i — ) < e — . ke — m=l PROOF OF INEQUALITY 8. Now Let t, be a point where | r ' ( ? ) | = ||r' [x-6,x+<5] r ' ( ? ) P'(S) q(?) + q'(C) q(?) r ( C ) By Inequality 2 - 14 -| p | ' [ s - f i . g+s ] |q(C) sup (z,weD(S,6) q(w) Since 6 < e/(k+l) and C £ [x-S,x+S], an a p p l i c a t i o n of Lemma 2 to the above in e q u a l i t y y i e l d s P'(ql < n || r|| e 2 n / k S i m i l a r l y , by Inequality 2 and Lemma 2, q'(g) 'q(C) r(?) < '[g-6,S+6] . , , n 2n/ki . , l y e |r(?) Thus [x-S,x+<5] - 6 2n 2n/k,, , < ^ — e r [x-e-6,x+e+fi] • COROLLARY 1. Suppose = p/q £ R^ where q has no roots within distance ^ of any point of [a,b] . Then 2 3 i i I l r n ' I [a+l/n^b-l/n] 1 8 e n " ' " [ a . b ] ' PROOF. e + 5 Let e - — r r " and l e t 6 = n+l (n+l) _ = 1 n+l ' , , 1 N2 - n+l ' n(n+l)~ n (n+l) 2 n+l " Then 1 + -± < ^ - + 1 and by Inequality 8 - 15 -I l r ' I I r O - T / K i / i < 2n(n+l) 2e 2| |r| | r , , I I 1 1[a+l/n,b-l/n] — M M [ a , b ] < 2 e 2 ( l + V n 3 | |r| | . , . - n 1 1 1 1 [ a , b ] Q 2 31 I I , ^ 8 e n l | r | l [ a , b . ] . • Q C o r o l l a r y 1 i s c l o s e l y r e l a t e d to Inequality 5 as demonstrated by the next remark. REMARK 1. If p e l ! has non-negative c o e f f i c i e n t s then p has no n roots i n the region {z| |arg(z)| < •^•} . n PROOF. Suppose p(z) = j a z n , a > 0, a > 0, and l e t t, be r r r ^ m n m — m=0 any point where 0 < arg(?) < ir/n . Then, f or each m <_ n, n 0 < arg(? ) < TT and hence, T a £ l i e s i n the region {z|im(z) > 0}. m=0 n In p a r t i c u l a r , £ a c, ^ 0 . Since p can have no p o s i t i v e r e a l roots m=0 m we are f i n i s h e d . • G.G. Lorentz i n [7] considers uniform approximations by polynomials with p o s i t i v e c o e f f i c i e n t s i n x and 1 - x ; that i s , by polynomials P n ( K ) = I \l x k ( l - x ) £  n k+£<n k J i k£0,JL >0 where a > 0 . We s h a l l r e f e r to such polynomials as P.P.C. of K.X/ degree n . REMARK 2. If p i s a P.P.C. of degree n then p has no roots i n r n n the region A = {z | | arg(z) | < y-} n {z | |arg(l-z) | < } . - 16 -v k Z PROOF. Suppose p (x) = I a x ( l - x ) , a, > 0, and suppose n k+l<n kJt Rji k C e A and arg(?) > 0 . Then, i f k,£ <_n, £ l i e s i n the quadrant % {im(z) ^ 0} n {re(z) > 0} and (1-?) l i e s i n the quadrant k a {im(z) £ 0} n {re(z) > 0} . Thus, £ (1-?) l i e s i n the region re(z) > 0 and p (c) ^  0 . Since p has no roots i n the unit *n n i n t e r v a l we are f i n i s h e d . We are now i n a p o s i t i o n to apply Inequality 8 to r a t i o n a l functions whose denominators are P.P.C. COROLLARY 2. Suppose r = p/q e R and suppose q i s a P.P.C. of n degree n . Then, i f 0 < y < \ > 2 3 24e n_ M i l [ Y . l - Y l ~ TT Y [0,1] " YTT PROOF. By Remark: 2 no root of q l i e s within distance of ' L e t £ = 4nlnT2T a n d l e t 6 = n ^ = ^ f e l ' T h e n e + 6 = and by Inequality 8 (note that k = n) " r ' " [ Y , l - Y ] ' ' r' '[Y-e,l-Y+el 2 2n 4n(n+2)e i i i i - YTT l | r M [ 0 , l ] 2 3 24e n i i i i - TT Y M r M [ 0 , l ] • F i n a l l y we derive an ine q u a l i t y that allows us to bound r a t i o n a l functions i n pole free regions i n the complex plane. Let Ep(x,e), p >_ 1, be the closed e l l i p s e w i t h f o c i x - e, x + e j . e , - 1, £ i l i and semi-axes y'P"*"^/' 2"lP~"pi ' INEQUALITY 9. (S.N. BERNSTEIN ([11, p. 42)). I f p e IT then n l p ( z ) l l E p ( x , E ) ^ p n l l p M [ x _ e ) X + E ] • INEQUALITY 10. Let r = p / q e R n and suppose that q has no roo t s i n D(x,(k+l)e) . Then, i f 6<_e, n I | i I 2n/k ! l r ( z ) l l D ( x , e ) n E p ( x , < S ) - P ll r |'[x-5,x+63: ' 6 PROOF. i | p | | [ x _ 6 > x + ( S ] £ | | r | | [ x _ 6 > x + 6 ] l ] q | l [ x - 6 , x + 6 ] - 'I l r ' I [x-6,x+6] I l q ' ' D ( X , E ) • Thus, by I n e q u a l i t y 9 l p H E p ( x,6) - P N N R H [ X - 6 , X + 6 ] 'I I D C X . E ) By lemma 2, since q has no roots i n D(x,(k+l)e) min | q ( z ) | 1 I I q | l D ( x e ) ' T h u s ' zeD(x, E ) I I I ni I I | 2n/k | r | lD(x,e)nE p(x,p) - P l | r | 1 [x-6,x+6] ' 6 - 18 -3 - INVERSE THEOREMS FOR C [a,b] . The prototype inverse theorem i s the " i f " d i r e c t i o n of the f i r s t r e s u l t of the introduction. This can be transformed to a r e s u l t about algebraic polynomials by a simple change of v a r i a b l e (x = cos 0) . We state one form of the theorem: THEOREM 1. ([11] p. 61). If there e x i s t s p e IT such that n n OO V k-1 I I I , J . n H f - P n l l [ a , b ] < ~' n=l then f i s k-times continuously d i f f e r e n t i a b l e on (a,b) . This i s not quite the complete r e s u l t . For examples THEOREM 2. ([11] p. 75). Suppose 0 < a < 1 . A function f i s p-times d i f f e r e n t i a b l e on [-1,1] with f ^ 3 ) e l i p a i f and only P i f there e x i s t s a sequence p e II and a constant C such that n n |f(x) - p (x) | l C [ m a x ( V ^ X , ^ ) ] V + a for each x e [-1,1] . The type of theorems we derive i n t h i s section are theorems that state f i s k-times d i f f e r e n t i a b l e i f f can be approximated by r e s t r i c t e d r a t i o n a l s with order — \ r for some B . We derive these n g k inverse theorems from the i n e q u a l i t i e s of the previous section by the same basic techniques as those used to pass from Bernstein.'.s i n e q u a l i t y to Theorem 1. Our f i r s t r e s u l t i s : THEOREM 3. Suppose there i s a sequence r n e R n[a,b] such that OO J J I ^ J I l a . b ] n 2 k _ 1 < 00 ' 19 -Then f i s k-times continuously d i f f e r e n t i a b l e on (a,b] . PROOF. We may assume that I If - r I I r , 1 i s monotone non-increasing. ' 1 n ' ' L a , b J Consider the expansion: (1) f(x) = r (x) + I (r- (x) - r (x)) n=0 2 2 and observe that ' r 2 n + l r 2 n l l [ a , b ] - " f r 2 n + l ' l [ a , b ] + " f ' [a,b] ^ 2 i l f - V M [ a , b ] Formal d i f f e r e n t i a t i o n of the r i g h t side of (1) y i e l d s (2) r< k>(x) + I ( r ( k > (x) - r ( k ) ( x ) ) n=0 2 2 where, by Inequality 4 b), for fixed e there i s a C independent of n such that | | r ( k ) - r ( k ) | | < C [ 2 n + 2 ] 2 k | | r - r I I l | r2 n + l r 2 n 1 1[a+e,b] - L L Z J l | r 2 n + l ^ n 1 1 [ a , b ] -Thus, I | | r ( k i . - r ( k ) | | r . < •% C [ 2 n + 2 ] 2 k 2 | | f - r |L M k=0 2 n + 1 2 n " [ a + e ' ^ - n = 0 2 n M [ a , b ] < 2C4 2 k I [ 2 n ] 2 k | | f - r || • n=0 2 n L a ' b J which converges because f - r r . , i s monotone and 1 1 n''[a,b] - 20 -y | | f - r M r ,-.n converges. Thus, (2) converges uniformly ^ 1 1 n 1 'La,bJ n=l on [a+e,b] and f e C [a+e,b] . An immediate c o r o l l a r y of Theorem 3 i s : t COROLLARY 3. Suppose there i s a sequence r ^ e R^ta,!)] such that, for fixed A and 6 > 0, A • f - r r _ < n 1 1[a,b] - 2k+<5 n then f e C (a,b] . If f i s k-times continuously d i f f e r e n t i a b l e on [a,b] then, by Theorem 1, f has polynomial order of approximation at l e a s t l i k e — k • Thus, except for endpoint phenomena, approximations by r e s t r i c t e d n r a t i o n a l s cannot provide spectacular increases i n the speed of approximation for such functions. From Inequality 5 and Inequality 6 we can derive, by the same techniques as employed i n Theorem 3, the following r e s u l t s : THEOREM 4. Suppose 0 <_ a and suppose there i s a sequence e R* such that | k - l E I I F - R I I r UT  N < ° ° . t, 1 1 n M [ a , b ] n=l Then f e C k(a,b) . THEOREM 5. Suppose 0 < a and suppose there i s a sequence e R^ such that - 21 -V I U II 3 k " 1 , ) f - r r u n < °° ^ 1 1 n M [a,b] n=l Then f e C k[a,b] As a c o r o l l a r y to Theorem 2 and Theorem 5 we ob t a i n COROLLARY 4. Suppose 0 < a . A f u n c t i o n f i s an element of oo + C [a,b] i f and only i f there e x i s t r e R such that L ' J n n J l l f - r n H [ a , b ] n=± k n < oo f o r each k C o r o l l a r y 4 underlines the s i m i l a r i t y between polynomial approximations and approximations from R*[a,b], a > 0, s i n c e C o r o l l a r y 4 remains true as a theorem about polynomials. We need only to change R + to II i n i t s statement. J n n An example of an inverse theorem obtainable from I n e q u a l i t y 8 and i t s C o r o l l a r y i s the f o l l o w i n g . THEOREM 6. Let c > 0 and suppose there i s a sequence r = p /q e R n n n n so that no root (complex or r e a l ) of q^ l i e s w i t h i n d i s t a n c e c/n of [a,b] . I f T I I f - r I I r n 2 < o o /; 1 1 n' ' [a,b] n = l then f e C 1(a,b) c PROOF, r s a t i s f i e s the c o n d i t i o n s of I n e q u a l i t y 8 w i t h e = — — n n+1 and 6 = ( n + j j 2 • Thus, f o r each n, 2 II 'II 2n(n+1) 2 I I • i 1 | r J 1 [a+e,b-e] 1 c 6 1 ' V 1 [a,b] ' - 22 -The r e s u l t now follows analogously to Theorem 3. The f i n a l r e s u l t i n t h i s vein i s a consequence of Co r o l l a r y 2. THEOREM 7. If there i s a sequence r = p /q e R , with q a n n n n n P.P.C. of degree n, such that OO E J l f - r n l l [ 0 , l ] ^ K 00 ' n=l then f e C k ( 0 , l ) . • In neither Theorem 3 nor Theorem 4 can we deduce d i f f e r e n t i a b i l i t y at the l e f t endpoint of the i n t e r v a l of convergence. A modification of an example due to A.A. Goncar ([11] p. 91) i l l u s t r a t e s that no speed ++ + f of convergence from R^ [0,1] (and hence from R^tOjl] and R n [ 0 , l ] ) guarantees d i f f e r e n t i a b i l i t y at 0 . EXAMPLE 1. Let 6 and e be two sequences monotone decreasing n n to zero. Then