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Power series properties invariant under various permutation semi-groups Wick, Darrell Arne 1972

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POWER SERIES PROPERTIES INVARIANT UNDER VARIOUS PERMUTATION SEMI-GROUPS by • DARRELL ARNE WICK B.S., San Diego' State College, 1963 M.S., San Diego State College, 1965 THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of MATHEMATICS We accept t h i s thesis as conforming to the required standard The University of B r i t i s h Columbia A p r i l , 1972 In p r e s e n t i n g t h i s t h e s i s i n 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 a t the U n i v e r s i t y o f B r i t i s h C olumbia, I agree t h a t 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 o r 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 t h a 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 l n o t 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 M a t h e m a t i c s The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date A p r i l 25, 1972 i i Chairman: Professor Z. A. Melzak ABSTRACT A global a n a l y t i c function i s uniquely determined by one of i t s function elements. This function element i s i n turn completely determined by the c o e f f i c i e n t s of the Taylor series expansion about some point. Therefore, we should be able to determine a l l of the properties of the function from those of the c o e f f i c i e n t s and by the formal properties of Taylor s e r i e s . Detection of these function properties from those of the c o e f f i c i e n t s i s the cent r a l problem of the theory of Taylor s e r i e s . Unfortun-ately t h i s has proved to be an extremely d i f f i c u l t problem and, with the exception of simple cases, very l i t t l e of the nature of a function i s known from the properties of the c o e f f i c i e n t s . In Chapter One of t h i s t h e s i s , the central problem of the theory of Taylor series i s approached. However, instead of a f i x e d sequence of c o e f f i c i e n t s , c e r t a i n rearrangements of the order of the c o e f f i c i e n t s are also considered. Hopefully t h i s relaxation w i l l allow additional information to be detected. In p a r t i c u l a r , the P-preservation set consists of those rearrangements of the order of the c o e f f i c i e n t s which preserve a property P. This P-preservation set i s i i i maximal i n the sense that each non-member 'maps' a function with property P onto a function without property P. The preservation sets (b^ten groups—always semi-groups) for several function properties are found. This concept of property preservation also permits examination of those properties which are invariant under . •various permutation groups. This y i e l d s a d i v i s i o n of the t o t a l i t y of a l l power series into subdivisions deter-mined by subgroups of S , the permutation group on the p o s i t i v e integers. In Chapter Two various summation processes, V, are used to sum the c o e f f i c i e n t s of a power s e r i e s . For these V-summation processes, permutations which map V-summable series onto V^summable series are discussed. The program of Chapter One i s continued, and the V-summability preser-vation sets are discussed. I t i s also shown that these V-summability preserving permutations leave the V-sum invar i a n t . F i n a l l y , the r e s u l t s are c o l l e c t e d i n a dual l a t t i c e c o n sisting of a n a l y t i c function properties on one side and the corresponding preservation groups and semi-groups on the other side. This l a t t i c e c e r t a i n l y represents only a beginning. Hopefully i t could lead to a d d i t i o n a l i n s i g h t into a n a l y t i c function theory or possibly even lead to information about the subgroups of S . 0) i v TABLE OF CONTENTS Page Number Introduction 1 Chapter One 5 Preservation Groups 5 Rational Functions 6 P o s i t i o n and Number of Poles 1 0 Algebraic Functions 1 2 Radius of Convergence 1 6 Entire F U H U L i o n a 2 3 Order of Entire Functions 2 8 Type of Entire Functions 3 2 Chapter Two 35 Summability 35 Conditionally Summable Series 36 Cesa.ro Means 4 1 Preservation Groups 5 5 C o n c l u s i o n = 5 6 Bibliography 6 0 V ACKNOWLEDGEMENT I w o u l d l i k e t o t h a n k my s u p e r v i s o r , P r o f e s s o r Z.A. M e l z a k , f o r s u g g e s t i n g t h e t o p i c o f t h i s t h e s i s . I am e s p e c i a l l y g r a t e f u l f o r t h e e n c o u r a g e m e n t , a d v i c e , a n d w o r d s o f w i s d o m he g a v e me d u r i n g t h e p r e p a r a t i o n o f t h i s t h e s i s . The f i n a n c i a l s u p p o r t o f t h e N a t i o n a l R e s e a r c h C o u n c i l o f C a n a d a a n d o f t h e 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 s g r a t e f u l l y a c k n o w l e d g e d . INTRODUCTION The development of the theory of an a l y t i c functions has followed two basic approaches: the global Cauchy-Riemann approach and Weierstrass' approach through power series and function elements. In t h i s thesis we begin with Weierstrass' approach. Consider the sequence of Taylor series c o e f f i c i e n t s of the power series f(z) = E f n z n expanded about the o r i g i n for s i m p l i c i t y . (We s h a l l r e f e r to the terms of th i s sequence as the c o e f f i c i e n t s of f or simply as the c o e f f i c i e n t s . ) The c o e f f i c i e n t s of f completely determine a corresponding global function. We should therefore be able to detect a l l properties of the function from those of the c o e f f i c i e n t s and by the formal properties of Taylor s e r i e s . Detection of these function properties from those of the c o e f f i c i e n t s i s the central problem of the theory of Taylor s e r i e s . I t i s surp r i s i n g how l i t t l e of the nature of the function i s known from the properties of the c o e f f i c i e n t s . For example, how can an ana l y t i c function given by a Taylor series be represented by a single formula? A r e s u l t by Hadamard gives the r e l a t i v e p o s i t i o n of polar s i n g u l a r i t i e s l y i n g within the c i r c l e of meromorphy. Beyond Hadamard's 2 r e s u l t , not much i s known, yet the position of singular points i s of paramount importance i n function theory. (An e n t i r e function i s characterized by the fac t that 0 0 i s i t s only singular point, meromorphic functions have only poles i n the f i n i t e complex plane, etc.) In Chapter One of t h i s t h e s i s , the central problem of the theory of Taylor series i s approached, but instead of a fixed sequence of c o e f f i c i e n t s , c e r t a i n rearrangements of the order of the c o e f f i c i e n t s are also considered. Hopefully t h i s relaxation w i l l allow additional information to be detected. J - J - „ „ _ . j. * w.. ^ J ^ ^ J - V.J- J. . .XW.. can the order of the c o e f f i c i e n t s of t h i s function be altered so that the new function s t i l l enjoys property P? That i s , which elements of S , the permutation group on the p o s i t i v e integers, map the