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On the uniqueness of multiple trigonometric series Cross, George Elliot 1958

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`Wat  Pttfoerstig  of  British  Oloiuxnbia  Faculty of Graduate Studies  PROGRAMME  FINAL ORAL  OF T H E  EXAMINATION  FOR T H E D E G R E E O F  DOCTOR O F PHILOSOPHY of G . E . CROSS B.A., Dalhousie University, 1952 M.A., Dalhousie University, 1954 FRIDAY, SEPTEMBER 26, 1958, at 3:30 p.m. IN ROOM 315, B U C H A N A N  BUILDING  C O M M I T T E E IN C H A R G E DEAN  G. M . SHRUM,  Chairman  R. D . J A M E S  BARNETT  F. M . C . G O O D S P E E D  AVRUM  W. H . SIMONS P. S. B U L L E N  -  SAVERY STROLL  B. C . B I N N I N G R. E . B U R G E S S  External Examiner: R. L . J E F F E R Y Queen's University  ON  T H E UNIQUENESS  MULTIPLE  OF  TRIGONOMETIC  SERIES  ABSTRACT  The multiple trigonometric series 2 c xv(Hm,x)), e  M  m = (m  v  ...  , m ), x = (x n  v  . .. , x ), (m,x) =  where  m x^ +  n  ...  x  + m x , and mi > 0 is said to be summable (T, k) if the series 2 Cp(x) is summable (C, k), where Cp(x) denotes the triangular sum 2 p(*( > ))> /»i . . . m = p. The series is said to be bounded (T, k) if the series 2 Cp(x) is bounded (C, k). n  n  c  ex  m  x  +  +  m  n  Using the fact that the triangular summation of multiple trigonometric series considered is equivalent to the Cesaro summation of a single series of a particular form^ this thesis obtains uniqueness theorems for multiple trigonometric series by first proving the required theorems for the single series. It is shown that if the series 2 c exp(int) is summable (C, k) then the coefficients c„are given in terms of the p integral defined by James [Trans. Amer. Math. Soc, vol 76, 1954, pp. 149-176, Section 8]. When the series is bounded (C, k) a Fourier representation is obtained in terms of Burkill's C £+i-P-integral [Proc. London Math. Soc. (2), vol. 39, 1935, pp. 541-552]. n  k  +  2  It is shown that if f(t) is periodic and C P-integrabIe, then the definite C P-integral is a constant multiple of the definite fl +l -integral of f(x). This gives a Fourier representation of the coefficients in terms of the jo k+2 -integral when the series is bounded (C, k). r  r  r  These results are then extended to multiple trigonometric series. A representation for the coefficients in terms of the C k+i P-integral is demonstrated if the series is bounded (T, k). Finally, a uniqueness theorem is proved where the summability set is a countable set of «-tuples of the form ( * , . . . , x ,x . ) for fixed x.q,..., x_ and i ^rrl. fll  n  x  0  fl  n  G R A D U A T E  FIELD OF STUDY:  S T U D I E S  MATHEMATICS  Modern Algebra Integral Equations  Rimhak Ree T . E. Hull  Theory of Functions  W. H .  Functional Analysis  R. R. Christian  OTHER  Simons  STUDIES  Symbolic  Logic  Contemporary  Philosophy  Avrum Peter  Stroll  Remnant  ON THE UNIQUENESS OF MULTIPLE TRIGONOMETRIC SERIES •  by  GEORGE ELLIOT CROSS M. A., Dalhousie University, 195*+  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of MATHEMATICS  We accept t h i s thesis as conforming to the standard required from candidates for the degree of Doctor of Philosophy::  THE UNIVERSITY OF BRITISH COLUMBIA October,  1958  ii  Abstract The multiple trigonometric series E where m = (m-^, ... ,m ), x = (x^,  c exp(i(m,x)),  x ) , (m,x) = m^x+- ...  