there e x i s t s a function f so that a) R + + ( f : [0,1]) < 6 for a l l n and n — n b) IT ( f : [0,1]) > e for i n f i n i t e l y many m . m — m PROOF. Pick Y + 0 so that (a) for each n, f < S . Let n , K — n k=n OO A = £ Y and pick p 4- 0 so that (b) for i n f i n i t e l y many m there Q n n 1 1 2 exis t n so that e < -^ -br Y - 4Am p ] . Define f by m — 2 2 n m °° Y n P n f(x) = T —; . I t follows from (a) that f i s continuous on k=0 X + P n co y p CO [0,1] and that R ^ ( f : [0,1]) < III ^ 1 I [ 0 > 1 ] 1 I Y n £ « n • k=n -n k=n - 23 -n Y k P k Since f(0) = I y, = A and f(p ) <_ I +, I y, we have k=0 k k=0 p k p n k=n+l K f(0) - f (p ) >_ Y n • Let p^ be the best polynomial approximant of degree m to f on [0,1] . Then l | P m l I [Q ]_] — 2 A a n d b Y 2 2 Inequality 3 | | p j | [ Q i ; ] £ 4m A . Thus, |p m(P n> - P m(0)| <_ 4Am p n and n ( f : [0,1]) > i[(f(p„) - p m ( p j ) + (p(0) - f (0))] m — / n m n > TL"T Y ~ A^m p ] > e — 2 2 n n — m for i n f i n i t e l y many m . • If e tends to zero more slowly than — the function of the m n previous example f a i l s to be d i f f e r e n t i a b l e at zero. This i s a consequence of Theorem 2 (or of d i r e c t computation). We s h a l l see i n section 10 that /x on [0,1] can be approximated by r a t i o n a l functions from R with order thus providing perhaps, a more e natural example of the f a i l u r e of our preceding theorems at the endpoints. EXAMPLE 2. [4]. Let 6^ be any sequence tending to zero. There e x i s t s f continuous and monotone on [0,1], i n f i n i t e l y d i f f e r e n t i a b l e on (0,1], such that R ( f : [0,1]) > 6 . n — n As these two examples i l l u s t r a t e no necessary and s u f f i c i e n t conditions, such as those of Theorem 2, e x i s t for approximations from 24 -the classes R + or R 1+ n on [0,1] The exact order of approximation from R^[0,1] needed to assure d i f f e r e n t i a b i l i t y on (0,1) i s , l i k e the exact power of n i n Inequality 4, an open problem. More p r e c i s e l y , what i s the smallest constant a so that the existence of a sequence r e R s a t i s f y i n g n n OO ni f - ' n i i[o, i , ° k a _ 1 < -n=l implies that f e C k ( 0 , l ) ? We have shown that 1 < a < 2 . In p a r t i c u l a r , i t would be i n t e r e s t i n g to know whether a i s s t r i c t l y greater than 1 or not. t This would t e l l us whether approximations from R can d i f f e r n fundamentally from approximations from n on the i n t e r i o r of the i n t e r v a l under consideration. - 25 -4,- INVERSE THEOREMS AND ANALYTIC FUNCTIONS. I f a f u n c t i o n i s C k[a,b] but not C k +^(a,b) then, as we saw i n the previous s e c t i o n , r e s t r i c t e d r a t i o n a l f u n c t i o n s provide s u r p r i s i n g l y l i t t l e improvement over polynomial approximations. We can make no such a s s e r t i o n f o r r e s t r i c t e d r a t i o n a l approximations to a n a l y t i c f u n c t i o n s . To achieve reasonable i n v e r s e theorems we must e i t h e r f u r t h e r r e s t r i c t the c l a s s e s of approximants or consider smaller c l a s s e s of a n a l y t i c f u n c t i o n s . For comparison purposes we f i r s t mention a fundamental theorem of B e r n s t e i n ([2] v o l . 1, p. 41) concerning polynomial approximation to a n a l y t i c f u n c t i o n s . THEOREM 8. (S.N. BERNSTEIN). I f f i s a n a l y t i c i n Ep(0,l) then l i m n / n ( f ) = — and conversely, i f l i m 'Vn ( f ) =• — + e f o r some n P 1 1 P e > 0, then f i s a n a l y t i c i n Ep(0,l) . i + ++ No speed of convergence from any of R^jR^ or R^ i s s u f f i c i e n t to guarantee a n a l y t i c i t y . EXAMPLE 3. Let 6^ be any sequence decreasing to zero. There e x i s t s a f u n c t i o n f so that f i s not a n a l y t i c i n any neighbourhood of any point of [0,1] and so that R + + ( f : [0,1]) < 6 . n — n I | PROOF. I t s u f f i c e s to construct a g so that R (g: [0,2]) < 6 n — n and so that g i s not a n a l y t i c i n any neighbourhood of 1 . Then, oo f o r s u i t a b l e a + 0 and b dense i n [0,1], f (x) = / a g(x + b ) n n — n n. n=l i s the d e s i r e d example. - 26 -Define g by k=l x R + (1 where a, > 0 i s chosen so that k a k (a) I —r =—j- <_ 6 and so that k > / n - l x k + (1 - ^ ) K 0 0 a, r k (b) for each m, I — r ;—rr converges uniformly i n some k=i x k + ( i - ± r m X m neighbourhood of C > where C i s a root of x + (1-7-) c l o s e s t 6 m m k to 1 . I t follows from condition (a) that 00 a. R ^ ( g : [0,2]) < J T k ^ 6 n ' n W n - 1 x k + From condition (b) we deduce that g i s not a n a l y t i c i n any neighbourhood of 1 . If g were a n a l y t i c i n a neighbourhood of 1 then we could i n f e r the contradiction that f o r a l l s u f f i c i e n t l y large k, — r ;—7- i s also a n a l y t i c i n t h i s neighbourhood. x k + • I f , however, the class of approximants has a l l i t s poles bounded away from the i n t e r v a l of approximation, then exponential rates of approximation do ensure a n a l y t i c i t y . THEOREM 9. Suppose c > 0 and suppose there i s a sequence r = p / q e R so that no root ( r e a l or complex) of q l i e s within n n n n n distance c of [a,b] . If I|f-r II, < p n for some p > 1, 1 1 n' ' [a,bJ — - 27 -then f i s the r e s t r i c t i o n to [a,b] of some function F a n a l y t i c i n a region containing [a,b] . -1/2 4/K C PROOF. Choose K > 1 so that p e = 3 < 1 • Let 6 = 2(K+1) and l e t £ = 26 . Then applying Inequality 10 to r n - * e R 2 n y i e l d s l r n - r n - l H D ( x , 2 6 ) n E 1 / 4 ( x , 6 ) 1 I l r n " r n - l l '[x-S,x+6] ' for each x e [a+6,b-6] . Set T = u {D(x,26) n E ,,(x,6)} . xe[a+6,b-S] p 1 Then, since Mr - r , M r , n < 2p ^ n , i t follows that 1' n n-1' 1[a,b] — i / \ / \ i * -n n/2 4n/K _ „n r (z) - r (z) < 2p • p p e < 2p • 3 ' n n - l — — f o r each z e T . Hence, converges uniformly on T, the i n t e r i o r of T contains [a,b] and the r e s u l t follows. Many n a t u r a l l y a r i s i n g functions are more e f f e c t i v e l y approximated by r e s t r i c t e d r a t i o n a l s than by polynomials. For example, i f f i s any function a n a l y t i c i n the e n t i r e plane except at -c where f has a r a t i o n a l pole, then i f we f i r s t subtract out the pole we can deduce from Theorem 8 that a) l i m n/R +(f:[0,1]) = 0 while n n-x» b) lim n/n +(f:[0,1]) = - where n p n-*>° - 28 -f(p+£-)-c. However, for other classes of e n t i r e functions approximations from are no more successful than approximations from 11^  . In p a r t i c u l a r , we prove that R+(e X:[0,l]) = II ( e X : [ 0 , l ] ) . n n Before we prove the above r e s u l t , we s h a l l e s t a b l i s h an inverse theorem for approximations to a n a l y t i c functions from . The f i r s t theorem we prove i s a gen e r a l i z a t i o n of the following theorem of Bernstein ([2] Vol. 1, p. 63). THEOREM 10. (S.N. BERNSTEIN). Let f and g be n + 1 times continuously d i f f e r e n t i a b l e on [a,b] and suppose that | f ( n + 1 ) ( x ) | < g ( n + 1 ) ( x ) , for a < x £ b . Then n n(f:[a,b]) < n n(g:[a,b]) . The proof of Theorem 10 ([11] p. 38) can be simply adapted to r a t i o n a l approximations to y i e l d : THEOREM 11. Suppose that f and g are n + 1 times continuously P n d i f f e r e n t i a b l e on [a,b] and suppose that there ex i s t s — e R m such that g - P has n + 1 zeros on [a,b] . . Then, i f | ( q m f ) ( n + 1 ) (x) | < (q^g) ( n + 1 ) (x) on [a,b], there e x i s t s . p n e lT n so that for each x e [a,b] P (x) P ( x ) I f < * > - ^ 1 i l g ( x ) - ^ - j -- 29 -PROOF. Let x ,...,x be the n + l p o i n t s where g - p /q_ = 0 . o n n T n t i l Let p^ be the n degree polynomial which i n t e r p o l a t e s f at x Q , . . . , x n . I n i t i a l l y , we assume that (q mg) ^ n +"^ i s never zero on [a,b] . Define T and S by T(x) = q m ( x ) f ( x ) - P n ( x ) and S(x) = q m ( x ) g ( x ) - P n ( x ) • L e t x be any f i x e d p o i n t of [a,b] so that x i {x ,...,x } . Define 0(y) by o n 0(y) = T(x)S(y) - S(x)T(y) . "Then 0(y) has zeros at the n + 2 d i s t i n c t p o i n t s x,x ,...,x n, and hence, by R o l l e ' s theorem, Q^U+^ (t.) = 0 at some p o i n t £ e (a,b) Now <|> ( n + 1 )(?) = [ q m ( x ) f ( x ) - P n ( x ) ] ( V g ) ( n + 1 ) a ) [ q m ( x ) g ( x ) - p * ( x ) ] ( V f ) ( n + 1 ) U ) = 0 . Since | | q m f ) ( n + 1 ) ( C ) | 1 ( q m ' g ) ( n + 1 ) ( ? ) and ( ^ g ) ( n + 1 ) (?) t 0, we have (1) | q m ( x ) f ( x ) - P n ( x ) | < |q m(x)g(x) - p n ( x ) | . We have now shown that i n e q u a l i t y (1) holds f o r any x I {x ,...,x } and s i n c e , (1) i s an e q u a l i t y f o r x e {x ,...,x } a l l we are l e f t ' " o n to prove i s that we can r e l a x the assumption that ( q ^ ) ^ n +"^ i s never zero on [a,b] . I f ( q m S ) ^ n + ^ has a zero consider n+l EX g e ( x ) = g(x) + ( n + 1 ) ! q m ( x ) - 30 -Then, on [a,b], , v(n+l) . . ( qmge) • (v*> (n+1) + e > 0 Let p e II be such that n,e n n n P n e We note (see Lemma 3, p. 31) that g 1 — must have at le a s t £ Sn n + 1 zeros and hence, by the f i r s t part of t h i s proof, for each e there e x i s t s p so that n, e i * « - i ^ r i . i i . . w - i ^ r The r e s u l t now follows by taking the l i m i t as e —>- 0 . • COROLLARY 5. Suppose that f,g e C n + 1 [ a , b ] , a ^ 0 . If | f ^ ( x ) | < _ g ^ ( x ) for every x e [a,b] and for k = 0,1,. . . ,n + 1, then R + (f :[a,b]) < R* (g:[a,b]) . n,m — n,m p PROOF. Suppose that —°- i s a best approximant to g on [a,b] from m the c l a s s R + [a,b] . Then n,m k=0 < T ( n I 1 )q B 0 0 '» < " f l " k > " < V ) ( " + 1 ) • k=0 k ^ m - 31 -P n If we know that — interpolates g at n + 1 points then we in can apply Theorem 11 and conclude that R + (f:[a,b]) < R + (g:[a,b]) . n,m — n,m P n The fa c t that interpolates g at a least n + 1 points i s a q n consequence of the next lemma. P n LEMMA 3. Let f e C[a,b] then i f — i s a best approximation to + t f from R [a,b] (or R [a,b]) then there e x i s t at le a s t n + 2 n,m n,m points a <_ < a^ .... < a ^ + ^ <_ b so that P (a.) p f(a.) 7 — r | = I If I L ,-, and rtt ^ P n ( a i \ ff, , P n ( a i + l \ ( f ( a ^ ) " W ) = " ( f ( a i + l ) - ^ i + 7 ) ' PROOF. The proof of t h i s lemma i s based on assuming the lemma f a l s e and then perturbing p^ to derive a better best approximant. This i s achieved analogously to the proof that best polynomial approximants (the case m = 0) have t h i s w ell known Chebyshev a l t e r n a t i o n property ([11], p. 30). We omit the d e t a i l s . The next lemma w i l l be discussed i n more d e t a i l i n section 5. Part a) i s due to Chebyshev and part b) i s due to Reddy [19]. - 32 -LEMMA 4. a) n ( x n + 1 : [ 0 , l ] ) = 1 n v ~ - i " * - " 2n+l b) R + ( x n + 1 : [ 0 , l ] ) = n n ( x n + 1 : [ 0 , l ] ) We are now i n a p o s i t i o n to prove the following approximation theorems for a n a l y t i c functions with p o s i t i v e c o e f f i c i e n t s . n 0 0 a x THEOREM 12. Suppose g(x) = Y n , , a > 0 and Ln n! n — n=0 8 = min (a (n+l-k)!) . Then k=0,...,n+l + B n 1 R n ( S : [ 0 ' 1 ] ) ^TnTiTT ' ^ n+T ' 3 x n + 1 PROOF. Consider f (x) = , . If 0 <_ k <_ n + 1 then, for x e [0,1] and k = 0,1,...,n + l , ,(k), , V , (k), * f ( X ) - (n+l-k) ! - \ - S ( X ) ' Thus, by Corol l a r y 5 and by Lemma 4, R+(g:[0,l]) >;R*(f:[0,l]) n > - (n+l)! 22n+l ' Q n a x v n COROLLARY 6. Suppose g(x) = I j— where a^ i s any p o s i t i v e 1 non-increasing sequence. Then k=0 n ! , + , r -t T \ . a n + l R;<g:[0,l]) > 2n+l • n (n+l)! 2 / n ^ ± - 33 -n °° a x COROLLARY 7. Suppose g(x) = £ , where 0 <_ a <_ a < b, then n=0 a) II <g:[0,l]) < n (n+1)!2 n L b) R+(g:[0,l]) 1 § - 2 ^ I n (n+1)!2 PROOF. Part b) i s an immediate consequence of Corol l a r y 5 and Co r o l l a r y 6. , n+! bex (n+1)' 1We prove a) as follows: Let f(x) = . ,, . Then, by Lemma 4 n (f : [o , i ] ) = — b e 2 n + 1 ( n + l ) ! 2 2 n + 1 and since for x e [0,1] (k-n+1) °° a, x g (n+l). ( x ) = j J c k=n+l (k-n+1)! ^ u K ^ u ii(n+l) , \ <_ be <_ be = h v 7 (x) , we deduce from Theorem 9 that n n(g : [o, i ] ) 1 — b e 2 n + 1 • n ( n + l ) ! 2 Z n + 1 • The next theorem shows that f or a n a l y t i c functions of the form Ya x 1 1 where a i s monotone decreasing, approximations from R +  L n n n reduce to polynomial approximations. oo v k THEOREM 13. Suppose f(x) = \ a x where {a,} i s a p o s i t i v e non-k=0 k k increasing sequence. Then, i f 0 <_ a < b <_ 1, R+(f:[a,b]) = n n ( f : [ a , b ] ) . - 34 -P n + PROOF. Let — be a best approximant to f from R . We can ^n n k write q = T b, x where b, > 0 and q i s normalized so that k=0 q n(b) = 1 Now n ( I a,x k: [a,b]) = II ( f : [a,b]) n n+l k n > R+(f: [a,b]) = ||f - P n / q n l l [ a 5 b ] > f-q - p , , , — 1 1 n n n 1 1[a,b] OO > n ( I c x k: [a,b]) n k=n+l k th where c i s the k c o e f f i c i e n t of f«q . We observe that for k n k > n, c, = a, b + a, 1 b 1 + ... + a. b k k o k-1 1 - • k-n n n > a k ( Z V = a k q n ( 1 ) ^ a k n=0 Thus, for x e [a,b] ( I ckxVn+1>>( I a , h i n + ± ) k=n+l k=n+l and by Theorem 9 , n ( I c t x k : [a,b]) ( 1 a,x k: [a,b]) n k=n+l k n k=n+l R We deduce from (1) and (4) that - 35 -R+(f: [a,g]) = n ( f : [a,g]) . • 00 COROLLARY 8. Suppose f ( x ) = £ a,x where {a, } i s a p o s i t i v e k=0 k k decreasing sequence. Then, i f 0 <_ a < b <_ 1, the best approximation to f from R +[a,b] i s unique and i s a polynomial, n PROOF. We note that i f b Q f 1 i n (2) of the proof of preceding theorem then c^ > a^ and i n e q u a l i t i e s (3) and (4) become s t r i c t . This c o n t r a d i c t s (1). Thus, we deduce b = 1 = q o n COROLLARY 9. I f 0 < a < b < l then R 4"(e X: [a,b]) = IT ( e X : [a,b]) n n I t f o l l o w s from C o r o l l a r y 7 that R 4"(e X: [0,1]) > 1 ... . ~ ( n + l ) ! 2 2 n + 1 This should be compared w i t h the r a t e of convergence of u n r e s t r i c t e d r a t i o n a l s . Newman and Reddy [20] have e s t a b l i s h e d that R ( e X : [-1,1]) < 6 ( n ! ) 2 ~ 2 2 n ( ( 2 n ) ! ) 2 I t i s not always true that u n r e s t r i c t e d r a t i o n a l s approximate a n a l y t i c f u n c t i o n s b e t t e r than polynomials. We i l l u s t r a t e t h i s i n phe next example. EXAMPLE 4. ([11] p. 35). Let a^ be any sequence s a t i s f y i n g 0 < a n +-^ < a n / n - • I f T n i s the nth Chebyshev polynomial then • - 36 -f 00 = I a T J x> _ n 0 n n=0 3 i s e n t i r e and s a t i s f i e s , f o r each k, R ( f : [-1,1]) = IT ( f : [-1,1]) = I a 3 k 3 n=k+l n Boehm [3] shows, however, that i f R n ( f : [a,b]) = P n ( f : [a,b]) f o r a l l n then f i s a polynomial. The f i n a l example of t h i s s e c t i o n shows that w h i l e r a t i o n a l f u n c t i o n s provide a u s e f u l technique f o r approximating meromorphic fu n c t i o n s w i t h c e r t a i n types of poles, the presence of a s i n g u l a r i t y can c o n s i d e r a b l y impede the approximations. EXAMPLE 5. Let 6^ be any sequence tending to zero. a) There e x i s t s an e n t i r e f so that l i m f ( x ) e x i s t s and x-x=° so that R ( f : [0 o o ) ) > •§ . n — n b) There e x i s t s an f e n t i r e , except f o r a s i n g u l a r i t y at 0, so that f e C[0,1] and so that R ( f : [0,!]) > 6 .. n — n We need two lemmas f o r the proof of Example 5. The f i r s t lemma i s a de La V a l l e e - P o u s s i n theorem f o r r a t i o n a l f u n c t i o n s and the second i s a t e c h n i c a l lemma concerning the growth of a n a l y t i c f u n c t i o n s . - 37 -LEMMA 5 ( [ 1 4 ] , p. 161). Suppose there e x i s t s r n e R^ and 2n + 2 poi n t s a < a, < .a„ < . . . < a„ , „ < b so that f - r a l t e r n a t e s r — 1 2 2n+2 — n at these p o i n t s . That i s , s i g n ( f ( a ± ) - r ( a ± ) ) = - s i g n ( f ( a ± + 1 ) - r n ( a ± + 1 ) ) » 1 = l,.-.,2n + Then, R ( f : [a,b]) > min {|f(a ) - r (a )|} n • - i „ „ ' l n l i=l,...,zn+2 LEMMA 6. Let a and 3 be any two sequences tending monotonically n n to °°, a^,g^ >_ 1 . Then there e x i s t s an e n t i r e f u n c t i o n f so that f ( 0 ) = 0 f ( 6 ) > c f ° r a l l n and f i s r e a l valued on [0,°°] . ' n — n PROOF. We assume that the a are i n t e g e r s . Define numbers y by n n Y l ' " " Y ( a -1) = 1 / 3 1 ^ ' " • ^ ( a ^ - l ) 1 / 3 2 Y ( a ; L + . .+a ) ' * , Y ( a +. . + a k + 1 - l ) 1 / B k + 1 and consider f ( z ) = I (y ) z • n=l Then f i s e n t i r e s i n c e ( ( Y ) n ) ^ y ' r i = y —> 0 and n • n f ( 3 k ) = I ( T n ) n ( 3 n ) n n=l (a^+...+a^-l) - A + A 6 « > n n=(a^+. . •+a-^_^J - 38 -PROOF OF EXAMPLE 5. Pick 8 '+ » so that 6 > 4- and pick, by n " n 6.ri Lemma 5, g en t i r e so that g(log B n) >. 4lln . s i n C 2 C z ) ) Consider f ( z ) = 2 H & . It i s immediate that f i s en t i r e z e and that l i m f (x) = 2 . Since g(log 8 ) >_ 4IIn, sin(g(z)) o s c i l l a t e s x-*>° between -1 and +1 at le a s t 4n times on [0,log 3 ] • Thus, there e x i s t 0 < a_ < a, < a 0 < ... < a. < log 8 so that — 0 1 2 4n — n f(a„.) = 2 + — > 2 + . 1 . = 2 + ~ -2i a. — log 8 8 i n n e e and x n We conclude, by applying Lemma 5 with r =2, that R n ( f : [O.log 8 n]) i f = 6 n . n We deduce part b) from part a) by considering f ( ~ ) • • This preceding example provides a counterexample to the following problem posed by Reddy and Erdos. PROBLEM ([7], Problem 9). Let f(x) be any en t i r e function s a t i s f y i n g the assumption that lim f(x) i s f i n i t e . Then there e x i s t r a t i o n a l r e R for which, for each e > 0, there e x i s t n n i n f i n i t e l y many n such that - 39 -< exp (-' f ( x ) n ' l[0,») - . , l + e (log n) We should note that the previous i n e q u a l i t y holds for e n t i r e functions with non-negative c o e f f i c i e n t s ([8] Theorem 7). The exact r o l e of r e s t r i c t e d r a t i o n a l s i n approximating a n a l y t i c functions on an i n t e r v a l i s unclear. What r o l e the c l a s s R plays n J i s even more problematical. It does not a r i s e nearly as n a t u r a l l y i n the a n a l y t i c case as the cla s s R + does. In l a t e r sections we w i l l J n examine some s p e c i f i c approximations that may help to c l a r i f y t h i s s i t u a t i o n . - 40 -5 - APPROXIMATING x 1 1 . How c l o s e l y can x 1 1 be approximated by polynomials or r a t i o n a l s of lower degree? Chebyshev answered t h i s question for polynomials as follows: THEOREM 14. ([11], p. 30) n n ( x n + 1 : [0,1]) - - ^ . •TU ^ - • n + 1 Tn+1 , * . th The best approxxmant i s x 2n+l ' w n e r e ™ i s the n sh i f t e d Chebyshev polynomial defined by T n ( x ) = ^ n ^ x - 1) • The Chebyshev polynomials and the above theorem are of fundamental importance i n approximation theory and i t has been natural to examine the e f f i c a c y of replacing II by if 1", R , R + etc. i n t h i s n n n n r e s u l t . For the class R Newman proves: n THEOREM 15 [17] R n ( x n + 1 : [0,1]) n , n+1 r /-. - i T \ 1 1 R (x : [0,1]) > n v " ' L " ' ^ J / - 22n-3 (3n+2. ' n /2n This error behaves roughly l i k e ^— and thus, the best - ( 3 / 3 ) 2 n approximation from R^ to x n +"^ i s considerably closer than the best approximation from II . The s i g n i f i c a n c e of t h i s r e s u l t i s not e n t i r e l y c l e a r . The n o n - l i n e a r i t y of R^ prevents us from employing Theorem 15 as we would Theorem 14. Achieser [1] derives the exact sol u t i o n to R^Cx11"1"^: [0,1]) i n terms of a determinant equation. - 41 -Reddy considers the approximation of xn+"'" by r a t i o n a l f u n c t i o n s from R + and e s t a b l i s h e s -n THEOREM 16 [19]. R 4 ' [ x n + 1 : [0,1]] = n [ x n + 1 : [0,1]] . n n We see that the r e s t r i c t e d r a t i o n a l approximation i n t h i s case reduces to polynomial approximation. This i s a l s o true f o r approximations I | from R^ . Newman and Reddy prove the f o l l o w i n g two theorems. THEOREM 17 [18]. The best approximation p k e ITk to x on [0,1] i s d, x where d, s a t i s f i e s k k and , • .