function f onto a new function, f *, both enjoying property P? (The functions f and f* have the same set of c o e f f i c i e n t s but t h e i r c o e f f i c i e n t s appear i n a d i f f e r e n t order. The permutation from S i s act u a l l y a map on t h i s order. This abuse of language w i l l be r e c t i f i e d later.) A permutation, TT , preserves property P i f for a l l f enjoying P, T r(f) also enjoys P. The set of a l l permutations preserving P i s the P preservation set. Thus to each property P, there 3 corresponds a preservation set (often a group). The concept of property preservation allows examination of those properties which are invariant under various permutation groups. This y i e l d s a d i v i s i o n of the t o t a l i t y of a l l power series into subdivisions determined by sub-groups of S . Such d i v i s i o n s or c l a s s i f i c a t i o n s are c e r t a i n l y not unusual. Riemann c l a s s i f i e d a nalytic functions by t h e i r associated Riemann surfaces. Today we know that each Riemann surface of an algebraic function i s t o p o l o g i c a l l y a sphere with p handles, and that the algebraic function i s a single-valued function of the points on t h i s surface. The Erlanger Programme i s another example. K l e i n considered groups of transformations i n space. The invariant theories of these groups each y i e l d a d e f i n i t e kind of geometry, and every possible geometry can be obtained i n t h i s way. K l e i n obtained metric, a f f i n e , and projective geometry and also topology through t h i s program. In Chapter Two various summation processes, V, are used to sum the c o e f f i c i e n t s of a power s e r i e s . For these V-summation processes, permutations which map V-summable series onto V-summable series are considered. The program of Chapter One i s continued, and the V-summability preser-vation sets are discussed. I t i s also shown that these V-summability preserving permutations leave the V-sum invariant. There are already two re s u l t s i n t h i s f i e l d . F i r s t , the absolute sum i s invariant under a l l rearrangements. Second, Levi [22] considered and found the preservation set for conditional summability. A new, shorter proof of Levi's r e s u l t i s presented. Levi's theorem i s then generalized and proved for Cesaro K-summability. Many res u l t s i n related areas may be found i n the l i t e r a t u r e . In a sort of antipodal r e s u l t , Bagemihl and Erdos [ 4 ] considered the V-sum set (The set of sums of a l l possible convergent rearrangements of a series.) The sum set for absolute convergence i s a single point, for conditional convergence (not absolute) i t i s the continuum, and for C-L sums i t i s either of the form {a + bj} , j = 0, ±1, ±2, ••• or i t i s given by Riemann1s theorem. 5 PRESERVATION GROUPS Consider the function oo f ( z ) = I f Z n=Q N n ana l y t i c at zero, under rearrangements of the order of the c o e f f i c i e n t s . Denote by S^ the permutation group on the p o s i t i v e integers. I f TT i s a permutation belonging to S , then i s a new function obtained from f by a rearrangement of the c o e f f i c i e n t s of f. For convenience of notation, denote the function f * ( z ) by Tr[f(z)] or simply by T r ( f ) . The permutation TT preserves property P i f , f o r a l l f enjoying property P, TT (f) also enjoys property P. The set of a l l permutations TT & S^ which preserve property P i s c a l l e d the P-preservation set. Any preservation set i s necessarily closed under composition and i s therefore at l e a s t a semi-group. oo f*(z). = I f n=0 n •rr (n) 6 RATIONAL FUNCTIONS The following are two well-known equivalent descriptions of the set R of r a t i o n a l functions a n a l y t i c at zero [26]: a) R i s the set of quotients P(z)/Q(z) of polynomials with complex c o e f f i c i e n t s and with Q(0) ^  0. b) R i s the set of power series E f n z n that are regular at z = 0 and whose c o e f f i c i e n t s s a t i s f y a l i n e a r recurrence r e l a t i o n N J c f ,. = 0 (n>nn) . . L~ i n+i 0 j = u -We s h a l l use the. following r e s u l t of Szego [12]: Theorem 1.1: If among the c o e f f i c i e n t s f of a Taylor series f(z) = £ f n z n , there are only a f i n i t e number of d i f f e r e n t numbers, then e i t h e r f(z) = P ( z ) / ( l - z m ) , where P(z) i s a polynomial and m a pos i t i v e integer, or else f(z) can not be continued beyond the c i r c l e of convergence of Z f n z n . An arithmetic sequence i s a sequence •l\n^ )> with a common difference p = n^.+^ - n^. When the value of the common difference i s relevant, we use the term p-arithmetic  sequence. 7 Consider a permutation TT £. .whose domain can be written as a union of arithmetic sequences i n such a way that the image under TT of each of these arithmetic sequences i s again an arithmetic sequence. Refer to these sequences as the domain and range sequences, respectively. For such a permutation, the union of arithmetic sequences can be considered d i s j o i n t without any loss of generality. I f there were two d i s t i n c t , i n t e r s e c t i n g , domain arithmetic sequences, they must have unequal common differences, say p and q. The f i r s t sequence could be written as the union of q sequences with common difference pq, the second as the union of p (pq)-sequences. The two o r i g i n a l p, q-sequences can then be replaced with the (pq)-sequences. These are d i s j o i n t since no two d i s t i n c t (pq)-sequences i n t e r s e c t . If TT can be characterized by the mapping of a f i n i t e number of domain sequences, TT i s an arithmetic permutation. Since the domain of TT i s s p l i t into a f i n i t e number of d i s j o i n t arithmetic sequences, l e t p 1 be the least common multiple of these common differences. If p i s the minimum of a l l possible p' , we say IT i s a p-arithmetic permutation. Theorem 1.2: Arithmetic permutations preserve r a t i o n a l functions. <Pj?oof S Let f(z) = Ef z 1 1 be a r a t i o n a l function and IT n be a p-arithmetic permutation. Then, p-1 i ir(f) = ir I- I f o -Z— ] i=0 l - z p p-1 i = I [TT( f o - 2 — ) ] i=0 l - z p where f° [ z 1 / ( l - z p ) ] represents the Hadamard product of f and z 1 / ( l - z P ) . The Hadamard product preserves r a t i o n a l i t y , hence the products f° [ z 1 / ( l - z P ) ] are a l l r a t i o n a l and t h e i r c o e f f i c i e n t s appear i n a p-arithmetic sequence which TT maps onto another arithmetic sequence. We thus have i zk IT c f ° - — p ) = g ° 1 - 5 8 l - z * where the non-zero c o e f f i c i e n t s of f o [ z 1 / ( l - z p ) ] and k cr of g o [ z /(l-zH)] are i d e n t i c a l i n value and order. Since f°[z 1/(l-z p)] i s r a t i o n a l , there are c^ s a t i s f y i n g N ^ j p(n+j)+i 9 Hence N : i 0 Dyq(n+])+k k k and g o [z /(1-z )] i s r a t i o n a l . Theorem 1.3: If the permutation TT preserves r a t i o n a l i t y , then rr must be arithmetic. Proof: Assume TT i s not arithmetic. Then e i t h e r 1 or 2 holds: 1. Each integer i n the domain of TT belongs to some arithmetic sequence whose image i s again an arithmetic sequence, but rr cannot be characterized by the mapping of a f i n i t e number of domain sequences. Then TT cannot preserve r a t i o n a l i t y because E C j f n + j i s a f i n i t e property. 