n  n  q  1  +• m x , and m. 0, i s said to be summable (T.k) i f the n n' j series z~ C (x) i s summable (C.k) where C (x) denotes the ° P P triangular sum ZI c exp(i(m,x)), m, + ... 4-m = p. The m 1 oo series i s said to be bounded (T,k) i f the series ^ ^ 7  7  n  c  x  p  i s bounded (C,k). Using the fact that the triangular summation of the multiple trigonometric series considered i s equivalent to the Cesaro summation of a single series of a p a r t i c u l a r form, t h i s thesis obtains uniqueness theorems f o r multiple trigonometric series by f i r s t proving the required theorems for the single series. It i s shown that i f the series o  c exp(int) n  is summable (C,k) then the c o e f f i c i e n t s c^ are given i n terms of the ? °  k + 2  - integral defined by James [Trans.  Amer. Math. Soc. v o l . 76 (195^) pp. l +9-176, section 8] . 1  When the series i s bounded (C,k) a Fourier representation is obtained i n terms of Burkill's C -^P - i n t e g r a l [Proc. k+  London Math. Soc. (2) v o l . 39 (1935) pp. 5^1-552]. It i s shown that i f f ( t ) i s periodic and C P r  integrable, then the C P - (definite) integral i s a constant r  multiple of t h e ^  r + 1  - (definite) integral of f ( x ) . This  gives a Fourier representation of the c o e f f i c i e n t s i n terms of t h e ^  k + 2  - i n t e g r a l when the series i s bounded (C,k).  iil  These r e s u l t s are then extended to multiple trigonometric s e r i e s . in terms of the C^^P  A representation f o r the c o e f f i c i e n t s - Integral i s demonstrated  series i s bounded (T,k).  i f the  F i n a l l y a uniqueness theorem i s  proved where the summability set i s a countable set of n - tuples of the form ( x ^ o ' 2 0 ' x  fixed  X^Q,X2Q,  . . . » fl_]_ 0 ' X  **•  ^ "  , x  n-10 ni^' , x  f o r  ^> 3) ••• •  In presenting  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 of  the r e q u i r e m e n t s f o r an advanced degree a t t h e  University  of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t freely  a v a i l a b l e f o r r e f e r e n c e and  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 t h e s i s f o r s c h o l a r l y purposes may of my  study.  I further  copying of  this  be g r a n t e d by the  Department or by h i s r e p r e s e n t a t i v e .  Head  I t i s under-  stood t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r financial  g a i n s h a l l not be a l l o w e d w i t h o u t rny w r i t t e n  permission.  Department of The U n i v e r s i t y of B r i t i s h Columbia, Vancouver #, Canada.  iv  Table of Contents  Chapter  1  Introduction  1  Chapter  2  Notation and Definitions  3  Chapter  3  The Integrated Series .  9  Chapter  k  The Expression of C o e f f i c i e n t s i n Terms of the £°  Chapter  5  6  fc  +  .  ^-P - i n t e g r a l . .  +  2  15  21  23  - integral  Uniqueness Theorems f o r Multiple Trigonometric Series  Bibliography  - integral  +  k  7  2  The C^ .]P.- i n t e g r a l and the , ?°  Chapter  +  The Expression of C o e f f i c i e n t s i n Terms of the C  Chapter  k  . * .  . . . . . . .  25 30  V  Acknowledgment  The author wishes to acknowledge with gratitude the generous assistance given by Dr. R. D. James i n the preparation of t h i s thesis.  1  1.  Introduction  Using a process of spherical summation introduced by Bochner [l] , Cheng  has proved that i f a multiple  trigonometric series (1.1)  c exp(i(m,x)) m  i s everywhere (C,l) summable to zero by spherical means and s a t i s f i e s a c e r t a i n condition, then i t vanishes i d e n t i c a l l y . Shapiro [l3, l *] has extended these r e s u l t s to allow ex1  ceptional sets of capacity zero.  More generally Cheng has  shown that i f the s e r i e s i s everywhere (C,l) summable by spherical means to a function f ( x , x , ... ,x ) then under 1  2  n  c e r t a i n conditions the series i s the Fourier series of f . This thesis i s concerned with uniqueness theorems for multiple trigonometric  series of a p a r t i c u l a r form  summed by a "triangular" method.  This method reduces the  summation of the multiple series to that of a single series of the form (1.2) where c„ = a - i b . n n n  ^ o  c exp(inx) n  James [?] has shown that i f the r e a l part of series (1.2) i s summable (C,k) and an additional condition involving the imaginary part i s s a t i s f i e d , then the c o e f f i c i e n t s are given i n modified Fourier form, where the i n t e g r a l involved is h i s P  - integral.  It i s shown here that i f series (1.2)  2  is summable (C,k) then the c o e f f i c i e n t s have a s i m i l a r  70 it + 2 Fourier representation i n terms of James 0  - integral.  1  B u r k i l l [3] has shown that, i f (1.2) (C,0)  i s bounded  except on a countable set and i f the series obtained  by integrating series (1.2) once converges everywhere, then the c o e f f i c i e n t s can be written C]_P - i n t e g r a l .  i n Fourier form using the  An analogous r e s u l t i s shown to be true In  t h i s thesis when (1.2)  i s bounded (C,k).  The proof of t h i s  depends on a powerful r e s u l t by Marcinkiewicz and Zygmund [8J  , and on generalizations  of theorems by Verblunsky and  Zygmund. James has proved that i f f(x) i s C P - integrable r  i t i s also f° of the P  r  +  -  1  integrable  T  and has given a representation  - ( i n d e f i n i t e ) Integral i n terms of the C P r  (indefinite) integral.  The two preceding r e s u l t s suggest  the relationship between the ? °  r  ^ and C P - ( d e f i n i t e ) r  integrals given i n the theorem of Chapter 6. Series  (1.1) i s said to be bounded (T,k) i f the  corresponding series (1.2) Is bounded (C,k).  I t i s shown  that i f (1.1) i s bounded (T,k) f o r a l l values (xj + t , x +- t) then the c o e f f i c i e n t s are given by repeated integrals n  of dimension n+-l,  the inner i n t e g r a l being a C^ ^P - i n t e g r a l . +  It i s then shown that i f there i s a countable set of n - tuples such that series (1.1)  i s summable (T,k) to 0  for a l l values ( X + t , ••• » - l 0 *" * ^ n i " ^ ' x  1 0  4  n  then the series vanishes i d e n t i c a l l y .  