-, i \ /• k xn+l-k ,n+l-k ( n + l ) ( l - d k ) = (n+l-k) (^j-) d k n k(x n + 1V [0,1]) = 1 - d k THEOREM 18 [18] n + ( x n + 1 : [0,1]) = R k ^ ( x n + 1 : [0,1]) Newman and Reddy [19] note that n ! ( x n + 1 : [0,1]) * c n v ^ ' n+l We show that Theorem 18 can be extended to any p o s i t i v e i n t e r v a l . THEOREM 19. I f a > 0 then, f o r k < n, n k ( ( x - a ) n : [a,a+l]) = R 4 4 " ( ( x - a ) n : [a,a+l]) - 42 + k where a best approximating polynomial from U.^  i s of the form ax I?k | | ri PROOF. Let — e L be a best approximation to x on [a,a+l].. q k k h k n Suppose p, (x) = £ a x and q, (x) = £ b x where — i s k h=0 h R h=0 h q k £ h normalized so that ) b, (a+1) = 1 . , « n h=0 If e = R^fCCx-a) : [a,a+1]) then |p^(a+l) - l | <_ eq^Ca+l) and Now P k(a+1) >_ 1 - e n p v ( x ) | |( x - a ) n - -^r-r\ I r < e q k(x) 1 1[a,a+1] implies f or x e [a,a+1] that q k ( x ) ( x - a ) n - P k ( x ) | < e Since ^ ( x ) £ q^lu+l) = 1> for x e [a,a+1], q k(x) • ( x - a ) n <_ ( x - a ) n Also, k ^ k h x h P k ( x ) = ^ V = ^ a h ( a + 1 ) <^i> k h = 0 • h=0 n a 1 >- ^k jU<«">h h=0 where the l a s t i n e q u a l i t y follows from (1). For x e [a,a+1], ( x - a ) n <_ ( ^ j ) ^ a n a x t follows from (3) and - 43 -(4) that q k ( x ) • ( x - a ) n < ( x - a ) n < d - a ) ( ^ I ) k + s C ^ ) " < Pk<*} + ^ Thus, | | ( x - a ) n - a - e X ^ I I ^ ^ ±z • REMARK 3. From the l a s t l i n e of the previous proof we see that H ( x - a ) n - ( ^ T ) k H [ a , a + l ] £ 2 R ^ ( ( x - a ) n : [a,a+l]) . I f we s u b s t i t u t e i n x = a + 1 - — we ob t a i n n In p a r t i c u l a r , -1 - i - r>++// Nn+1 r , i n s 1, a+1 -1, l i m R ((x-a) : [a,a+1]) > -^[e - e ] n — z n-x» and we see t h a t , i n con t r a s t to the case a = 0, f o r a > 0 l i m R + + ( ( x - a ) n + 1 : [a,a+1]) > 0 . n REMARK 4. The geometric c o n s i d e r a t i o n that a best approximant 3x to ( x - a ) n on [a,a+1] must o s c i l l a t e at l e a s t twice a c h i e v i n g a minimum d i s t a n c e below ( x - a ) n at a + 1 and a maximum d i s t a n c e above at some y e (a,a+1) leads to a system of equations: (a) g ( a + l ) k = 1 - e (b) n ( Y - a ) n _ 1 = ( k - l ) B Y k _ 1 - 44 -(c) By k " ( Y - a ) n = e where I | ^ e = ((x-a) : [a,a+1]) . From these equations ((x-a) : [a,a+1]) can be computed f o r any p a r t i c u l a r k,n and a . Another g e n e r a l i z a t i o n of Theorem 18 i s r-« \H THEOREM 20. Let f ( x ) = ) e x . where c > 0 . Then L ,, m m — m=n+l R ^ ( f : [0,1]) = n+(f: [0,1]) . P n ++ PROOF. Let — be a best approximant to f from R where q n n n k n k P n ( x ) = J Q a k X a n d q n ( x ) = J 0 V ' L e t e = N f " P n / q n H [ 0 , l ] and suppose that ^ ( 1 ) = 1 • Then, 1) 0 < q n ( x ) f ( x ) < f ( x ) 2) p ^ ( l ) > f ( l ) - £ = ( I c^ - £) and n - ^ m 3) p (x) = [ Y > ( I a ) x n  n k=0 k k=0 k = p n ( l ) x n ( y c - £ ) x n > n+l > f ( x ) - £X Thus, on [0,1] q (x)f (x) <_ f (x) <_ ( I c m - e ) x n + EX 1 1 <_ p (x) + EX 1 1  n n+l' - 45 -from which we conclude that l f ( x ) " ( J i C m " e ) x I 1 I l ro , i ] - e We note that Theorem 18 i s a s p e c i a l case of the above r e s u l t . We now consider the problem of approximating xn+^~ by r a t i o n a l f u n c t i o n s w i t h monotone decreasing denominators. The approximations from t h i s c l a s s , i n con t r a s t to approximations from R^, are considerably more e f f i c i e n t than polynomials. THEOREM 21. There e x i s t s p /q e R , q monotone-decreasing on *n TI n n [0,1], so that I 1 1 + 1 / I I A n' Hn ' 1 [0,1] -, 28n/3 where A i s a constant independent of n . PROOF. Let T n be the n Chebyshev polynomial. Then ([22] p. 14) n , . n-k T (z) = I a, z k k=0 R where, a, = 0 i f k i s odd and k -(n-k+2)(n-k+1)  a k k(2n-k) \-2 We show f i r s t that i f k < n ( l - /2/3), then - 46 -Now , . I(n-k+2)(n-k+1)• i i | a k ! 1 k(2n-k) 1 1 ak-2 1 > < n- k> 2 la I -k(2n-k) 'k-2 1 * Thus, i f . fe~k,\ > 2, then la. I > 2|a. J . We observe that k(2n-k) — 1 k 1 — 1 k-2 1 • 2 : i ^ k?'\ = 2 exactly when k(2n-k) n 2 - 6nk + 3k 2 = 0 which has solutions: k = n ( l ± /27J) . From (3) we deduce (2). We consider T (/z) for fixed even n . n TAG) = I d, ( v ^ ) n - k n k=0 k n/2 . (n/2-1) . . (n/2-m) = a z + a„z +....+ a_ z o 2 2m (n/2-m+l) + a(2m+2) Z + ... + a n . Pick the largest even m so that 2m <_ (1 - v/273)n . Consider (n/2-m) m m-1 , z [a z + a 0 z + ... + a„ J o 2 2m m Now - 47 -^ t / x m-1 . , , N m-2 P (z) = ma z + (m-l)a„z m o z , , „. m-3 , , 0 s m-4 + (m-2) a, a + (m-3)a,z 4 o + 2 ( a ( 2 m - 4 ) ) z + a(2m-2) Since a 4 n > 0, a ^ n + 2 < 0 a n d b y ^ ' l a k ' - 2 ' a k - 2 ' f ° r k - ' 2 m > we have ••'P'(z) < 0 f o r z € [0,1] . (6) m A l s o , since | a k| >_2|a k_ 2| and m i s even P,(l) - a o + ( a 2 + a 4 ) + .... + <a 2 m_ 2+a 2 n> We are now i n a p o s i t i o n to approximate z1^ m . We have, by ( 4 ) , (6) and ( 7 ) , T (/z~) = z ( n / 2 - m ) P ( z ) + Q n mv x(n/2-(m+l)) where i f m s a t i s f i e s (5) P m i s monotone decreasing on [0,1] and P (1) > 2 m ~ 1 a = 2 m _ 1 2 n " 1 . Thus, s i n c e | |T /Z"| | R , = 1, m — o n IU,1J I, (n/2-m) Qn/2-(m+l)•• 1 m M z + P I I [0,1] - 9 m - l 9 n - l ' ( 8 ) m Z Z By (5) m s a t i s f i e s n ( l - S2/3) >_ 2m > n ( l - J2/3) - 4 . - 48 -' I f we s e t h = (n/2 - m) t h e n and h = (n/2 - m) < I S2/3 + 2 o r n >_ 2/3/2 h - 4/372 m - 1 + n - 1 >|(1 - J2/3) - 4 + n >_ i i(3/2 - \ S2/3) - 4 >_ 2/3/2 h(3/2 - I /2T3)- 4/372(3/2 - I /T/3) - 4 = h(3v^/2 - 1) - 2(3/372 - 1) - 4 . S i n c e (3/3/2 - 1) = 2.67.. •> 8/3, t h e r e s u l t i s co m p l e t e d by s u b s t i t u t i n g t h e above i n t o (8) . COROLLARY 10. There i s a c o n s t a n t A so t h a t R n ( ( l - x ) n + 1 : ^ 7 ( ^ 3 ) • -49 -6 - POLYNOMIALS AND THE CLASS R'' . The c l a s s R'' does not n n con t a i n the polynomials. Thus, the d i r e c t theorems of polynomial approximation do not immediately apply. In f a c t , we cannot even a p r i o r i a s s e r t that any reasonably l a r g e c l a s s of fu n c t i o n s can be r e a l i z e d as the l i m i t s of r a t i o n a l f u n c t i o n s w i t h p o s i t i v e c o e f f i c i e n t s , CO The uniform c l o s u r e of { u II } i n C[0,1] incl u d e s only those n=l n f u n c t i o n s a n a l y t i c i n the u n i t d i s c whose power s e r i e s expansions have only non-negative c o e f f i c i e n t s . However, the uniform c l o s u r e 00 I j of { u R } i n C[0,1] i s e x a c t l y the continuous f u n c t i o n s which n=l are non-negative on [0,1] . This l a t t e r f a c t i s a consequence of a r e s u l t of E. Meissner. LEMMA 7 [15]. I f p e II and p (x) > 0 f o r x > 0 then n n n p = r e R f o r some m . n m m In t h i s s e c t i o n we w i l l bound the m of the above lemma i n terms of n and some knowledge of the roo t s of p n . We w i l l develop estimates f o r how c l o s e l y polynomials can be approximated by p o s i t i v e c o e f f i c i e n t e d r a t i o n a l functions»and a l s o estimates f o r how c l o s e l y c e r t a i n r a t i o n a l f u n c t i o n s can be approximated by polynomials. 2 LEMMA 8. Suppose a, 3 > 0 and suppose x - ax + 3 has no p o s i t i v e 2 | | roo t . Then x - ax + 3 = r (x) where r (x) e R and where m m m m < 32 ( <T) ~ 4-^ 3 2 PROOF. The quadratic x - ax + 3 has no p o s i t i v e root i f and only 50 -2 i f a < 48 . 2 We set c = a /8 and note that c->< 4 . Consider: (x - ax + 8)(x + ax + 8) = x + (28 - a )x. + 8 4 2 2 = x + 8(2 - c)x + 8 • If c < 2 we have the desired f a c t o r i z a t i o n . If c > 2, consider ( x 4 + 8(2 - c ) x 2 + 8 2 ) ( x 4 - 8 (2-c)x 2 + 8 2) 8 2 2 4 4 = x° + 8 (2 - (2 - c ) Z ) x ^ + 8 2 If 2 - (2 - c) > 0 we are f i n i s h e d . In general we proceed as follows: Let „n+l „n-l 9 „ ? „n „n P n(x) = x Z + 8 (2 - (2 - (2 - . . . ( 2 - c ) Z ) Z . . . ) Z ) x Z + 8 „n+l „n-l „n „n = x + 8 c x + 8 n where c has n nested terms, n Let _ 9n+l 9 n - l „n „n P (x) = x - 8 c x + 8 n n 2 Note that, since c = (2 - c ) n+l n - 51 -_ „(n+l)+l „n „ n+1 „n „n+l „n+l P • P = x - f3 c x + 26 x + (3 n n n „n+2 „n ? n + l ~n+l - x 2 + s2 + 8 Consider the sma l l e s t n ( i f i t e x i s t s ) so that c i s non-negative. n Then by (3) and ( 4 ) , ( x 2 - ax + B)(x 2 + ax + 3) = P., and P • P • P_«. . . «P -. = P 1 1 2 n-1 n where P . . . -P ' e n + , 0 s i n c e each c < 0 f o r k < n 1 2 n-1 ( 2 - 2 ) and where P e IT+ since c > 0 . Thus, we have n 2 n _ (x +ax+B)P 1'P 2-...