2. There i s an integer b i n the domain of TT such that the image of each arithmetic sequence <an+b)> , a > 1, f a i l s to be an arithmetic sequence. Then by Szego, TT (f) cannot be r a t i o n a l where 4 7 1 \ v an+b f (z) = 2 z 10 POSITION AND NUMBER OF POLES A permutation TT 6 i s c a l l e d f i n i t e i f TT (n) = n for a l l n larger than some N. I f TT i s a f i n i t e permutation then f and TT (f) d i f f e r only by a polynomial. I f TT i s a f i n i t e permutation, then the inverse permutation, TT ^ R i s also f i n i t e . The f i n i t e permutation set i s a group. Theorem 1 . 4 : If F i s the group of f i n i t e permutations and i f f(z) = Ef z 1 1 i s a Taylor s e r i e s , then Proof: Each f i n i t e permutation obviously preserves a l l singular points on the c i r c l e of convergence. Hence i f TT moves an i n f i n i t e number of c o e f f i c i e n t s , i t i s only necessary to exhibit a power series f(.z) = E f n z n so that f and Tr(f) do not have the same singular points. I f TT i s not an arithmetic permutation, then i t maps a r a t i o n a l function onto a function with a natural boundary Hence l e t TT be arithmetic with period p and l e t h n f and Tr(f) have the same singular points on the c i r c l e of convergence for a l l f, } = { TT £ S : f and TT (f) have the same singular points within the c i r c l e of meromorphy for a l l f. 11 b e t h e m a x i m u m d i s t a n c e a n y a r i t h m e t i c s e q u e n c e i s m o v e d ( h = s u p | TT ( n ) - n | ) . F u r t h e r , TT m u s t m a p p - a r i t h m e t i c n s e q u e n c e s o n t o q - a r i t h m e t i c s e q u e n c e s . C o n s i d e r f ( z ) = 1 / ( l - z ^ + ^ ) , t h e n q ; x n k k=Q [ 1 _ z ( p h + l ) q ] w h e r e n^ < h . S i n c e + z ( p h + i ) ( q - i ) h a s d e g r e e (ph+1) C3-1) w h i c h i s g r e a t e r t h a n t h e d e g r e e o f t h e denominator IT-Z t h e f u n c t i o n i r ( f ) h a s p o l e s n o t p o s s e s s e d b y f . 12 ALGEBRAIC FUNCTIONS D e f i n i t i o n : The function f(z) represents an algebraic  function i f f s a t i s f i e s an algebraic equation P(z,f) = 0 where P(z,f) i s a polynomial i n z and f over the complex number f i e l d . Lemma: Let f(z) = E f n z n be algebraic. Then for a l l p and a l l m = 0, 1, (p-1), the power series m pn+m _ C q Z y f z p n ™ = ^ TTn+ m L_ pn+m , p n=0 ^ l - z ^ represents an algebraic function. Proof: We define an eraser operator g ^ : g h [ f ( z ) ] = I g n z n = f(.z) - u hf(03z: y n=0 where ooP = 1. This operator has the following properties: 1. g =0 i f and only i f f =0 or n = -h(mod p); ^n 1 n ^ 2. For each n there i s a polynomial P n such that g = P (to) f and P (co) = P , (co) . ^n n n n ' n+pv 13 3. I f f(z) i s algebraic, then g, [f(z)] i s also np algebraic. These properties are e a s i l y v e r i f i e d by expanding g h p [ f ( z ) ] : g [f(z)] - f(z) - a) f(wz) = I (1 - w n + h ) f z n . n p n=0 n Now to prove the lemma, consider the operators gj where h = 0, 1, (p-1) and h ^ m. Apply the f i r s t operator to f ( z ) , the second operator to the function just obtained, and so on. Each ap p l i c a t i o n y i e l d s an algebraic function while replacing p-arithmetic sequences of c o e f f i c i e n t s with zeros. After (p-1) a p p l i -cations we obtain n-1 CO { n g. • } {f (z)} = • I P . (u)f , z P n + m h=0 h P 1 n=o P n + m P n + m h^m P(u) I f , z P n + m which i s algebraic, Denote by the set of permutations which map 14 algebraic functions onto algebraic functions. We then have the following theorem. Theorem 1.5: = Algebraic preservation set = Arithmetic permutation group. Proof: Pick any arithmetic permutation TT with period p. Then TT maps p-arithmetic progressions onto q-arithmetic progressions, f o r some q. Let f(z) = T.f^zn be any algebraic function. Then, P7 1 T r t f t z ) ] = TT{ I I f z k=0 n=0 p n + K pn+k p-1 <» ,, k=0 n=0 ^ k=0 n=0 n q K where the integer a^ i s dependent on k and represents the ' s h i f t distance'. This i s a f i n i t e sum of functions, each algebraic by the previous lemma. If TT i s not arithmetic, then Szego (Theorem 1.1) provides an algebraic (rational) function which, i s mapped by TT onto a function with a natural boundary. 15 The degree of ir(f) has an upper bound which i s a function of the o r i g i n a l degree of f and of the permu-tatio n TT. If f s a t i s f i e s an i r r e d u c i b i l e polynomial i n z and f, P(z,f) = P 0 ( z ) f n + P 1 ( z ) f n " 1 + ••• + P n ( z ) where each ^ ( z ) i s a polynomial i n z over the complex numbers, then f i s algebraic of degree n. We then have the following: 1. If a) i s complex, then f ( u z ) i s algebraic of degree n. O T-F Vt n c a nnoi f i t T O i nf orrar f h o n /,\ •£ ( rr \ - i c algebraic of degree n. 3. The product z^fCz) i s algebraic and of degree at most n. 4. If g i s algebraic and of degree m, then f + g i s algebraic and of degree at most nm [18, pl02]. 5. I f g, i s the eraser operator, then the degree P 2 p _ 1 of g, (f) i s at most n and [ I I g, ] (f) has degree p h=0 P , ' h^ m 2 p - l r at most n 2 p - l 2 p - l 4p--l 6. The degree of TT Cf) i s at most (n ) = n 16 RADIUS of CONVERGENCE Lemma; Let A = {TT : IT Q. and TT (n) ^ n}. Then A i s a non-abelian group. We s h a l l r e f e r to A as the asymptotic group. Proof: It i s only necessary to show closure and the existence of an inverse. If TT. £ A, then there exists an e. > 0 such that n ( l - e±) < 7r i(n) < n ( l + e±) , i = 1, 2 and hence for the composition T r ^ [ T r 2 ] we have n ( l - e') < n ( l - e") < n ( l - e 2) (1 - e±) < < TT 2 (n) (1 - e x) < •n1 [ T T 2 (n) ] and T r 1 [ T f 2 ( n ) ] < T r 2 ( n ) ( l + e ) < n ( l + e 2 ) (1 + e±) = n ( l + e' ) . 17 To show the inverse i s i n A pick e so that 1 - e < 1/(1 + and 1 + e > 1/(1 - . Then f o r the inverse of TT^, we have m(l - e) < m/(l + < T T " 1 (m) < m/(1 - e 1) < m(l + e) and T T ^ 1 6 A whenever TT^ <£ A. Theorem 1.6: The Asymptotic group, A fTT £ S : TT preserves radius} L CO c of convergence. Proof• Let f (z) = If z n where l i m I f | n = R 1 . F i r s t we n 1 n 1 show that i f TT i s asymptotic, then TT preserves radius of convergence. We break the proof into two cases depending on whether a l l the c o e f f i c i e n t s are eventually less than one or not. Case I. | f | <_ 1 for a l l s u f f i c i e n t l y large n. This case follows from one s t r i n g of i n e q u a l i t i e s . The v e r i f i c a t i o n s for the two not so obvious i n e q u a l i t i e s 18 are given i n a and b below. -.1 _ e ) l / ( l - e ) f ,1/nd-e) m | ,1/Tr(n) —(a) 1 n 1 — 1 n 1 and ra|fn|1A(n) < H H | f n | 1 / n { 1 + e ) < ( b ) (R _ 1 + e j V d + e ) Now to v e r i f y a and b: l i m | f n | 1 / / n = R _ 1 i f and only i f a) given e >0 and N, there exists <% n > N such that f j 1 / n > (R _ 1 - e) or equivalently f n , l / n ( l - £ ) > ( R-1 _ £ ) l / ( l - e ) and b) given e > 0, there exists N such that n > N implies f j 1 / n < CR"1 + e) or equivalently j f |l/n(l+e) < ( R-1 + e ) l / U + e ) 19 Hence lim I f l-*-/1^11) = R 1 for case I. 1 n 1 Case I I : For each p o s i t i v e integer N, there i s an n > N such that ]f | > 1. This proof follows the same outline as case I. -1 _ ,1/U+e) ra,f ,1/nd+e) < ^ ,1/TrCn) v ' —(a) 1 n 1 — 1 n 1 and T i m | f | 1 / l T ( n ) < I T 5 | f | 1 / N ( 1 " E ) < , . . ( R " 1 + e ) 1 / ( 1 - £ ) . 