1  =  1  » >3? ... > 2  3 2.  Notation and Definitions  Consider the multiple trigonometric (2.1)  series  exp(i(m,x)),  where c  may be complex, m - (m^, ... ,m ), (m,x) = i i + ••• m  m  x  n  +- m x , and the summation i s over a l l non-negative integers m-j.. n  n  DEFINITION 2.1.  The series i s said to be summable (T.,k)  to sum f(x) i f the series (2.2)  £l  C (x)  P s 0  V  i s summable (C,k) to f ( x ) , where C ( x ) denotes the triangular p  sum (2.3)  Z I c exp(i(m,x)), m  where the summation i s over a l l non-negative integers m^ such that m^ + ... +• m•••= p and (x) =• (x^, ... >x ). Q  n  DEFINITION 2.2  The series (2.1)  at (x) i f the series (2.2)  (T,k)  DEFINITION 2.3. interval  [a,b] .  g ( t + h) = c 0  i s said to be bounded  i s bounded (C,k) at ( x ) .  Let g(t) be a function defined  I f . f o r a given t  Q  i n [a,b] ,  + c h + c h 2 / 2 I + ... + c ^ / k l  Q  x  i n the  2  (h ), k  0  where the numbers c^ = c ^ ( t ) are independent of h, then c Q  k  Is c a l l e d the k - t h de l a Vallee Poussin derivative of g at the point t  Q  and i s denoted by ^ d ^ C ^ ) .  DEFINITION 2.If. I f g ( ) ( t ) exists f o r 0 < k i n - 1, k  define  V ( t , h ) by. n  Q  Q  n -1 r ( t , h ) = g(t +  h /nl n  G  0  h) - g  0  ( o )  (t ) - ^ [ h V k l ]  g  0  ( k )  (t ), D  and l e t ^Srn^tJ 6  DEFINITION 2.5. f(x  0  Cn)  =  l  i  sup V  m  h-> 0  o  Let CP(t) = * ( x  1 0  + t ) = u ( x + t ) + i v ( x -t- t ) .  CP(t  0  0  -ft) « • *  . *  for 1 < r < k, where constants  +°C f p^t"*  t 2 2  /  0  +  * •••  2 1  with t  0  n  h)  >  «" t ) =  x n 0  I f for a given t  0  0  (t  C^p  +  £  O  J  r,t)t /rl r  and the < ^ are complex  T  Independent of t , then °f w i l l be c a l l e d the k - t h k  generalized derivative of U> at the point ( X -*• t ) and w i l l be q  denoted bv ^ ( ) C k  + t )«  x 0  o  I t w i l l be convenient to write  0  C0(k)(t ) for 6 0 ( ) (  t ) wherever i t w i l l cause no  x  k  Q  o  Q  misunderstanding. Thus i f U(t) s u ( x ••- t ) and V(t) = v ( x 0  =  U  (k)^o>  +  1  v  (k)^o)  w  h  e  r  e  (k><*o>  u  a  n  d  v  0  t) then  (k) o> ( t  denote the k - th de l a Vallee Poussin derivatives of U(t) and V(t) at ( x + t ) . Q  Q  For reference purposes some d e f i n i t i o n s and theorems required for the development of the ^°  n  - i n t e g r a l [6] and the  C P - i n t e g r a l [2] are given below. n  Let u(t) be a r e a l valued function of the single variable t defined i n [a,b] and l e t a^, 1 - 1 , 2 , ... ,n be fixed points such that a - a, < a~ < ... < a_= b.  5 DEFINITION 2.6. The functions Q(t) and q(t) are called major and minor functions, respectively, of u(t) over (a^) =  Q(t) and q(t) are continuous i n [a.b] , and, f o r  (2.^.1)  1 < k 5 n - 1, each Q(j )(t), ^ ( k ) ^ exists and Is f i n i t e c  in (a,b)j QCa^ = q C a ^ ="0, 1 = 1 , ... ,n, where a = a-^ <  (2.^.2) ...  <a =b; n  (2A.3) fQ( )(t)  2 u(t)  n  (2.