•P n_ 1 2 where n i s the smallest i n t e g e r so that c > 0 . n Now, suppose c,,...,c , ,c < 0 (6) . V 1 1 n-1 n Then c = 2 - ( c T ) 2 < 0 n n-1 im p l i e s (c , ) 2 > 2 or -c . > (il + 2 ) 1 / 2 n-1 n-1 and by i t e r a t i o n c > 2 + (2 ... (2 + (2 + 2 1 / 2 ) 1 / 2 ) 1 / 2 . . . ) 1 / 2 = 6 (7) n where (7) contains n twos. - 52 -By assumption c < 4 and since 6^ -> 4 as n -> 0 0 i t i s clear that for some n (6) i s not s a t i s f i e d and eventually some i s greater than zero. We now analyze the speed of convergence of 6^  . We note that 1/2 6 = 2 + 6 ., and hence n n-1 (6 - 6 1 ) = 6 1 / 2 - 6 1 / 2 n n-1 n-1 n-2 6 . - 6 _ n-1 n-2 fil/2 + ,1/2 • n-1 n-2 Since 6 > /2 and 6 > 6 . for a l l n, n n n-1 n n ~ 2/2 /2 4 < ^=1 <  ( 2 / 2 ) n _ i ~(2/2)n Thus |4 - 6 I < I 4 . < 1 »- k — n k=n (2/2) k (2/2) n If h s a t i s f i e s (2/2) h 2 < 4 - c = 4 - x then, by (6) and (7) for some n < h, c^ > 0 and by (5) i t follows that we may choose m i n the statement of the lemma so that ,h+2 m = 2 where h s a t i s f i e s h > l o g 2 2/2 - 53 -2 1 In p a r t i c u l a r , we may choose h = -j(log2(^_ c)) + 3 • LEMMA 9. Suppose p e II has no roots i n the region r r n n = {z: |arg(z)| <_ ^ } and p n(x) > 0 for x > 0 . Then , -i^T.4/3 p = r e R where m < Ion n . n m m — 1 PROOF. Let x - ax + y be a quadratic factor of p n . We assume 2 that a,Y > 0 since otherwise either x - ax + y has p o s i t i v e c o e f f i c i e n t s or i t has a p o s i t i v e root. We proceed to replace, using I | • lemma 8, each such factor by an element of R^ 1 1 2 1 1 2 We write y = - ^ ( — ^ + l ) a + 6 and set g = - ^ ( — ^ + l ) a . Since h • h 2 1 x - ax + y has no roots i n ^(77), i t follows that and / , 2 1 -/4y - a _^ TT a 1 2 4y >_ (—? + l ) a h from which we deduce that 6 > 0 Consider x - ax + 3 with 1 r 1 1 -1 \ 2 h By lemma'8 x - ax + where k < 32 4 - % 2/3 = 32 4, -+ 1 2/3 = 32 2 1 h Z ( - ? + 1 ) -•• h 2/3 < 32h 4/3 - 54 -We now repl a c e x - ax + y by r ^ + 6 . Since there are a maximum of n/2 such terms that need r e p l a c i n g we have p = r e R n m m where m < 32h 4/3 n 16h n • The 6 defined by statement (7) of lemma 8 tend to 4. n i f we f i x e > 0, then i n (8) of lemma 7 f o r n > N £, Thus, 4 - 6 < n — ( 4 - e ) ' We can deduce from t h i s the f o l l o w i n g v a r i a t i o n of lemma 8. LEMMA 10. For each e > 0 there i s an A so t h a t * i f f o r any a ,B > 0 2 -•the quadratic x - ax + 2 has no p o s i t i v e roots then x - ax + r (x) e R m m where m < A 4 -From lemma 10 we can deduce a strengthened form of lemma 9. LEMMA 11. For each e > 0 there i s a constant A so that i f e p ell has no roo t s i n fi(-r) and p (x) > 0 f o r x > 0 then n n h n T , + + , A T.(1+E) p = r e R where m < A h n . n m m — e REMARK 5. Lemma 10 i s " e s s e n t i a l l y " c o r r e c t i n the sense that we cannot re p l a c e the exponent y + e by any exponent n < TJ- . To see - 55 -t h i s we consider 2 2 1 2 1 q(x) = x - ax + 3 = (x-1) + —„ = x - 2x + 1 + —, — i i TT Then q (x) has roots at 1 H and since a r g ( l -i ) < — we deduce ^ n n n I | from Remark 1 that, i f q = r e R , then m > n . Since m m 1/2 ' 1 2 4 - « [ B J I 1 + 2"' n 4(1 +- 2) - 4 n 1/2 n < — we see that the exponent i n Lemma 10 must be at l e a s t as large as The next three lemmas provide estimates for approximating poly-nomials whose roots are bounded away from a region containing [0,1] Let A(<5) be the half plane defined by A(6) = {z|re(z) >_ 6} and l e t W(^, 6) be the wedge-shaped region defined by Wfckfi) = {zlarg(z) < h n {x|re(z) < 6} . n — n — 2 LEMMA 12. Let 6 > 1 . Suppose x - ax + 3 has both roots i n the ha l f plane A(S) = (z|re(z) ^.6} .' Then 2 R ! + ( ( X 2 - ax + 3 ) : [0,1]) < 2 6 2 m U A M y • - , m + l o - 1 2 PROOF. Suppose that x - ax + 8 has two complex roots / 2 y ± . Then >_ 6 > 1 and 8 > 0 and we can write, f o r some non-negative y, - 56 -Now x - ax + B = x - ax + + y = ( y - x) + Y .a. ,2 ,ou2, ,2, 2 ( y x ) = (1 " <->* ( - ) 2 (r ( f x ) k ) 2 k=0 f o r x e [0,1] C o n s i d e r r„ (x) = 2m m ( « ) 2 V -+ ( I ( f x ) V k=0 a Then, r„ (x) e R„ and on [0,1] Zm zm r 2 m ( x ) - (x - ax + B) ,ou 2 2, v2 = O (1 " Ox) - m „ . I £ x ) k k=0 a ,ctN 2 , 2 m „ , I (- x ) k k=0 a (1 - 2 x) + * a ' m X (- x ) k k=0 a < (7) (1 - - x) ? 2 1 a 1 - ( f x ) m + 1 ) a - | x ) ( l - ( f x ) m + 1 ) - ( 1 - f x) 2 1 ^ 2 1 - (- x) a; ,2-m+l JLm-1 2 V - 2 V 28 < < : - < /2.m+l — . ,l x m + l — .m+1 1 - (—) 1 - (-7) 0 - 1 a 0 I f x - ax + g has r e a l r o o t s we p e r f o r m a s i m i l a r a p p r o x i m a t i o n f o r each l i n e a r f a c t o r . In f a c t , i f r (x) = - f o r y > 6, then n n , h=0 T Il(x-Y) - r ( x ) | | r n < 1 n ^ " [ 0 , l ] - .n 1 6 LEMMA 13. Suppose, f o r i = l , . . . , n , that a^ > 0 and | - a^ | < Then n n n I II a. - H b . l < ( H a.)ene n . i = l i = l i = l PROOF. n n n n b. I l a . - n b. I = ( n a.) I i - n — | i = l i = l i = l 1=1 l n n b. - a. = ( n a.) Ii - n (i + — ) I . , l . -> a. 1 i = l i = l i n n |b . - a . | < ( n a ) [ II (1 + — - ± - ) - 1] i = l 1 i = l a i n n < ( n a ) [(1 + e ) n - 1] < ( n a.)'(e e n - 1) i = l 1 i = l 1 n oo , .k n i ( n a ) [ ^ < ( n a . ) e n e £ n . i = l 1 k=l K- i = l 1 2 25 1 LEMMA 14. Let 8 > 1 and suppose that n ' < — -( 6-l ) 2 ( 6 m + 1 71) - n P n e has no roots i n s i d e the wedge W(-j^,6), p^ > 0 on [0,1] then 2 2 : « P n > : [0,1]) 1 ||p n l l r o ±]l 2 £ f m + 1 " 1. (16hZ+2m)n n n L U ' 1 J ( S - l ) A ( & m + 1 - 1) PROOF. We w r i t e p = p II t . IT t . where p has no ro o t s i n r n a , , x . x *a k=l . k=b+l ), each t . , i = l , . . . , b , s a t i s f i e s Lemma 12 and each n x t . , i = b + l , . . . , c , i s of the form a. - x w i t h a. > 6 . x 1 x I | 2 By Lemma 9, there e x i s t s r ^ e R^ w i t h d <_ 16h a so that = p^ . By Lemma 12 (and the l a s t l i n e of the proof) 2 2 6 R 2 m ( V [ ° ' 1 ] ) ± f im+l _ ± Let s. be an element of R„ that s a t i s f i e s (1) f o r each t . and x 2m x consider r = r,sns„...s . Then d 1 2 c I | (a) r e R ^ where c <_n - a 16h a+2mc (b) l i t , - s " ' ^ i " i 1 1 [ 0 , 1 ] - xm+l • and f o r x e [0,1] 2 - , - 2 6 2 | t . ( x ) | (c) t . C x ) > ( 6 - l ) Z and t.(x) - s . ( x ) | < ^-1 ( 6 - l ) 2 ( 6 n i + 1 - 1) I f x e [0,1] then, by lemma 13 and (c) above, c c | p ( x ) - r ( x ) | = |p ( x ) | | n t , ( x ) - n s ( x ) | n a k=l 1 k=l 1 c < | P ( x ) | I n t ( x ) | a k=i 1 2en6 2 l (5 - l ) 2 (6 m + l - 1)J - I ' p n ( x ) I' ro i i •• T i f f i — ~ n L U ' J (6-1) (6™ - 1 ) The next lemma concerns the approximation of polynomials w i t h p o s i t i v e c o e f f i c i e n t s i n x and (1-x) . - 59 -LEMMA 15. Suppose p (x) = £ a x (1-x) i s a P.P.C. of degree n k+£<n ^ n then R ^ C p C x ) : [0 ,h) < p n(|) n 2 n _ m . mn n / — n z PROOF. We observe, as i n the proof of Lemma 12, that (1-x) (1-x) , = (1-x) -m 1 1 ] 1 1 v ' m+1 1 1 1 l+x+. ..+xm [ 0 , y ] 1 - x m + 1 [ 0 , j ] m+1... . = I|X (1-x)I| 1 1 m+11 1 r 1-. 1 - x [ C y ] . 1 < - 2m+l Since a^ - b^ = (a-b) (a^ + a^ 2*b + ... + a'b^ 2 + ^) (1-x)" - ( i -) ill I . _ A 1+X+...+X11' "[0,|] - 2 m + 1 ' Let r = and consider s(x) = Y a, „x k(r )^ . m l+x+...+xm k+£<n k A m We note that each term of t h i s sum can be brought to the common i n IT j j denominator (l+x+ +x ) and hence, s(x) e R. . Also, by (m+l)n (1) p (x) - s(x) || <_ I a J |x k| n W [0, i] " k 4<n U " [OJ ] 2 m + 1 n r _1 - 2 m + 1 k+i<n 2 k" Since p ( i ) = £ a —, • — we deduce from (2) that n k+£<n k l 2 K 236 - 60 -|p (x) - s(x)|| l P n ( | > ^ • We observe that a reasonable f a c s i m i l e of Lemma 15 can be deduced from Lemma 14 and Remark 2 (p. 15). The f i n a l series of r e s u l t s concerns the approximation of r a t i o n a l function with no poles i n the half plane {re(z) > 0} . In p a r t i c u l a r , we show that i f q has no roots i n {re(z) > 0} then —_ can be n q n e f f i c i e n t l y approximated by polynomials with p o s i t i v e c o e f f i c i e n t s i n x and (1-x) . LEMMA 16. Suppose a >_ 0 and 1 > 6 > 0 then there e x i s t s a poly-nomial p e n + so that n n II n _ < k _ - ^ l l • 1 (l-< S ) n + 1 ( l - 6 ) n + 1 H p n U x ; x+a1 1 [6,2-6] - 1+a 6 - 6 PROOF. We expand —^— i n powers of (1-x): x+a 1+a 1_ y ,1-x, k x+a 1 (1-x) 1+a . L n v l + a y 1 " U+oO k = 0 1 n 1-x k We l e t p (1-x) = y — £ O r r - ) a n c * observe that k=0 - 61 -l pn ( 1 x ) x+ J '[6,2-6] 1+a I l k = J + 1 (l+a ) I I[6,2-6] 1 + a k = n + l 1 + a L a k=n+l 1 ( l - 6 ) n + 1 1+a LEMMA 17. Suppose a,8 > 0 and suppose 1 > 6 > 0 . Then t h e r e / e x i s t s p^ n a P.P.C. of, degree 2n so that ( x + B ) 2 + a " P 2 n ( X ) M ^ > 1 ] £ ( l + B ) 2 + a ' 1 ( l - 6 ) n 4 : 1 PROOF. We w r i t e (1+B) 2 (x+B) 2 + a rx+Bx 2 , a 1 + B d + B ) 2 A 1 + c y + c ! _ (IzZ) l+c' 00 = A y ( i z Z ) k 1 + c . L n ' k=0 1/2 "4" 6 2 We observe that i f x e [6 ,1] then C j ^ g ) e [<S»1] and as i n the preceding lemma, i f we set k=0 then - 62 -7 — J — ~ q n ( 1 " W } 1 1 [ ^ , 1 1 - lHte . * W (x+g) + a k=n+l : A ( l - 6 ) n + 1 - 1+c 5 1 ( l - 6 ) n + 1 (1+g) 2 + a We complete the proof of t h i s lemma b y n o t i c i n g that f-t ,x+g^2N _ r. x+g, . 