1 n 1 — 1 n 1 —(b) Now lim | f | 1 / / N = R 1 i f and only i f a) given e > 0 and N, there exists .; n > N such that I f J 1 / " > (R- 1 - e) or equivalently | f J 1 / n ( 1 + £ ) > ( R - 1 - s ) 1 / ( 1 + £ ) and b) given e > 0, there e x i s t s N such that n > N implies I f | 1 / N < ( R _ 1 +e) n • or equivalently 20 |l/n(l-e) K { R - 1 + e ) l / U - e > Hence T~-—i sz i l/fr (n) „-l lim f \ ' = R n and therefore asymptotic permutations preserve the radius of convergence. F i n a l l y , we show that the set of radius of convergence preserving permutations i s a subset of the asymptotic group. Suppose IT preserves radius of convergence. We want to show lim T r(n)/n = 1. n Suppose not. Then lim Tr(n)/n = a > 1, Consider oo n z n=0 2 where -1 R = lim _-n11/n _ 1 1 1 2 Now, from the d e f i n i t i o n of lim Tr(n)/n = a, we have a) given e > 0, there e x i s t s an N such that n > N implies < (a + e) or u(n) < n(a +e) 21 and b) given e > 0 and N, there exists * n > N such that IT (n) n > (a - e) or Tr(n) > n(a - e) Using these two expressions to evaluate lira] 2 n|-'-/'IT(n) we obtain n £ | 2 - n | l / i r ( n ) ± ^ n 5 | 2~n j 1/n (a+s) = m 2-l/(a+e) =  = 2-l/(a+e) and i 3 H | 2 - n | 1 / i r ( n ) > ( b ) T i ¥ | 2 - n | 1 / n ( a - e ) = 2 _ 1 / ( a - e ) . Thus H E | 2 - n | l A ( n ) = 2 ~ l / a which has the same radius of convergence only when a = 1. Remark: The above counterexample i l l u s t r a t e s a power series E f n z n with radius of convergence 2, mapped onto a new power series with radius of convergence 2 1 / / a. I t i s i n t e r e s t i n g to note that the radius:of convergence cannot be moved across |z| =1. Suppose there was a permutation TT and a function f where R(f) > 1 while R[Tr(f)] < 1. Then 22 f ( l ) = Ef i s a b s o l u t e l y c o n v e r g e n t and hence 7 r [ f ( l ) ] = f ( l ) i s a b s o l u t e l y c o n v e r g e n t w h i c h i s a b s u r d . 2 3 ENTIRE FUNCTIONS We have been formulating a dual l a t t i c e composed of function properties and preservation groups. The present entries enable pred i c t i o n of information about possible new e n t r i e s . As an example, consider the entire functions preservation set. For an entire function, Cauchy's n-th root t e s t equals zero: The function f(z) = Ef^z 1 1 i s enti r e i f R _ 1 = Tim" I f ! 1 / n = 0. 1 n 1 Obviously the set of permutations preserving entire functions w i l l contain the set preserving radius of conver-gence . With any function, Cauchy's n-th root t e s t associates a unique R 1 which i s non-negative. Now given any entire function f, permuting i t s c o e f f i c i e n t s can only make the R 1 associated with f larger. From t h i s i t would appear the only r e s t r i c t i o n on a member of S g, the e n t i r e function preservation set, i s that there i s some upper bound on the forward t r a v e l of the permutations. Following t h i s reasoning, TT could have unbounded t r a v e l i n the reverse d i r e c t i o n . But these are not symmetrical t r a v e l conditions. This tends to indicate that the set S w i l l not contain a l l inverses and w i l l e only be a semi-group. We s h a l l follow t h i s o u t l i n e , proving that the members of S g are characterized by an upper bound [TT (n) = 0 (n) .. or simply TT = 0(n)], and that S g does not contain inverses and i s consequently not a group. Theorem 1.7: I f S g i s the set of permutations preserving en t i r e functions, then s e = I71 ^ S W : TT (nl = 0 (n)_ 1 . Proof: F i r s t , {TT <£" S : TT = 0 (n) } C S . ————— Q Let f be any entire function, f(z) = Z f n z n , where R"1 = U S |f | 1 / n = lim|f | 1 / n = 0 i n i n and l e t TT = 0 (n) lequivalently l e t TT Cn) < nk ^ J ;0' Now, given e n > 0, l e t e CTT) = £„. Since l i m | f n | 1 / / n = 0, there exists an such that n > implies I f I 1 / n k £^ = e or equivalently | f 1 1 / n k ~ < e for n > N 1 n 1 • e 25 Rewording: Given e > 0, there exists an N £ such that I f j 1 / / n k < e whenever n > N . Hence lim I f 1 1 / / n k = o. 1 n 1 e 1 n 1 F i n a l l y , using t h i s l i m i t i n addition to |f | 1 / 1 T W < |f | x / n K for |f | < 1 1 n 1 1 n 1 1 n 1^ ^ /TT (n) „ i^. ,1/ k y i e l d s T . i £ i 1/TT (n) . lim f 7 v ' = 0. ' n 1Second, S C J ^ G S : IT = 0 (n) } . e — 1 co * Suppose not. Then there must be some u e SG where TT f- 0(n). This means, for each N and k, there e x i s t s an n > N such that TT (n) > nk. Let N = k. Then there e x i s t s n, > k such that k TT(nk) > n kk > n kg[n k) where g(n^) i s a p o s i t i v e increasing function of n k, n . , -A-n'g(n) n and A i s a constant. Here Now consider f(.z) = Ef z where f = e n n R"1 = l i m | f n | 1 / n = lim|e A*9 ( n>| = 0 (Hence entire) Next, for TT (f) = Ef z T t ( n ) ' n R 1 = lim n l/iT(n) - s - i— -A-n*g(n) 1/TT (n) ^ ' = lim e ^ ' > > lim -A*n«g(n) l/n«g(n) -A This contradicts TT <£ S . e Theorem 1.8: The set = {TT £ S^: TT (n) = 0 (n) } forms a semi-group with i d e n t i t y . Proof: Closure follows immediately from the d e f i n i t i o n of S g, hence i t i s a semi-group. The set S e does not form a group. The following i s an example of a permutation which does not preserve e n t i r e t y , yet i t s inverse does. Let and define TT as follows: For a, b ^ 10 x and a < b, require that TT (a) < Tr(b). For n = 10 , l e t TT (n) = m where m i s picked to s a t i s f y 10 X! 1/m 1 - x This i s equivalent to picking m larger than 27 ra > logiW l o g ( l - i ) > 0. F i r s t , observe that TT (f) i s not entire because R 1 = l i m I f I 1 n 1 l / i r ( n ) lim 1 1/m = 1. 10 ! -1 However, TT (EL S : e Tr - 1(m) < m for those m which are the images of powers of 10 and TT -^Cn) < n + p(n) where p(n) i s the highest power of 10 which i s less than n. 28 ORDER OF ENTIRE FUNCTIONS D e f i n i t i o n : The entir e function f (z) i s of order p i f — log [log M(r)] lim — — -—— log r p , 0 < p < where M(r) denotes the maximum modulus of f(z) for |z| = r. A constant has order 0, by convention. Asymptotic Theorem 1.9:. {Permutation} Group ITT <~ S 11 P r e s e r v e s t n e order i " t o * of enti r e functions ' The rollowing lemma gives the order of an entire function as a function of the sequence of c o e f f i c i e n t s [7]. Lemma: The entire function f(z) = Ef z 1 1, f + 0, i s of n ' n ' f i n i t e order i f and only i f -r-.— n• log (n) y = lim ~ — -log 1 f n i s f i n i t e ; and then the order p of f(z) i s equal to y. For the proof of Theorem 1.9, we show in c l u s i o n i n both d i r e c t i o n s . F i r s t , members of the asymptotic permutation group preserve the order of entir e functions. 