^.lf) ^ Q  ( n  >Aq  ) ( t ) ^ - oo, A q  ( n )  ( n )  ( t ) , In (a,b);  ( t ) 1= + co i n (a,b).  DEFINITION 2.7. For each major and minor function of u(t) over (a^), the functions defined by Q*(t) = ( - l ) - Q ( t ) , q (t) = C - l ) * q ( t ) , a r +  n  r  ft  n  p  < t < a „ r  l f  are c a l l e d associated ma.lor and minor functions, r e s p e c t i v e l y , of u(t) over (a^). DEFINITION 2.8. Let c be a point of ( a ^ a ) such that c * a^,I = 1,2, ... ,n. Then u(t) i s said to be fr° n  lntegrable over (a^;c) i f for every  € > 0 there i s a pair  Q ( t ) , q ( t ) s a t i s f y i n g conditions (2.1+.1) - (2 A A )  such that  |Q(c) - q(c)| V 6 • THEOREM 2.1.  I f u(t) i s T  n  - lntegrable over ( a ^ c )  there i s a function U*(t) which i s the inf of a l l associated ma.jor functions of u(t) over (a^) and the sup of a l l associated minor functions.  6  DEFINITION 2.9.  I£ u(t) I4 f °  n  - lntegrable over (a ;c) jL  and I f U*(t) i s the function of Theorem 2 . 1 . define U(t) by ( - l ) * U ( t ) = U ( t ) , when a r  If a  s  < c <a  s+ 1  n  A  , the ? °  n  < t < a  ^  r  - i n t e g r a l of u(t) over (a^jc) Since ( - l ) * ^ ( a ^ - 11(8^ =  i s defined to be ( - 1 ) * U ( c ) . s  p  n  s  0, the integral i s defined to be zero i f c = 8 ^ 1  =1,2,  ... , n,  The notation i s (-l) THEOREM 2.2.  s + n  j u C t ) d„t.  U(c)=  I f u(t) i s £ ° - lntegrable over ( a ^ c ) , n  i t i s also ? ° - lntegrable over (a^;t) f o r every t i n [a,b] . n  If U(t) i s the function of D e f i n i t i o n 2.9. then, f o r a  s *  t  <  a  s+l»  f  t  (-l)  s+ n  U(t) =  u(t) d t .  The C P - i n t e g r a l i s defined by induction.  Suppose  n  that for n > 1 the C  _ ^P - i n t e g r a l has been defined taking  n  as the C P - i n t e g r a l the Perron i n t e g r a l [9, p. 20l] . 0  Assuming that u(t) i s  c n  C ( u , t , t *- h) = ( n / h ) C n  n  DEFINITION 2.10. C  n  - continuous at t  n  _ i  p  - lntegrable, l e t A* h / (t + h - J )  _ P X  n  " \ ( f  )d f  The function u(t) i s said to be 0  DEFINITION 2.11.  i fC (u,t ,t n  0  Q  + h)  The upper and lower C  u ( t ) as h Q  n  - derlvates of  u(t) denoted by C D u ( t ) and C D u ( t ) , r e s p e c t i v e l y , are 4  n  n  A  0.  7 defined to be the l i m SUP and the l i m i n f . respectively, as h -> 0 of the expression (n + l/h)(C (u,t,t + h) - u ( t ) ) . n 4  DEFINITION 2.12.  I f C D u(t) = C D u ( t ) , t h e i r common Q  value i s defined to be the C  n  4  - derivative of u(t) and i s  n  denoted by C D u ( t ) . n  DEFINITION 2.13. C  Q  The functbn M(t) i s said to be a  - ma.ior function of u(t) over fa.b]  (2.5.1)  M(t) i s C  (2.5.2)  M(a) = 0;  (2.5.3)  C D M(t)  (2.5A)  C D M(t) > -oo  n  n  A C  n  A  n  if  - continuous:  > u ( t ) , p.p i n \a,b]  4  ;  i n [a,b] .  - minor function m(t) i s defined i n a  similar way. DEFINITION 2.1k, I f , f o r every  € > 0, there Is a pair  M(t), m(t) s a t i s f y i n g the conditions of D e f i n i t i o n 2.13 and such that  )M(b) - m(b)| < e , then u(t) i s said to be C P n  Integrable over  [a,b]  DEFINITION 2.15.  . £et 1(b) = lower bound of a l l M(b)  and J(b) = upper bound of a l l m(b).  For a C P - integrable  function u(t) the bounds have a common l i m i t  n  [2] which i s  c a l l e d the d e f i n i t e C P - i n t e g r a l of u(t) over fa b1 . n  Suppose that v(t) a r e ^  n  r  ^ ( t ) = u(t) + 1 v(t) where u(t) and  - integrable and i n the notation of Theorem 2.2  8  (- l ) * "iKc)= J  u(t) d t  s  (- l ) If a < c < a g  s+  n  V(c)=  /  v(t) d t .  the T ° - i n t e g r a l of ^ ( t ) over (a ;c) n  s+  1  ?  i  i s defined to be (- 1 ) * ^ ( c ) + i ( - 1 ) *" 'VCc), i . e . S  /  S  ^ ( t ) d t - / ° u(t)d_t + i / ° v ( t ) d t = n  - (a ) /  n  - (a )  n  /  1  • (a )  n  /  1  (- l ) * s  n  n  1  [u(c) +-1 V(c)]  =  (- l ) * s  n  \$(c).  The C P - integral of CP (t) i s defined  i n an  n  analogous way. When a series of the form oo  c exp(int)  2> (in)  nTT  j  i s summable (C, r - j - 1) to a function, say, F  r  " (t) J  where r and j are positive integers, i t w i l l be convenient to write simply, (2.6)  y^cnexpdnt) = F  r  " Ht)  - i G " Ct), p  (C, r - J -  3  n = 1 (In)1 without stating e x p l i c i t l y that F defined  by ( 2 . 6 ) .  r  " ^(t) - 1 G  r  " (t) is J  9  3.  The Integrated  Series  A solution of the representation problem f o r the series (3.1)  S_  k  n  =1  n in terms of the £°  c exp(int)  2  - i n t e g r a l , when the series i s  summable (C,k), involves the following theorem stated by Zygmund [l6, p. 226, problem 12J to be proved as an exercise: THEOREM 3.1. for a fixed  Let the series (3.1) be summable (C,* ),  * = 0, 1, 2, ... and t = t , to sum s, |s/ < co . Q  Let r be an integer  > ' « + 1.  Then the series integrated  term by term r times converges uniformly f o r a l l t to a continuous function f ( t ) , and * P ( ) ( t ) exists and i s equal r  0  to s. When the series (3.1) i s bounded (C,k) a Fourier representation f o r the c o e f f i c i e n t s i n terms of the C  k +  ^P  - i n t e g r a l can be obtained using a generalization of Theorem  3.1. THEOREM 3.2.  Let the series (3.1) be bounded (C, « ),  for a f i x e d « = 0, 1, 2, ... and t 6 E, /E/ > 0. <* +• 2, then for each t e E,  (3.2.j)  If r =  \  c exp(int) n  n F  = r  1 (in)3  " Ht) - i G  r  ~ (t), J  .  (C,r - j - l ) ,  10  where j = 1, 2, ... , r - 1, and (3.2.r)  y ~  c e x p ( i n t ) = F(t) - i G(t) n  = 1 (in)  n  CP(t),  r  where the l a s t series i s absolutely for a l l t . In addition.  and^uniformly convergent  ^ ( ) ( t ) exists and i s f i n i t e f o r s  0 < s < r - 1, t e E, and ^ ( t - h) = L£(t) + h ^  (3.3)  [h  r  ' V(r  ( 1 )  - l)l\& _ {t  1 }  where u> (t,h) - 0(1) as h -» 0. (  (P^  3-^)  +  2  - j )  (  t  )  (t)  +  ... -  (t) +  [^(t,h)/rl] h  r  Furthermore.  " Y~  c  <C,  n *P(l > e  n t  n =l (in)  •< + 1 - j )  3  f o r 0 < j < r , and ^ ( ) ( t ) exists p.p. i n E. ?  r  Proof.  I t i s clear that c e x p ( i n t ) - 0(n** ) and t h i s i s n  s u f f i c i e n t to guarantee the convergence property of series (3.2.r).  The summability (C, * ) of series (3.2.1) and the  summability of series ( 3 . 2 . 2 ) , ( 3 . 2 . 