1 x+g,-, q n ( 1 " W } " q n [ ( 1 " 1 + I ) ( 1 + i T g ) ] 1 + B (1+g) 2 and hence, that q^ i s a P.P.C. of degree 2n . LEMMA 18. Suppose 6 > 0, suppose that - <_ — , and suppose that q e II , q has leading c o e f f i c i e n t 1 and q has no roots n n n n i n the half plane {re(z) > 0} .. Then, for each j , there e x i s t s P n.j a P-P.C. of degree n*j so that f o r x e [/<5,1] I 1 _ /• \ I <- en ( l - ( 5 ) j + 1 lq n(x) " P n - j W l - |q n(x)| 6 1 h PROOF. We write — 7 — r - = II t. (x) where h < n and where each t . q n(x) . = 1 1 i s either of the form • (a) -J— , a. > 0 x+a. x x or ( b ) ^-Tp where g. > 0 and a. >_ 0 . (x+g.) + a. 1 1 x x If t . i s of the form — 7 — then b y Lemma 16 there e x i s t s x x+a. - 63 -p. e if!" so that If t . i s of the form - then by Lemma 17 there 1 ( x + 8 ± ) + a e x i s t s a P.P.C. p of degree 2j so that (2) || \ V*) l l [ / 6 - ±] 1 — — 2 ^ ^ f ^ ( x + 8 . ) + a. J L ' ° ' 1 J ( 1 + 8 . ) + a. 0 X X X X We consider r . = IT p.- where p. i s an approximation to t. nj j = 1-3 J 3 which s a t i s f i e s , as applicable, (1) or (2). By construction r i s a P.P.C. of degree n j . We note that on / 0 . 1 1 , 1 1 (3) — o a n d -ZZZ- > / . o \ 2 . — / 1 , o \2 , x+a. — 1+a. (x+B.) + a. ( 1 + 8 . ) + a. x x x x x x Hence, by (1), (2), (3) and Lemma 13 i f x e [/cS,l] then I — i 1 1 < e n ( 1 - 6 ) J + 1 1 q (x) r . (x) • — r-^r 7 -M n v nj q n ^ - 64 -7 - APPROXIMATIONS FROM THE CLASS R'" . We expl o i t the theorems n of the previous section and demonstrate that functions i n C[0,1] can be approximated quite e f f i c i e n t l y by r a t i o n a l functions with p o s i t i v e c o e f f i c i e n t s . Our cotnparitive standard of e f f i c i e n c y i s unre s t r i c t e d polynomial approximation. We l i s t two known r e s u l t s that we require: th Let f e C[0,1] . We define the n Bernstein polynomial by B ( x ) = B (f:x) = I f A ( ^ ) x k ( l - x ) n _ k . n n , u - n k k=0 We define, i n keeping with [12], the class A = A (m,M n,M„,M 0,...M ), where m § 0, M. > 0 i = 0,2,3,...r r,a r,a 0 2 3' r x — and 0 < a <_ 1, to be the class of f e C [0,1] which s a t i s f i e s (a) m <_ f (x) <_ MQ f o r x e [0,1] , (b) M f ( i ) M [ 0 , l ] l M i 2 1 1 1 r (r) and (c) f e Lip a . THEOREM 22 ([21] p. 15). If f e C[0,1] then ||f(x) - B n ( f :x) | | [ Q j < 2a>(f,l//n") . THEOREM 23 (LORENTZ [12]). I f f belongs to the clas s A (m,M„,...,M ) then there e x i s t s p a P.P.C. of degree n r,a ' O r ' cn . such that ||f(x) - P n ( x ) | | [ 0 j l ] l C ^ r + a - 65 -where C depends only on r,m and the . We are now i n a p o s i t i o n to prove the following approximation theorems: THEOREM 24. If f e C[0,-|] and f ^ 0 on [0,-|] then R ^ C f : [Q,h) < | | f | | , n 2 n _ m + 2a>(f,—) . mn Z — r „ ± , j— [O.j] /n PROOF. We extend f to a continuous function on [0,1] by r e f l e c t i o n . That i s , by se t t i n g f (x + y) = f (x - \) for x e [0,|] . Then the modulus of continuity of f on [0,1] i s the same as the modulus of continuity of f on [0,-|-] . Consider B the n ^ - B e r n s t e i n polynomial f o r f . Then, since n f i s p o s i t i v e on [ 0 , 1 ] , B^ i s a P.P.C. of degree n and B ("r) < I I f I I m -n • Thus, by Lemma 15 and Theorem 22 n z — ' ' ' ' LU,1J < I |f M n 2 n - m + 2 u ( f . L U , I J VTT • COROLLARY 11. If f e C[0,-|-], f > 0 for x e [0,~] and 6 > 0, then there e x i s t s A^ depending only on 6 so that 1 > . \ (f: [ O ^ ] ) i A f . ( f , ~ T ~ 4+6' n PROOF. We extend f to [0,1] by r e f l e c t i o n and observe that, i f f(a) = 0 for some a e [ 0 , 1 ] , - 66 -2*(f,-j) > 2«B(f >|. I | f | l [ 0 , l ] • We note that i t s u f f i c e s to prove the c o r o l l a r y under the assumption f(a) = 0 since otherwise we could consider f(x) - i n f {f(x)} . xe[0,l] lrl"*5 / 2 Set m = n . Then by Theorem 24 and (1), R 2 + 2 6 ( f : [0,j]) < | | f | n [0,1] (U f i / 2 + 2 w ( f ' T } „n(n . -1; Yn n ,n(n -1) + 2 o > ( f , — ) Yn < A. 03 ( f , — ) - 6 Yn If h = 2+6/2 then h = vn and R ^ ( f : [0,f]) < A, -(f: - V - ) I t i s worth noting, since m and h need not be i n t e g r a l , that the extra error introduced by keeping m and h i n t e g r a l can be subsumed by the constant A, . r 1 THEOREM 25. If f e C [0,j] and f belongs to the clas s Ar^a(m,MQ,M2,...,Mr) then R n ( f : [O.j]) <_A(—j-) 4+6 - 67 -where A depends only on r,m and the . Theorem 25 i s proved analogously to Coro l l a r y 11 s u b s t i t u t i n g the use of Theorem 22 by Theorem 23 where appropriate. We omit the d e t a i l s . The l a s t two theorems, coupled with the r e s u l t s of section 3, show that for an f e C k [ 0 , l ] , R^"(f: [0,1]) i s "roughly" bounded 3/2 X / (\ between (II (f)) and (IT ( f ) ) . The i n t e r e s t i n g question n n that remains unanswered i s : Do there e x i s t functions f exactly k-times d i f f e r e n t i a b l e whose respective ' orders of approximation from ++ 1 R and II d i f f e r by a f r a c t i o n a l power of — ?. n n n As noted e a r l i e r , R' i s sometimes much more e f f i c i e n t than n II for approximating a n a l y t i c functions. Our next r e s u l t s i n d i c a t e that we can always employ suc c e s s f u l l y i n approximating p o s i t i v e a n a l y t i c functions. THEOREM 26. If f i s a n a l y t i c and never zero i n some open region containing [0,1] and f > 0 on [0,1] , then there ex i s t s y > 1 so that R ^ ( f : [0,1]) < — Y where A i s a constant independent of n . PROOF. Since f i s a n a l y t i c and zero free i n some region containing [0,1], i t follows from Inequality 9 and Theorem 8 that there e x i s t s a region Q P [0,1] , a number n. > 1 and a sequence of polynomials p so that n I, M A 1 f - p L < A — 1 n 1 1[0,1] — n n where A i s independent of n and each p^ has no roots i n ti . We note that for some fi x e d h and 1 < 6 <2 ^1(^,8) c Q n and hence, by Lemma 14, 2 + + / - 1 1 \ l I l l 2en6 R (p : [0,1]) < | | P | L , (16hZ+2m)n n n [ U ' ± J ( l - 6 ) Z ( S m + 1 - 1 If we set m = n i n (2) and combine t h i s with (1) we obtain 2 4-4- 1 , , , , 2pnrt R M « 2 + , ^ [ 0 ' 1 D - n + M P n M [ 0 , l ] n 8 , h ^ + 1 1) (16h +2n)n n (l-o) (o - 1) Thus, there e x i s t constants B,C and y > 1 so that for each n, R ^ 9 ( f : [0,1]) < B - n „ z — n Cn y and the r e s u l t follows. Under stronger assumptions, including the a n a l y t i c i t y of f a di s c containing [0,1], we can obtain exponential rates of approximation. LEMMA 19. Suppose f i s a n a l y t i c on a dis c D(0,f3) and suppose that a < g . Then ins i d e D(0,a) f ( z ) = - 6 9 00 S (f:a) 1 ( j L - ¥ - ) z k  k=0 a 00 k. 1 \ k=0 a th where S, (f:a) i s the k Taylor polynomial of f evaluated at a . v k PROOF. Suppose f ( z ) = • Consider oo k 8 ( z ) = ^ T = I \ • 1 - - k=0 a Then °° k f( Z)g(z) = I d a m - i - ) , „ n m k-m; k=0 m=0 a k z OO k r 1 , r m. k k=0 a m=0 oo S (f :a) V / k s k k=0 a and the r e s u l t now f o l l o w s . THEOREM 27. Let 1 < a < 3 . Suppose that f i s a n a l y t i c i n 'D(0,B) and f ( x ) > 0 f o r x e [0,a] . Then, there e x i s t s 6 > 1 so that R ^ ( f : [0,1]) < \ n —- „n o • where A i s independent of n . PROOF. Since f ( a ) > 0, f o r n s u f f i c i e n t l y l a r g e 8 = S (f,a) > 0 n n - 70 -By Lemma 19 i n s i d e D(0,a) 'k, k k=0 a We p i c k N so t h a t , f o r n > N 3 n > 0 and so that N B k k ( £ (—j-)z ) has no zeros on the i n t e r v a l [0,a] . We now consid e r , k=0 f o r m > N the r a t i o n a l f u n c t i o n r e R defined by m m N 3, . m 3 , . k=0 a. , k=N+l a r (z) = — 1 + : . (2) m n r k m z I ~~j7 I ~~i7 k=0 a k=0 a Then V , k. k. v, ( k, k [f (x) - r m ( x ) | | = | I — ^ — ^ — — I I [ 0 j l ] I \ I (\) k=0 a k=0 a OO p CO l l I ^ ) l + l | f | l [ 0 ) 1 ] l I \ \ • O) k=m+l a k=m+l a For k >_ N, 3 k £ f ( a ) and we deduce from (3) that , , , , k a > I + I . K I l . r o n ' f - r r n n <• .m L U > 1 J . (4) m C ° ' 1 ] _ ••. ( a - l ) a m We note that i n (2) the second term has p o s i t i v e c o e f f i c i e n t s and that the f i r s t term has no zeros on the i n t e r v a l [0,a] . Hence by - 71 -Lemma 14, since N i s f i x e d , there e x i s t s a constant A so that for I | each h there i s an r. . e R, - so that hN . hN k=0 a a Consider m 3 r ^ v k mN k=N+l a (mN+m) m Y m ^k. I k I "IT k=0 a k=0 a Then s „, e R „. and by (4) and (5) .mN+m mN+m J If - s ||rn < D 'mN+m1 1 [0,1] - Am ' n • a U 72.