2 9 Let f(z) = Ef^z 1 1 be an entire function of f i n i t e order p. Denote the order of Tr(f) by p , where TT i s i n the asymptotic permutation group. Then p = lim og TT (n) K j-^ n (1+e) ° l o g [n (1+e) ] TT (n) • ] L  log 1 f n log n since TT i s asymptotic to n. Now, given e > 0 and for a l l s u f f i c i e n t l y large n, 1 , log(1+e) , 1+e (1+e)-log(n) Equivalently, log[n(l+e)] < (1+e) • log.(n) and hence P i r 1 TiTa nd+e)log[n(l+e)] £ U + e ) 2 . m n-log(n) < p ( 1 + £ log n log n Since Tr(n) > n(l-e) for large n, there i s a s i m i l a r lower bound: TT = YTrn7 1 ( n ) 1 ? g ( n ) ] > T i m n (1-e) log [n (1-e) ] log log n "n 30 Now, f o r e > 0, 1 , l o g (1-e) ^ , _ T T T + — ?— ;—- > 1 f o r a l l l a r g e n. 1 - e (1-e) • l o g (n) E q u i v a l e n t l y , and t h u s l o g [ n ( l - e ) ] > ( l - e ) - l o g ( n ) ^ T - i — n (1-e) l o g [n (1-e ) ] ^ ,, ,2 •=-:— n * l o g ( n ) .,, > >_ l i m —- — I £ i- • >_ (1-e) - l i m 1 £ j = p.(l-e) l o g n l o g n T h i s y i e l d s p = p. TT F i n a l l y , i f e i t h e r f o r TT (f) has unbounded o r d e r , t h e n t h e y b o t h have unbounded o r d e r . Second, we show t h a t a l l o r d e r p r e s e r v i n g p e r m u t a t i o n s a r e i n t h e a s y m p t o t i c group. C o n s i d e r f Cz) n y £ L _ n - l o g ( n ) The f u n c t i o n f ( z ) i s an e n t i r e f u n c t i o n : R •1 -r-.— i -n«log(n) . 1/n —. - l o g ( n ) i _ = l i m j e ^ | ' = l i m j e ^ | = 0. The o r d e r o f f ( z ) i s f i n i t e : 31 P = ITH — n , 1 ° g ( n ) , . = i , l o g | e n , l o g ( n ) I Also If TT i s asymptotic, then lim [IT (n)/n] = a > 1, and p = H i lJn)Iog [ T r(n)] < m n(a+e)log[n(a+e)3 TT n*log(n) — n«log(n) = ITS {<a+e) + IS±£il£2l5±£l}= ( a + e ). = X l m ( n ) l Q 9 tit (n) ] > h (a-e) l o g [n (a-e) ] n - l o g ( n ) — n « l o g ( n ) 1 l o g ( n ) Thus, i f TT i s not contained i n the asymptotic permu-tat i o n group, then TT does not preserve order. 32 TYPE OF ENTIRE FUNCTIONS D e f i n i t i o n : The entire function f(z) of p o s i t i v e order p i s of type x i f H m { r _ p - l o g [M(r) ] } = T , (0 < T £ »)• r;->-oo where M(r) denotes the maximum modulus of f(z) for |z| = r. . , . Asymptotic Theorem 1.10: {TT € S : "Preserves, the type j = { p e J m £ t a t i o n } 1 co of entire functions J LJ^ J group The following theorem from Boas [7] provides a characterization of the type of an entire function i n terms of the c o e f f i c i e n t s . Theorem 1.11: I f v = lim(n[f | p/ n) , 0 < v < °°, then the n-M» function f(z) = Xf z i s of order p and type T i f and only i f v = exp. If v = 0 or °°, f(z) i s , respectively, of growth (p,0) or of growth not less than (p,°°); and conversely. Proof of Theorem 1.10: I f TT i s a member of the asymptotic group, then for e > 0, n ( l - e) < Tr(n) < n ( l + e) for s u f f i c i e n t l y large n, We may assume ] f | < 1 w i t h o u t any l o s s o f g e n e r -a l i t y . H e n c e , n ( l - e ) | f I P / 1 1 * 1 - ^ < T R ( N ) | F | P / n U - e ) K FF(N),F | P/TT ( n ) n 1 n 1 1 n 1 and i r ( n ) | f n | p A ( n ) < T r ( n ) | f n | p / n ( 1 + £ ) < n (1+e) | f R | p / n ( 1 + e ) Thus T • , \ i j r i P/TT (n) I _ | p/n l i m TT (n) f \ w / = n f H / n 1 n 1 n->°° and t h e t y p e o f t h e e n t i r e f u n c t i o n f (z) i s p r e s e r v e d by TT. I f TT i s n o t a member o f t h e a s y m p t o t i c g r o u p , t h e n TT d o e s n ' t p r e s e r v e t h e o r d e r o f e n t i r e f u n c t i o n s . Remark: I t w o u l d be i n t e r e s t i n g i f t h e r e was some b a s i c g e n e r a t i n g s e t f o r t h e p r e s e r v a t i o n g r o u p s m e n t i o n e d t h u s f a r . The a r i t h m e t i c p e r m u t a t i o n g r o u p i s g e n e r a t e d b y t h e s e t o f a l l s h i f t s , w h e r e a p - s h i f t i s a p - a r i t h m e t i c p e r m u t a t i o n w h i c h i n t e r c h a n g e s two a d j a c e n t p - a r i t h m e t i c s e q u e n c e s . A l t h o u g h t h e s e p - s h i f t s w e r e n o t s p e c i f i c a l l y m e n t i o n e d i n t h e p r o o f s i n v o l v i n g r a t i o n a l a n d a l g e b r a i c f u n c t i o n s , t h e y n e v e r t h e l e s s w e r e o f f u n d a m e n t a l i m p o r t a n c e . 34 The t r a v e l permitted by the asymptotic group and the bounded forward t r a v e l semi-group i s so random and the two sets are so large that i s doesn't appear there w i l l be a s i g n i f i c a n t l y smaller subset which would generate them. 35 C H A P T E R TWO S U M M A B I L I T Y L e t E f d e n o t e a s e r i e s o f c o m p l e x v a l u e d t e r m s . We s h a l l r e f e r t o s e v e r a l s u m m a t i o n p r o c e s s e s ; t h e s e w i l l b e d i s t i n g u i s h e d b y a n a t t a c h e d i n i t i a l . We s a y t h e s e r i e s E f i s V - s u m m a b l e t o S o r s y m b o l i c a l l y oo v-Tf = s. n=0 A p e r m u t a t i o n of t h e s e r i e s E f a g a i n m e a n s a s e r i e s , E f , w w h e r e TT i s a m e m b e r o f S , t h e s e t o f TT ( n ) to p e r m u t a t i o n s o f t h e p o s i t i v e i n t e g e r s . A p e r m u t a t i o n TT p r e s e r v e s V - s u m m a b i l i t v i f E f , . = - T r ( n ; i s V - s u m m a b l e w h e n e v e r E f i s V - s u m m a b l e . W i t h e a c h n t y p e o f s u m m a b i l i t y , V , we a s s o c i a t e a p r e s e r v a t i o n s e t c o n s i s t i n g o f t h o s e p e r m u t a t i o n s p r e s e r v i n g V - s u m m a b i l i t y . I n t h i s c h a p t e r , a b s o l u t e convergence;, c o n d i t i o n a l convergence, and Cesa.ro summability are considered. T h e p r e s e r v a t i o n g r o u p i s f o u n d f o r e a c h t y p e o f s u m m a -b i l i t y a n d , f u r t h e r , i t i s s h o w n t h a t t h e p r e s e r v a t i o n g r o u p a l s o l e a v e s t h e s u m i n v a r i a n t . oo T h e . p r e s e r v a t i o n g r o u p f o r t h e a b s o l u t e s u m , £ | f | , n=l o f a s e r i e s E f i s S . n to 36 C O N D I T I O N A L L Y S U M M A B L E S E R I E S T h e s e r i e s If i s c o n d i t i o n a l l y c o r w e r a e r v t i f N n ~ — — l i m Y f e x i s t s a n d i s f i n i t e . u n N+c* n = i L e v i [22] o r i g i n a l l y s t a t e d a n d p r o v e d t h e r e s u l t w h i c h g a v e t h e c o n d i t i o n a l c o n v e r g e n c e p r e s e r v a t i o n g r o u p . We b o r r o w s o m e o f L e v i ' s t e r m i n o l o g y i n o r d e r t o s t a t e h i s r e s u l t . A n e w p r o o f , c a p a b l e o f g e n e r a l i z a t i o n t o h i g h e r o r d e r C e s a r o s u m m a t i o n , i s p r e s e n t e d . L e t IT b e a p e r m u t a t i o n o f t h e p o s i t i v e i n t e g e r s w h i c h t r a n s f o r m s t h e i n f i n i t e s e r i e . s If. i n t o t h e r e a r r a n g e d s e r i e s If , . . TT ( n ) F o r a g i v e n p o s i t i v e i n t e g e r n , c a l l f a j u m p i n g - o u t t e r m i f m <_ n a n d TT (m) > n , a n d f a j u m p i n g - i n t e r m i f m > n a n d TT (m) <_ n . T h i s t e r m i n o l o g y a r i s e s n a t u r a l l y w h e n o n e c o n s i d e r s t h e n - t h p a r t i a l s u m o f t h e o r i g i n a l a n d o f t h e r e a r r a n g e d s e r i e s . A s e t o f c o n s e c u t i v e t e r m s o f t h e s e r i e s i s c a l l e d a b l o c k o f j u m p i n g - o u t ( j u m p i n g - i n ) t e r m s i f ( a ) t h e s e t h a s o n l y j u m p i n g - o u t ( j u m p i n g - i n ) t e r m s , ( b ) t h e t e r m i n t h e s e r i e s j u s t p r e c e e d i n g a n d j u s t f o l l o w i n g t h e s e t i s n o t a j u m p i n g - o u t ( j u m p i n g - i n ) t e r m . 37 T h e t o t a l n u m b e r o f t h e s e b l o c k s , f o r a n y g i v e n n , i s d e n o t e d b y k ( n ) . N o t i c e t h a t e a c h TT d e t e r m i n e s a f u n c t i o n k ( n ) , t h e v a l u e o f w h i c h d e p e n d s s o l e l y o n t h e p e r m u t a t i o n TT a n d i s i n d e p e n d e n t o f t h e t e r m s i n t h e s e r i e s . T h e o r e m 2 . 1 ( L e v i ) : A p e r m u t a t i o n TT p r e s e r v e s c o n d i t i o n a l convergence i f a n d o n l y i f t h e a s s o c i a t e d f u n c t i o n , k ( n ) , i s u n i f o r m l y b o u n d e d f o r a l l n . S u c h a p e r m u t a t i o n a l s o l e a v e s t h e s u m o f t h e s e r i e s u n a l t e r e d . We s h a l l u s e t h e f o l l o w i n g t h e o r e m b y K o j i m a a n d S c h u r [ 1 2 , p 3 8 5 ] . T h e o r e m 2 . 2 : T h e n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n s t h a t oo z ' = Y a , z , s h o u l d b e c o n v e r g e n t w h e n e v e r z , i s c o n -n Tl / Kl K l K l v e r g e n t a r e oo ( a ) I | a vI < M i n d e p e n d e n t l y o f n ; k = l n ' ~ ( b ) l i m a , = a, n , k k i . e . t h e l i m i t o f a , e x i s t s w h e n k i s f i x e d a n d n -> °°; n , k ( c ) I a = A k = l ' t e n d s t o a l i m i t a w h e n n -»- °°. M o r e o v e r , i f z^. -> z w e h a v e ( d ) z ' = l i m z ' = a z + ) a, ( z , - z ) n i -j k k n->-oo k = l 38 If a matrix (a , ) s a t i s f i e s conditions a, b, and n i K c i n Theorem 2.2, c a l l i t regular. k Proof of Theorem 2.1: Let = £f denote the n-th p a r t i 0 0 1 sum of the series T f and assume that lim z = z exists k n n 1 n->-°° and i s f i n i t e . C a l l S the C summation matrix where S i s an i n f i n i t e —o dimensional matrix with unit entries on and below the main diagonal and zeros above the main diagonal: 1 0 0 0 / 1 1 0 0 I 1 I 0 1 1 1 1 If (f ) denotes an i n f i n i t e column matrix i n which n the n-th row entry i s f , then the matrix 1 n S(f ) = (z ) n n i s also an i n f i n i t e column matrix whose entries are the n-th p a r t i a l sums z n We can now write the following: 1 0 0 • - 1 1 0 0-1 1 0 0-1 \ . . / s - 1 ( z k ) - ( f k ) P S _ 1 ( z k ) = P ( f k ) S P S _ 1 ( z k ) - SP(f k) . Let (SPS ) , denote the entry i n the n-th row, k-th column of SPS - 1. Referring to Theorem 2.2, l e t a^ ^ = (SPS v so that (an,k> = S P S -1 We s h a l l show that (a , ) i s regular. n f K. oo oo. (a) I |a J = .1 |(SPS - 1) | < M uniformly, k=l n , K k=l n , K ~ The matrix SP Is obtained from the matrix S by a rearrangement of the columns of the matrix S. There i s a 1 i n row n of SPS 1 corresponding to each 1,0 p a i r i n the n-th row of SP. There i s a -1 i n row n of SPS 1 corresponding to each 40 0, 1 p a i r i n the n-th row of SP. Thus each block of consecutive l ' s i n the n-th row of SP i s denoted by entries 1 and -1 i n SPS 1 . (If the f i r s t block i n any row begins i n the f i r s t column, associate only a 1 with that block.) These blocks of l ' s correspond to the jumping-out (jumping-in) terms. Now, Ela , I i s uniformly bounded i f and only i f the number of blocks i n the n-th row, k(n), i s uniformly bounded. (b) The terms i n the k-th column of SPS 1 are a l l zero for n > N, hence lim a , = a, = 0. ' n, k k N->OD (c) The entries i n the f i r s t column of SP are a l l 1 a f t e r some point. From (a) above we have CO I a. n ,k = A n = 1 for n > N k=l = 1. (d) From (b) and (c) , each a, = 0 and a = 1. = lim z' = z n n-5-°° and the sum of the series i s unaltered. 41 CESARO MEANS The re s u l t s of the preceeding section can be extended to Cesa.ro's summation by weighted means. We use a d e f i n i t i o n of Cesa.ro summability suggested by Hardy [16] . Write A° = f n + f, + ••• + f n 0 1 n and k -k 1 _.k 1 k —1 A = A A + A, + ••• + A n 0 1 n D e f i n i t i o n : If k. i s the le a s t p o s i t i v e integer so that lim k!n k A k = A n n->-°<> e x i s t s and i s f i n i t e , then we say t h a t J- f i s summable n=0 n (C ,k) to sum A and write •I f = A (C,k). n=0 If S i s the summation matrix, then S 2 = 1 0 0 ' 2 1 0 » 3 2 1 4 3 2 1 0 0 • * • 13 1 0 \ 6 3 1 10 6 3 42 and i n general, the term at the i n t e r s e c t i o n of the n-th k+1 row and j - t h column of S i s /• k + l - j _ (• ; -f K \ [ b j n , j " I ( n - j ) i (n-j) + ^where (ll? ^-T^} i s to be interpreted as zero i f (n-j) (n-j) i s negative. Further, /--2 1 0 0 0 - 2 1 0 0 1-2 1 0 0 1-2 1 0 0 1-2 .-3 1 0 0 0 3 1 0 0 3-3 1 0 -1 3-3 1 0-1 3-3 \ : \ / / and i n general, the term at the i n t e r s e c t i o n of the n-th row and the j - t h column of S ^ k +^^ is -(k+1) } • = C - l ) n + j ( k + 1 > where (^-j) t o b e i n t e r P r e t e d as zero i f (n-j) i s negative or i f (n-j) > k+1. Let D be the diagonal matrix D / V 2 O 1/3 1/4 \ o / 43 We can now write the column matrix which has the Cesaro (C,k) n-th p a r t i a l sum, k!n A , i n the n-th row: k! D S k „ k + l < fi» • The series Ef^ i s Cesaro (C,k) summable i f the k k+1 sequence of terms i n the column matrix klD S (f^) converge. Assuming t h i s sequence does converge, which permutation matrices, P, insure that k! DkSk+"'"P (f ^ ) also converges? Theorem 2.3: A permutation TT preserves Cesaro k-summability i f and only i f Such a permutation also leaves the Cesaro k-sum of the series unaltered. Remark: (An i n t u i t i v e approach to the C^-preseryation groups) If a series converges C Q , then the difference between p a r t i a l sums, i = l j=l I ik l I C-l) i + TT (.j) fn-j+ky ( n-j ' k+1 TT C j ) - i >| = . 0 (nk). . N' N n 44 must approach zero as N -> °°. The blocks of jumping-out (jumping-in) terms have th i s form and, therefore, also approach zero. Since the p a r t i a l sums of the rearranged series d i f f e r from the p a r t i a l sums of the o r i g i n a l series by a bounded number of blocks, the two p a r t i a l sums must have the same l i m i t . Bounded block t r a v e l may also be described by re-quiring the t o t a l number of unordered pairs of c o e f f i c i e n t s within the p a r t i a l sums to be bounded. (Note that each block begins and terminates with an unordered pair.) Since terms of the series Ef. are given a k-th order binomial weighting, ( ^ ^ j ^ ) , when they appear i n the n-th p a r t i a l sum, no simple characterization appears possible. However, a few comments might convey some insight into these permutations. For C^~summability, the f i r s t condition for r e g u l a r i t y 2 —2 -1 of DS PS D requires the absolute row sum to be uniformly bounded: 1 | a n k | < -M uniformly. Now, • I|a , | = i I i|b . | , L 1 n,k' n L 1 n,k1 ' 4 5 where the b , are zero for each ordered t r i p l e n , K ( t r i p l e i n o r i g i n a l order) and nonzero for each unordered t r i p l e ( s l i g h t o v e r s i m p l i f i c a t i o n ) . Thus, we require the expected value (or position) of the unordered t r i p l e s to be uniformly bounded: For C^-summability, we roughly require the expected value of the unordered (k+2)-tuples to be uniformly bounded. Proof of Theorem 2.3: We seek conditions on P such that (f.) i s C. -summable i f and only i f P('f.) i s C.