-8 - INVERSE THEOREMS FOR RATIONAL FUNCTIONS WITH NO POLES IN {re z > 0} We consider approximations by r a t i o n a l functions with no poles i n the ha l f plane H = {re z > 0} . We apply the approximation r e s u l t s of section 6 to derive stronger inverse theorems for t h i s c l a s s than those we obtained i n section 3 for the l e s s r e s t r i c t e d c l a s s R + . n We need the following converse to Theorem 23. THEOREM 28 (LORENTZ [12]). Suppose there e x i s t e > 0 and P.P.C.'s p so that n ' f P J I [0,1] - ,/-,k+£ • Then f i s k-times continuously d i f f e r e n t i a b l e on (0,1) . We now show that i f we r e s t r i c t our approximants so that they have no poles i n H then they are no more e f f i c i e n t than polynomials for approximating C functions. THEOREM 29. Suppose there ex i s t s e > 0 and a sequence e R^ so that each r has no poles i n (0 < re z < 1} and so that n Ilf - r || r„ < A n 1 1[0,1] - k+e ' n Then f i s k-times continuously d i f f e r e n t i a b l e on (0,1] . P n PROOF. Write r = — where q i s normalized to have lead n q n n c o e f f i c i e n t 1 . Fix a > 0 so that ~ - > k + ^  e • By .Lemma 18 (with j - = n") there exists, for each, q a polynomial S so that f or each n n ,.. . v l l +1 1 - S , . . , ( x ) | < ^ ^ 'q n(x) ° n ( l + a ) ^ " - |q n(x)| 6 Thus, for each x e [/<5,1], ( l - 6 ) n + I n " S /^xOOp ( x ) 1 r H r n n e n !q (x) " ( l + a ) V A ' 1 V A " - 1 1 n M [ 0 , l ] ^ 6 n n We deduce from the l a s t l i n e and the given conditions of the theorem that f o r each n there e x i s t s a polynomial p (j_+a) s o 2n that I l f ~ P (1+a) I' [/6,1] -~k+e ' 2n n If we set m = 2 n ^ + 0 ^ then II C D f pml l [ / 6 , l ] ±. k+e - k + l . .nul+a m z <.2; We appeal to Theorem 1 to complete the proof, We d i r e c t our at t e n t i o n to the subclass of R'' whose elements n have no poles i n H . Approximations from t h i s c l a s s are l e s s e f f i c i e n t than polynomial approximations. THEOREM 30. Suppose there e x i s t s e > 0 and a sequence of P n r = — e R so that each p i s a P.P.C, each q has no roots n q n n n n i n {re z > 0}, and so that - . 7 4 -H f " r n H [ 0 > l ] - (^k+s ' Then f i s k-times continuously d i f f e r e n t i a b l e on (0,1) . PROOF. The proof of t h i s theorem proceeds i d e n t i c a l l y to the proof of Theorem 29 . We observe that the polynomial p ( i + a ) o f ( x) 2n i s a P.P.C. and that the r e s u l t can be completed by appealing to Theorem 28 instead of Theorem 1. - 7 5 -9 - APPROXIMATING x ^ 2 AND e X 1/2 x"^ 2 and e X 1/2 FROM R . We show that n can be approximated u n i f o r m l y on [0,1] by r a t i o n a l f u n c t i o n s w i t h non-negative c o e f f i c i e n t s as e f f i c i e n t l y as by r a t i o n a l f u n c t i o n s w i t h u n r e s t r i c t e d c o e f f i c i e n t s . This r e s o l v e s questions 16 and 21 posed by Reddy i n [20]. We e x p l o i t D.J. Newman's w e l l known r e s u l t -1/2' THEOREM 31 (NEWMAN [16]). Let ? = e -n n-1 ,. p (x) = II (x+? ) and n k=l x(p (x) - p (-x)) r j x ) n P n ( x ) + P n(-x) , then, f o r n > 4, W - r n ( x ) H [ - l , l ] ± 3 6 -n 1/2 al s o R (|x| : [-1,1]) > | e 9 n n — z 1/2 The f o l l o w i n g i s the key observation: LEMMA 20. The r a t i o n a l f u n c t i o n r , defined i n Theorem 31, has non-n negative c o e f f i c i e n t s and has a l l i t s roots and poles on the l i n e r e ( z ) = 0 . PROOF. Since P n ( z ) a n a P n / - Z ^ have no common r o o t s , i t s u f f i c e s to show that i f |p n(a+ib)| P n ( - a - i b ) |P n<«)| l p n ( " a ) 1 then re(a) = 0 . Suppose = 1, then n-1 n k=l a+ib+? -a-ib+i; n-1 n k=l 2 x 2 k , 2 , , k a +g +b +2a^ 2^ 2k 2 k a +1, +b -2a^ 1/2 = 1 -'76 -Let T. = k 2.2k., 2,_ k a +£ +b +2a£; 2, 2k,,2 „ k a +C +b -2a? n-1 1/2 , then T. > 1 i f a > 0 and T < 1 ' k . k i f a < 0 . Thus, TI T = 1 i m p l i e s a = 0, as r e q u i r e d . k=l R • COROLLARY 12. For n > 4, R (|x|: [-1,1]) < 3e n — -n 1/2 THEOREM 32. For n > 4, R ^ C x 1 7 2 : [0,1]) < 3 e " ( 2 n ) n — 1/2 and _. , 1/2 r n . 1 -9(2n) R (x : [0,1J) > T e n — z 1/2 1/2 PROOF. Let r be defined as i n Theorem 31, then r . (x ) i s a n 2n r a t i o n a l f u n c t i o n of degree at most n that has non-negative 1/2 1/2 1/2 —(2n) c o e f f i c i e n t s and M r2 n^ X ) ~ x I I [Q 1] — 3 e " T n u s » R + V 7 2 : [0,1]) < 3 e " ( 2 n ) n — 1/2 Let p ( x ) , q(x) e then, by Theorem 31, | x 1 / 2 - p ( x ) / q ( x ) | | [ 0 j l ] =|| |x| - p ( x 2 ) / q ( x 2 ) 1 -9(2n) [-1,1] - 2 6 1/2 Thus, ++. 1/2 r n 1 1 N 1 -9(2n) R (x : [0,1]) > e n — / 1/2 • REMARK 6. A.P. Bulanov [5] proves that l i m [R |x|: [ - l , l ] ) ] n -1/2 = e —IT n-x» - 77 -The r a t i o n a l functions he uses to obtain h i s upper bounds can be shown, as i n Lemma 20, to have non-negative c o e f f i c i e n t s . From t h i s we can deduce that —1/2 —1/2 lim [ R ^ C x 1 7 2 : [ 0 , l ] ) ] n = * = lim [R ( x 1 / 2 ; [ 0 , l ] ) ] n n n n-x» n-*» THEOREM 33. For n > 4, -M- 1 / 2 - n 1 / 2 R ( e X : [0,1]) < 4e-e n n — and ^ / x r„ 1 7 , 1 -Ion e R n(e : [0,1]) > y e -PROOF. We write 1/2 oo 1/2 k ... «, 1/2 k x V (x ) f \ , r \ 1/2 , v (x ) / l V k=0 k=2n where p and q e II + and for x e [0,1], 1 < q (x) < e . Let n n n — n — 1/2 s„ = p (x) + q (x)r„ (x ), where r_ i s defined as i n Theorem 31. Zn n n zn zn Then s„ i s a r a t i o n a l function of degree at most 2n and s„ has 2n ° 2n ++ x 1 / 2 x 1 / 2 non-negative c o e f f i c i e n t s . Hence, R„ (e : [0,1]) < l i e - s 0 (x) I I r _ zn _ i i zn ' ' [U, * \ \ t \ \\ I I 1 / 2 < 1 / 2 ^ I I a. e , o - ( 2 n ) 1 / 2 ^ e < | | q n ( x ) | | [ 0 5 l ] | | x - r 2 n ( x ) | | [ 0 > 1 ] +J2njl 1 e 3 e + J^T • 1/2 1/2 Hence, R n ( e X : [0,1]) <_ 4e'Ae Let t (x) = U (x)/V (x), U (x), V (x) e n and l e t p (x) n n n n n n n and 1 n ( x ) he defined as i n (1) . Then 1/2 1 / 9 •» ,1/2. I E * " 'nW I I [0,1] = I + *n<*>* ' ~ V ( X ) + J 2 N V I I [0,1] Since q n ( x ) i l l o n t 0 ' 1 ! ' I, - 1 / 2 , M l II 1/2 , P n ( x ) - V X ) | | y 1 x 1 / 2 1/2 Thus, R n ( e X : [0,1]) > R 2 n ( x 7 : [0,1]) - - ^ y 1 - 1 8 n 1 / 2 e > w e 2 " (2n)! " 1/2 x 1 / 2 Neither x nor e are d i f f e r e n t i a b l e at zero. Thus by I | 1/2 Theorem 7, i f a > 0 then neither ((x-a) : [a,a+1]) nor ++ (x-a)"'"'''2 1 R (e : [a,a+1]) can converge more r a p i d l y than — 0 . This, n J n of course, i s considerably slower than the order of approximation for the case a = 0 . In f a c t , the following stronger r e s u l t i s v a l i d : LEMMA 21. Let 0 < a < 8 . If f(a) = 0 then (1 + ^ ) R ^ " ( f : [a,6]) > f ( 8 ) . a P n ++ PROOF. Let — = r be a best approximant to f from R on q n n ^n • [a,8] . Since p has p o s i t i v e c o e f f i c i e n t s , we have, as i n (1) of the n proof of Inequality 5, Pn(8) ±4pn^  a and hence, r n ( B ) ^4rn(a) ' a Thus, f(3) = f(3) - f(cO < |f(3) - r n ( 8 ) | + | f(a) - r n ( 3 ) | < 1 R ^ ( f : [a, 81) + ^ |r n(a)| a < (1 + ^ ) R ^ ( f : [a,31) • THEOREM 34. If a > 0, then (a) R" H"((x-a) 1 / 2: [a,a+1]) > n ~ ( l + e 1 / a ) ^ (b) R ^ ( e l X a j : [a,a+1]) > 6 n , > n ,, , 1/a. — , 1 , 1/a. T -(1+e ) (1+e )/n 1 6 n 1 n PROOF. We set 8 = a + — and note that — = (1 + — ) n < e n n an — a Both parts of t h i s theorem now follow e a s i l y from Lemma 21. We note that C o r o l l a r y 11 y i e l d s : R^CCx-a) 1 7 2: [a,a+1]) < 7-n — 1 n(8+6) and f a / 2 A ++. (x-a) r a R (e : [a,a+1]) < n — 1 n(8+6) 10 - SOME COMMENTS ON INTERVAL DEPENDENCE. Neither of the classes I | _l_ R or R are closed under a t r a n s l a t i o n x ->• x - a . Hence, we n n would expect approximations to f(x-a) on [a,a+1] to be less e f f i c i e n t than approximations to f(x-g) on [g,B+l] i f a > 3 . This, i n general, i s the case. THEOREM 35. If 0 < a < b then a) R+(f: [a,b]) > ll^f: [a,b]) - b " • 2||f|| a ' and b) R ^ ( f : [a,b])>l£(f: [a,b]> - b • 2| jf | | [ & > b ] . n k PROOF. Suppose q (x) = £ a,x , a, > 0, then i f b > a > 0 n k=0 fc fc n k n k=0 k n , , k n V k b ^ b , . = ^ a k a ' ~~k - TT Q ( A ) ' k=0 k a k an b n Thus, i f q (a) = 1 then q (b) < — . ' in n n — n a P n Now suppose that — i s a best approximation to f on [a,b] + ^ from the cl a s s R . Suppose q i s normalized so that q (a) = 1 n n n Then P n P n I l f " i^ l I [a,b] - I l f " P J I [a,b] " ' lPn ~ ' [a,b] P_ > n ( f : [a,b]) - | | l - q || ||-^| | — n 1 1 n M [ a , b ] M q n ' ' [ a , b ] > n n ( f : [a,b]) - " & n • 2| |f | L a , , . — n ^n ' 1 1 ' La,bJ - 81 -Part b) i s proved s i m i l a r l y . COROLLARY 12. Suppose f > 0 for x >_ 0 . Then f or each n a) l i m R + ( f (x-a) : [a,a+1]) = II ( f ( x ) : [0,1]) n n a-*00 b) lim R + + ( f ( x - a ) : [a,a+1]) = y( max (f(x)) - min (f(x)) a+~ n xe[0,l] xe[0,l] PROOF. Part a) i s an immediate consequence of the preceding theorem. I | To prove part b) we observe that, by Inequality 7, i f e then I l rn' I [a,a+1] - a ' ' r n I ' * a ++ Thus, i f r i s a best approximation from R to f(x-a) on [a,a+1] n n i i a i i 2n i i II 2n | | , | t h e n l l ^ ' U l a . a + l ] - I l | f | ' [ 0 , l ] • S i n c e a H f H [ 0 , l ] " ° a S a °° we deduce that as a ->• 0 0 the r tend to a constant. The n best constant approximant to f(x-a) on [a,a+1] i s •y[ max (f(x) - min f ( x ) ] and the r e s u l t follows. xe[0,l] xe[0,l] We now show that i f R (f : [0,1]) + R ( f : [0,1]) then n n R + + ( f ( x - 3 ) : [8,8+1]) i s s t r i c t l y l e s s than R + + ( f ( x - a ) : [a,a+1]) n n i f g < a . En route to t h i s r e s u l t we must employ the following c h a r a c t e r i z a t i o n theorem f o r best r a t i o n a l approximations. THEOREM 36. (CHENEY [6] p. 159). An element r e R i s a' best r n,m approximation to f I R^ m i f and only i f no element - 82 -<|> e l i n e a r span { l , x , . . . , x n , r , x * r ( x ) , . . . , x m , r ( x ) } has the same s i g n as f - r on the set of p o i n t s Y = {y: |f(y) - r ( y ) | = ||f - r||} . We e x p l o i t Cheney's proof of the previous theorem to prove: THEOREM 37. I j (a) I f r i s a best approximant to f from R then e i t h e r n,m i ) some