-summable. 1 k * l k Equivalently, we seek conditions on P such that the sequence of terms forming the column of k!DkS*C+"'" (f ) converges i f and only i f the sequence of terms forming the column of E(X) = - I i|b v | < M uniformly. n L 1 n,k k!D kS k + 1PCf.) = Ik!D kS k + 1pS Jk!D S k 0k+l = ICa,.)Jk!D"S k 0k+l converges. 46 Lemma: The n-th row sum of S PS K 'D i s asymptotic k to n . Proof: We proceed by induction over the Cesaro number k . Assume S k P S - k D ~ ( k - 1 ) = (b. . ) D " ( k ~ 1 ) has n-th row i f 3 sum asymptotic to n k _ 1 ; then S k + 1 P S ~ ( k + 1 ) D _ k has n-th k_ row sum asymptotic to n . The sum of the n-th row of S kPS _ kD~^ k _ 1^ may be rewritten as v .k-1, k-1 ) x b . ~ n v n, 1 x We wish to show . k n y x i y b . - y b. ._,,] ~ n ' y i i y b. . - y b. . _ j i i i j=l 3,x. j 4 i D ^ + l H co co =1 i = l 3 , : L i = l J ' ^ - 1 3 n a> j=l i = l 3 1 i=2 n I {*>, ! + I b. i I i k - C k - l ) k ] } =1 D , x i=2 1 1' 47 I [b + (-k) I b i^1 + I b .{o( 1=1 -]'J' i=2 J r i=2 3 ' ? [b - k j k _ 1 + °(j k- 2)] 3 n n , , n . „ I b - k [ j * ' 1 + I . ( j k - 2 ) =1 ^ j=l j=l . k k-1 , k. , n ,n . o(n ) - k - £ + o ( k _ 1 ) k n Now, returning to the proof of Theorem 2.3, the matrix (a^ ) i s regular: (a) I |a .| < M independently of n. Consider the n-th row of Ca .) = k!D kS k + 1PS ( k + 1 ) D " k n, i I |a • | = K I ^ II ( S k + 1 ) . ( P S - C k + 1 ) ) 4- 1 n, l 1 k 4- 1 4- v n ,T K I i k I I ( S k + 1 ) . ( s " ( k + 1 ) ) , . , . k 4- l4' v n,j ' TT (i) , l n l j ' J 48 - i y - i k iv f _ u i + T T C : ) tn~i+k\ < k + 1 , ~ k I 1 1 4 1 X) { n-j ; - k fO(n k)] n < - k [Mnk] n = M . (b) lim a. . = a. = 0 since a. . = 0 for i > I. • i , i i i , i CO £ a ± > . = A. ~ 1. 3= 1 F i n a l l y , the sum i s unaltered: (d) Since a = 1 and a k = 0, z' = lim z = az + T a, Czn - z) = z n . L-, k k n->-°° k=l Theorem 2.4: I f c^, k = 0, 1, 2, ••• denotes the Cesa.ro k-summability preservation semi-group, then C k ( ^ C k + i Ci = 1, 2, 3, • • • ) . Proof: (k > 1) . We exhibit a permutation i T k & C^ - C k+i' Let n+1 n-1 n k m i f n = 2 , m i n t e g r a l , k m i f n = 2 +1, otherwise. The permutation 7 i k interchanges two sui t a b l y t h i n subsequences and leaves a l l other terms unchanged. F i r s t , TTk £' C k- The permutation Trk i s contained i n C k i f and only i f D k S k + 1 P v S - ( k + 1 V k k i s regular, where P k i s the permutation matrix associated with TTk. It i s only necessary to show that I |a<k).|< 1=1 n ' 3 -M independent of n where a Ck) rJcJc+r ^-(k+l) -k n,D L k D } . 50 i f b C k ) . = ( s k + V s - ( k + 1 ) ) . then (k) 1 , (k) .k a . = — b . j n,D n k n,j Proof: b C k ) . = c s k + 1 P - s - ( k + 1 ) ) . n,_j nrj = i c-D i + j (n;i:rk) t k ! h i = l J k+1 = i ( - D i + j c n - ^ : : k ) c k : h , i-j=0 J since ^ ^ j ) = 0 f o r a 1 1 values of i - j except for i - j = 0, 1, 2, •••, (k+1). Lett i n g t = i - j y i e l d s k+1 = TT t I C - D 1 ! [ ( . n - j ) + k - t ' ] [ C n - j ) + l - f ] } ( k + 1 ) K - t=0 t 51 (-D t{( kJ: 1) C n - j ) k + J k k + l ^ f ) ( n - j ) k _ 1 + K ' t=0 + 0((n-j) k- 2}} 1 r . .. k k r " L / n t /k+l> , k, 1 (n-D) 1 (-D ( t ) •+ K* t=0 k+1 + ( n - j ) k _ 1 I |(k+l-2t') ( - D 1 " ^ 1 ) + t=0 + 0 ( ( n - j ) k _ 2 ) } The f i r s t summation i n the l a s t expression equals k+1 the binomial expansion of (1-1) = 0. The second summation can be rewritten as a f i r s t degree polynomial i n t ' . When the permutation matrix P 1 equals the i d e n t i t y permutation, t 1 = t and the summation equals zero. To obtain the permutation P" = P^ ., exactly two of the t 1 d i f f e r from t and hence the summation cannot equal zero. Returning to the proof of Theorem 2.4; ir^. & i f OO t \ and only i f \ |a ;|< M uniformly. j = l U ' J j=l n ' 3 k j=l n ' 3 n 3=1 , J 3 = 2 52 2k+l v , ..k-l.k 2K j=l 3 = 2 The l a s t two expressions follow from the lemma. For n ^ , the summation w i l l be maximum when n = j , hence I |a ( k )|< **±i I ( 2 k P - 2 k S ) ( 2 k S ) k j=l n , J " ^ k ^ k s=l 2k+l r P ^ 1 ,Jc P k-1 „ k s + 1 -. k p + 1 ,=1 2 { I (2JV )^ x 2^ } + (2k+l) o i , i P ~ l i S + l = 2k+l | Y 2k j + ( 2 f c + 1 ) „k p s=l 2 k + (2k+l) k P 1 J 2 k = 4k + 2 . •4 53 We next show TT^ ^  c k + i f o r ^ L 1 -CO ^r-v— v i (k+i) | Y - r — 1 v i,(k+i)|.k+i l i m JJan,j I = l i m -k+T .1 J b n , j 13 n->co j = i '-> n n j=l J i l i m - I T i I J b n , j 13 n n J=2 J lim n n" k+i I ( n - j ) k + i j k + i . ,k S >_ lim n 1 k p k+i OT ) P I s=l C 2 k P - 2 k S ) k + i ( 2 k S ) k + i > lim 1 P C 2 k P ) k + i ,kp 1 v k + i ^ k ^ . k + i ( 2 " - 2~ ) (2 J ) lim P ( 2 ^ ) kp k+i lcP k p - l k + i kP L ( 2 k - 2 k ). ( 2 ^ ) lim P (2 J - 2J -P-1 k+i <2*Y k p k+i = lim P C2~ ) k P k ( 2 ^ ) 54 k p, i lim (2 ) P Two cases remain; F i r s t , c i 9^  c i + i f o r 1 L 1 ] Define 2 P + 1 i f n = 2^ for i n t e g r a l p, TT-, (n) = < >P n P i f n = 2 P + 1, otherwise. Then i n £ C n - C. for i > 1. This proof i s omitted; 0 0 r — ^ i t i s almost i d e n t i c a l to the previous proof. Second, C Q r/_ C\ for i _> 1. Define P"2p + 1 i f n = 2 p , TTQ (n) = L 2 p i f n = 2p + 1. Here TTQ interchanges even and odd terms, There i.s at most one jumping-in and one jumping-out term, hence k(n) i s uniformly bounded and TTQ £ C Q . An argument s i m i l a r to the previous cases shows TT ^  c^ for i >_ 1, PRESERVATION GROUPS Number of poles on the c i r c l e of convergence Finite'permutations Number of poles within the c i r c l e of meromorphy F i n i t e permutations Rational functions Arithmetic permutations Algebraic functions Arithmetic permutations Radius of convergence Asymptotic permutations L V— W A A U Type of entire functions Asymptotic permutations Entire functions Bounded forward t r a v e l Polynomials A l l permutations Sum of absolutely convergent series A l l permutations Sum of co n d i t i o n a l l y convergent series Bounded block movement Sum of Cesaro C, summable k series Bounded (k+2)^tuple movement 56 CONCLUSION This thesis has barely touched the possible problems involved with rearrangements of c o e f f i c i e n t s and preser-vation groups. A d d i t i o n a l l y , there are the obvious ex-tensions to the dual l a t t i c e of function properties and preservation groups. There are other types of summation to be considered. The l a t t i c e of preservation groups points out an i n t r i g u i n g property of the character of the rearrange-ments. This property appears to be of a fundamental nature. Each permutation group which preserved a function property was described by a t r a v e l condition whereas those permutations which preserved d i f f e r e n t summabilities were described by smoothness conditions i n the sense that pairs (k-tuples) were kept together. There are several possible approaches toward obtaining an underlying characterization of these permutations. Possibly generating sets could do t h i s . I t seems more l i k e l y that the rearrangement 'type' i s basic, Three 'types' of rearrangements might be t r a v e l conditions, smoothness conditions, and density preserving conditions. 57 The permutations considered so far could a l l be represented by a 0-1 doubly stochastic matrix. Other types of permutations generated by doubly stochastic (but not 0-1) matrices might be considered. Also, t r a n s i t i o n matrices might be of i n t e r e s t . Other types of series could be rearranged, for example, D i r i c h l e t or Laurent s e r i e s . Several function properties have been considered. The tolerance allowed i n the order of the c o e f f i c i e n t s while s t i l l guaranteeing preservation of a p a r t i c u l a r property was discussed. A l o g i c a l question i s then: Are there properties of power series which are independent of the ordering? That i s , specifying s o l e l y the set of c o e f f i c i e n t s (and possibly the m u l t i p l i c i t y ) , can con-clusions be drawn about the properties of any power series composed of a l l the c o e f f i c i e n t s from t h i s set? There are two well-known res u l t s along t h i s l i n e . F i r s t , a theorem by Szego states that i f the c o e f f i c i e n t s form a set of f i n i t e c a r d i n a l i t y , then the power series represents a function which i s either r a t i o n a l . o r singular, depending on whether the c o e f f i c i e n t s form a periodic sequence or not. Second, i f the set of c o e f f i c i e n t s i s bounded away from the second and t h i r d quadrants by rays 58 emanating from the o r i g i n , then the power series with radius of convergence R has a s i n g u l a r i t y at z = R. A t h i r d r e s u l t may be obtained as a c o r o l l a r y of Weyl's c r i t e r i o n for uniform d i s t r i b u t i o n [28]: I f the c o e f f i c i e n t s of a power series are not uniformly d i s -t r i b u t e d on the unit c i r c l e , then some Hadamard power of f, f ( k ) z = I ( f n ) k z n , w i l l be singular at z = R. Unfortunately there i s no guarantee that f w i l l be singular. Functions with natural boundaries suggest another problem. A power serie.s i s singular i f i t s c i r c l e of convergence i s a natural boundary. Are there, then, functions which are singular and a l l of whose permutations are also singular? We r e c a l l that almost a l l decimals are i r r a t i o n a l and that almost a l l functions are nowhere d i f f e r e n t i a b l e . Perhaps almost a l l power series are singular. Consider the set consisting of the c o e f f i c i e n t s of such a singular function. The author attempted to show the existence of singular functions by specifying some set condition. I t was conjectured that i f the set of c o e f f i c -ients were suitably scattered, say dense i n some region, then any function formed with c o e f f i c i e n t s from t h i s set must be singular. An i n t e r e s t i n g counterexample with c o e f f i c i e n t s dense i n the plane and only two poles 59 i s the d e r i v a t i v e of f ( z ) = J e i a n ( c o s b n ) z n n=0 ^ 1 2^ \ i(a+b) 1 - e z , i ( a - b ) 1 - e z - In e i ( a + b ) n + e i ( a - b ) n - \ where a and b are p i c k e d so t h a t (a+b) ^ k(a-b) f o r any r a t i o n a l k. 60 BIBLIOGRAPHY Agnew, R. P., On Rearrangements of Series, American Mathematical Society Bulletin,46(1940). Ahlfors, Lars V., Complex Analysis, Second E d i t i o n , McGraw H i l l , 1966. Armstrong, Convergence Radius of Regularly Monotonic Functions, Duke Math Journal, Vol. 37 No. 1, March 1970. Bagemihl and Erdos, P., Rearrangements of C-, -summable Series, Acta Mathmatica, Vol. 92 (1954)7 Bieberbach, Ludwig, Analytische Fortsetzung, Springer-Verlag, 1955. B l i s s , G i l b e r t Ames, Algebraic Functions, American Mathematical Society Colloquium Publications Vol. XVI, New York, 1933. Boas, R. P., Entire Functions, Academic Press, Inc., 1954. Carlson, F r i t z , Uber Potenzreihen mit ganzzahligen Koeffizienten, Mathematische Z e i t s c h r i f t , Vol. IX, October 1919. Cartan, Henri, Elementary Theory of Analytic Functions  of one or several Complex Variables, Addison-Wesley Publishing Co., Inc., Palo A l t o , 1963. Cassels, J.W.S., An Introduction to Diophantine Approximation, Cambridge University Press, 1957. Davenport, H., Erdos, P., and LeVeque, W.J., On Weyl's  C r i t e r i o n for Uniform D i s t r i b u t i o n , Michigan  Mathematical Journal, Vol. 10 C1963). Dienes, P., The Taylor Series, Dover Publications, Inc. New York, 1957. Fine, N.J. and Schweigert, G.E., On the Group of Homeomorphisms of an Arc, Annals of Mathematics, Vol. 62 No. 2, September 1955. 61 [14] Ford, Walter B., Studies on Divergent Series and Summability, Chelsea Publishing Co., New York, 1960. [15] Guha, U.C., On Levi's Theorem on Rearrangements of Convergent Series, Indian Journal of Mathematics, Vol. 6, 1967. [16] Hardy, G.H., Divergent Series, Oxford at the Clarendon Press, 1949. [17] Hausdorff, F e l i x , Zur Verteilung der fortsetzbaren Potenzreihen, Matheroatische Z e i t s c h r i f t . Vol. 4, 1919. [18] H i l l e , Einar, Analytic Function Theory, Vols. I & I I , Ginn and Co., 19 62. [19] K l e i n , F e l i x , Elementary Mathematics from an Advanced  Standpoint—Arithmetic, Algebra, Analysis, Dover Publications, New York. [20] K l e i n , F e l i x , Elementary Mathematics from an Advanced  Standpoint—Geometry, The MacMillan Company, New York, 1939. L ^ -i- J A t i A V ^ j ^ j ^ ^ J.\. • f JL l i C U l J GilAKJ. f l ^ j-V J. X O Q U ± U X X W X. J.11 L J-i._l.J- U London, 1'92 8. [22] Levi, F.W., Rearrangements of Convergent Series, Duke Mathematical Journal, Vol. 13, 19 46. [23] Lorentz, Z e l l e r , Series Rearrangements and Ana l y t i c Sets, Acta Mathematica, 1958. [24] Mahler, K., On the Taylor C o e f f i c i e n t s of Rational Functions, Proceedings of the Cambridge  Philosophical Society, Vol. 52, Part 1, 1956. [25] Melzak, Z.A., A Countable Interpolation Problem, Proceedings of the American Mathematical Society, Vol. I I , No. 2, A p r i l 1960. [26] Melzak, Z.A., Power Series Representing Certain Rational Functions, Canadian Journal of Mathematics, Vol. 12, 1960. [27] Moore, Charles N., Summable Series and Convergence Factors, American Mathematical Society Colloquium  Publications, Vol. XXII, 1938. 62 [28] Niven, Iven, I r r a t i o n a l Numbers, Mathematical Association of. America, Carus Mathematical  Monographs, No. 11, 1956. [29] Ostrowski, A., On Hadamard's Test for Singular Points, Journal of the London Mathematical  Society, 1926. [30] Polya, G., Sur Les Series Entieres A C o e f f i c i e n t s E n t i e r s , Proceedings of the London Mathematical  Society, 1922. [31] Redei, L., Foundation of Euclidean and Non-Euclidean Geometries according to F. K l e i n , Pergamon Press, 1968. [32] Titchmarsh, E.C., The Theory of Functions, Second Ed i t i o n , Oxford at the Clarendon Press, 1960. 

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