c o e f f i c i e n t of r i s zero or i i ) r i s a l s o a best approximant from R n,m (b) I f r = —^ i s a best approximant to f from R + then e i t h e r 111 i ) some c o e f f i c i e n t of i s zero or i i ) r i s a l s o a best approximant from R^ m . P n t (c) I f r = — i s a best approximant to f from R [a,b] then v q r v n,m m ' e i t h e r i ) = 0 somewhere on [a,b] or i i ) r i s a l s o a best approximation from R^ m . PROOF. We prove part ( a ) . P a r t s (b) and (c) are s i m i l a r . Suppose that r = i s a best approximant to f from R ^ ^ f a j b ] and suppose that r s a t i s f i e s n e i t h e r c o n d i t i o n a ) i ) or a ) i i ) . Then, since r i s not a best approximation from R , by Theorem 36 there n,m' e x i s t s § = p + q r so that <j> and f - r have the same s i g n on Y = {y: |f(y) - r ( y ) | = ||f - r||} and so that p e II , q e II . Consider > 83 -r = P + ^P q - Aq Then, for s u f f i c i e n t l y small A, since neither p or q have any I | zero c o e f f i c i e n t s , r, e R A n,m We need only v e r i f y that f o r some small A ' l f rX'l I [a,b] < I l f r | 1 [a,b] to complete the proof by contradiction. Let a(y) = sign ( f ( y ) - r( y ) ) and l e t 6 = i n f a(x)cj)(x) . Let xeY e(x) = f(x) - r(x) and define A = {x e [a,b] : a(x)c(>(x) •> ^ 5 and |e(x)| > i l | e | | [ a f b ] } V B = [a,b] - A . Since B i s compact there e x i s t s y so that |e(x) < u < e L for x e B 1 1 1 ' [a,b] Thus, for x e B. (1) | f ( x ) - r x ( x ) | < |e(x)| + | | r - r x | | [ a f b ] < y + ||r - r J | [ a > b ] and for x e A, i f A i s small enough so that f - r ^ and f - r have the same sign on A, then (2) |f(x) - r x ( x ) | = o(x)(f(x) - r(x)) + a(x)(r(x) - r x ( x ) ) < || e|| Aa(x)Kx) [ a ' b ] (q(x) - Xq*(x)) From (1) and (2) we can deduce, as required, that f o r a l l s u f f i c i e n t l y small A - 84 -l f - . r x H[a,b] * H f - r H [ a , b ] * THEOREM 38. Let g > a > 0 a) I f R*"1" ( f ( x - a ) : [ a , a+1]) = R^" ( f ( x - B ) : [8,8+1]) n,m n,m then R (f(x-a).: [a,a+1]) = R ( f ( x - a ) : [a,a+1]) n,m n,m b) I f R + ( f ( x - a ) : [a,a+1]) = R"1" ( f ( x - g ) : [3,3+1]) n,m n,m then R + ( f ( x - a ) : [a,a+1]) = R ( f ( x - a ) : [a,a+1]) n,m n,m PROOF. To prove a) we observe that i f R ( f ( x - a ) : [a,a+1]) = R ( f ( x - 3 ) : [8,8+1]) n,m n,m and i f r i s a best approximation to f(x-B) on [8,8+1] from n ,m R , then r (x + (8~a)) i s a best approximation to f ( x - a ) on n,m n,m [a,a+1] which has s t r i c t l y p o s i t i v e c o e f f i c i e n t s . The r e s u l t now f o l l o w s from Theorem 37. Par t b) i s analogous. - 85 -REFERENCES [I] ACHIESER, N.I., Theory of Approximation, New York: Frederick Ungar Publishing Co., 1956. (Translated from the Russian.) [2] BERNSTEIN, S.N., Collected Works (Russian), Akad. Nauk SSSR, . Moscow, v o l . 1, 1952, v o l . 11, 1954. [3] BOEHM, B., "Functions whose best r a t i o n a l Chebyshev approxi-mations are polynomials", Numerische Mathematik, 6 (1967), 235-242. [4] BORWEIN, P.B., " A r b i t r a r i l y slow r a t i o n a l approximations on the p o s i t i v e r e a l l i n e " , J . Approximation Theory, To Appear. [5] BULANOV, A.P., "Asymptotics f o r l e a s t deviation of |x| from r a t i o n a l functions", Mat. Sbornik, 76 (1968), 228-303; Math. USSR - Sbornik, 5 (1968), 275-290. [6] CHENEY, E.W., Introduction to Approximation Theory, New York: McGraw-Hill, 1966. [7] ERDOS, P. and A.R. REDDY, "Problems and r e s u l t s i n r a t i o n a l approximation", Periodica Math. Hung., 7 (1976), 27-35. [8] ERDOS, P. and A.R. REDDY, "Rational Approximation", Advances i n Math., 21 (1976), 78-109. [9] GONCAR, A.A., "Inverse theorems of best approximation by r a t i o n a l functions", I z v e s t i a Akad. Nauk SSSR, 25 (1961), 347-356. [10] JACKSON, D., "Onf..approximation by trigonometric sums and poly-nomials", Trans. Amer. Math. S o c , 13 (1912), 491-515. [II] LORENTZ, G.G., Approximation of Functions, New York: Holt Rinehart and Winston, 1966. [12] LORENTZ, G.G., "The degree of approximation by polynomials with p o s i t i v e c o e f f i c i e n t s " , Math. Ann., 151 (1963), 239-251. [13] MARKOV, A.A., "On a problem of D.I. Mendeleev", St. Petersburg, I z v e s t i a Akad. Nauk, 62 (1889), 1-241 [14] MEINARDUS, G., Approximation of Functions: Theory and Numerical Methods, B e r l i n , Heidelburg, New York: Springer-Verlag, 1967. (Translated from the-German.) [15] MEISSNER, E., "Uber p o s i t i v e Darstellungen von Polynomen", Math. Ann. 70 (1911), 223-255. [16] NEWMAN, D.J., "Rational approximation to |x|", Michigan Math. J . , 11 (1964), 11-14. - 86 -[17] NEWMAN, D.J., "Rational approximation to x 1 1", J . Approximation Theory, To Appear. [18] NEWMAN, D.J. and A.R. REDDY, "Rational approximation to x 1 1", P a c i f i c J. Math., 67 (1976), 247-250. [19] REDDY, A.R., "On c e r t a i n problems of Chebyshev, Zolotarev, Bernstein and Achieser", Inventiones Math. 45 (1978), 83-110. [20] REDDY, A.R., "Recent advances i n Chebyshev r a t i o n a l approximations on f i n i t e and i n f i n i t e i n t e r v a l s " , J. Approximation Theory, 22 (1978), 59-84. [21] RIVLIN, T.J., An Introduction to the Approximations of Functions, Waltham, Mass., Toronto, London: B l a i s d e l l Publishing Co., 1969. [22] SNYDER, M.A., Chebyshev Methods i n Numerical Approximation, Englewood C l i f f s , N.J.: P r e n t i c e - H a l l Inc., 1966. [23] TIMAN, A.F., Theory of Approximation of Functions of a Real Variable, New York: Macmillan, 1963. (Translated from the Russian.) BIBLIOGRAPHY BOOKS ACHIESER, N.I., Theory of Approximation, New York: Ungar, 1956. (Translated from the Russian.) BERNSTEIN, S.N., Collected Works (Russian), Akad. Nauk SSSR, Moscow, v o l . 1, 1952, v o l . 11, 1954. BERNSTEIN, S.N., Lecons sur le s Proprietes Extremales et l a Meilleure Approximation des Fonctions Analytiques d'une Variable Reelle, P a r i s : G a u t h i e r - V i l l a r s , 1926. CHENEY, E.W., Introduction to Approximation Theory, New York: McGraw-H i l l Book Co., 1966. "FEINERMAN, R.P. and D.J. NEWMAN, Polynomial- Approximation, Baltimore: Williams and Wilkins Co., 1974. LORENTZ, G.G., Approximation of Functions, New York: Holt Rinehart and Winston, 1966. MEINARDUS, G., Approximation of Functions: Theory and Numerical Methods, B e r l i n , Heidelburg, New York: Springer-Verlag, 1967. (Translated from the 'German.) RICE, J.R., The Approximation of Functions, Reading Mass: Addison-Wesley, v o l . 1, 1964, v o l . 11, 1969. RIVLIN, T.J., An Introduction to the Approximation of Functions, Waltham, Mass., Toronto, London: B l a i s d e l l Publishing Co., 1969. SNYDER, M.A., Chebyshev Methods i n Numerical Approximation, Englewood C l i f f s , N.J." P r e n t i c e - H a l l Inc., 1966. TIMAN, A.F., Theory of Approximation of Functions of a Real Variable, New York: Macmillan, 1963. (Translated from the Russian.) WALSH, J.L., Interpolation and Approximation by Rational Functions i n the Complex Domain, v o l . XX, 2nd ed., Providence: Amer. Math. Soc. C o l l . Publ., 1956. PERIODICALS CHALMERS, B.L. and G.D. TAYLOR, "Uniform approximation with constraints' To Appear. - 88 -ERDOS, P., D.J. NEWMAN and A.R. REDDY, "Approximation by r a t i o n a l functions", J. London Math. Soc. (2), 15 (1977), 319-328. ERDOS, P. and A.R. REDDY, "Problems and r e s u l t s i n r a t i o n a l approximation", Periodica Math. Hung., 7 (1976), 27-35. ERDOS, P. and A.R. REDDY, "Rational approximation", Advances i n Math., 21 (1976), 78-109. GONCAR, A.A., "Inverse theorems of best approximation by r a t i o n a l functions", I z v e s t i a Akad. Nauk SSSR, 25 (1961), 347-356. GONCAR, A.A., "Estimates of the growth of r a t i o n a l functions and some of t h e i r a p p l i c a t i o n s " , Math. Sbornik, 72 (1967), 487-503; Math. USSR - Sbornik, 1 (1967), 445-456. GONCAR, A.A., "On the r a p i d i t y of r a t i o n a l approximation of continuous functions with c h a r a c t e r i s t i c s i n g u l a r i t i e s " , Math. Sbornik, 73 (1967); Math. USSR - Sbornik, 2 (1967), 561-568. GONCAR, A.A., "Properties of functions re l a t e d to t h e i r rate of approximability by r a t i o n a l functions", Amer. Math. Soc. Tr a n s l . , 91 (1970), 99-128. GONCAR, A.A., "Zolotarev problems connected with r a t i o n a l functions", Mat. Sbornik 78 (1969); Math. USSR - Sbornik, 7 (1969), 623-635. LORENTZ, G.G., "The degree of approximation by polynomials with p o s i t i v e c o e f f i c i e n t s " , Math. Ann., 151 (1963), 239-251. NEWMAN, D.J., "Rational approximation to |x|", Michigan Math. J . , 11 (1964), 11-14. NEWMAN, D.J., "Rational approximationtto x 1 1", J . Approximation Theory, To Appear. : NEWMAN, D.J. and A.R. REDDY, "Rational approximation to x 1 1", P a c i f i c J. Math., 67 (1976), 247-250. REDDY, A.R., "On ce r t a i n problems of Chebyshev, Zolotarev, Bernstein and Achieser", Inventiones Math., 45 (1978), 83-110. REDDY, A.R., "Recent advances i n Chebyshev r a t i o n a l approximation on f i n i t e and i n f i n i t e i n t e r v a l s " , J . Approximation Theory, 22 (1978), 59-84. REDDY, A.R. and 0. SHISHA, "A class of r a t i o n a l approximations on the po s i t i v e r e a l axis - a survey", J. Approximation Theory, 12 (1974), 425-430. 

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