3 ) , . . . , (3.2.r - 1) follows from two theorems by Hardy £5, Theorem 71, p. 128 and Theorem 76, p. 131, respectively} . To obtain  (3.3) and (3.^) i t may be assumed without  loss of generality that t = 0. Let X (u) =. exp(iu)/(iu) , r - 1 v \ rs ©(h) - y (ih) /» », A ( h ) - exp(ih) - 6 ( h ) . and f o r r  *=0 any  (ih)  r  sequence ( u ) l e t An = A.^u = u - u t m) n n n n + 1'  11  A  =  " u ).  [ l 6 , pp. 260 -  Then Zygmund's proof  1  n  , with condition s - o(n ) replaced by s = 0(n ) n  n  yields (3-5)  r - 1 <^(h)= . Z Z < A „ / v i h  where A, = 5 Z s * A. * " ( i n ) n # <  1  "  , /  r  + h R(h),  v  r  and R(h)= Z s  8  A " n  1  A(nh)  +1  both converge absolutely, and R(h) = 0(1) as h-» 0. (Pit f h) = 6?(t) + h ^ (h  r  ( 1 )  ( t ) + ...  " V ( r- l ) ! ^  r  where <*> (t,h) = 0(1) as h-» 0.  _  x  Thus  +  ) (t) H-  [co(t,h)/rl] h , r  I t follows from a theorem due  to Marcinkiewicz and Zygmund [8, Lemma 7, p. 15] that  ^( )(' ) t  r  exists p.p. i n E. Equation (3-5) gives -0,  4  ,(0) = A  vr- j ;  = 51 s* A ^ C l n ) " K r-j  and since the (C, * ) sum of the series  n  c /(n)^ equals the  1  (0,0) sum of the series (3.*+) i s established. THEOREM 3.3. I f under the hypotheses of Theorem 3.2, the set E i s an open i n t e r v a l , then (3.6)  C D F ( t ) - F. . ( t ) , C DG (t) - G. x(t), a? (a + 1) a (a + 1) a  a  n  0 < a < «< , 1 1 E, C  „I> F, 4  «* +• 1 C C  DP.  ^  , (t), C x  ,>(t),  ( OC +• 1) D*G,  n  ,x(t),  D G, , ( t ) are f i n i t e f o r te.E, and + 1 A (°< +- 1) x  12 (3.7)  C G  IJF *  ( )( )>  +  l ( t ) - F.  (t), C ^  _DG *  +  1  (t) -  P«P* i n E.  T  R  For the proof a lemma i s required. LEMMA.  y ° a n = 1  If  n  i s summable (C, r -t- 1 ) , where r > -  then a necessary and s u f f i c i e n t condition that i t should be bounded (C,r) i s that B £ - 0 ( n  + r  1  ) where b = n a fl  n  and  B°, B , B , ... are formed from the b as A , A , A , ... n* n' n' n n' n' n* are from the a ( c f . 5, p. 9 6 ) . 1  2  0  1  2  fl  Proof.  I t i s easy to v e r i f y that  (ri +- r +• l ) A = B n n and hence that r  r  +• (r + l ) A n  r  1,  1  B  A ^  (r - l / n + r  +  DCpf-y + A £ +  h,  and the lemma follows. The r e l a t i o n s (3.6) w i l l be proved by induction and i n v i r t u e of the symmetry of the enunciation only the f i r s t of the r e l a t i o n s w i l l be considered. The r e s u l t i s well known f o r a = 0 [ l 5 , Lemma 5, p. 206] . By the lemma, series (3.2.1) i s bounded (C, < - 1 ) , c  series (3.2.2) i s bounded (C,°<- 2 ) , ... , series (3.2.r - 1) i s bounded ( C , 0 ) .  Assume that the r e l a t i o n holds for a l l s, » s < °< and hence that F (t) i s the C P - i n t e g r a l of S  F  (s  + 1 ) ^  parts gives  f  o  r  a  1  1  s  <^ •  T n e t l  (°< - 1) integrations by  13  t + h lira  ( */h  p  )C  j "t  « - 1  h-> o  (t + h - / )  "  e<  c r )df - F * (t)  h/( * + 1)  lim [( « +• l ) ! h *  F ( t f h) - F ( t ) - XI (h /ki)F (t) k - 1 k  x  k  Hence C DF lim  h-*0 F  (t) =  [( <* + l ) J / h *  1  1  (<* +• l )  ( t )  >  b  v  T h e o r  1  [ F ( t +-h) - F ( t ) -  (h /ki)F  (t)  k  ( k )  3.2.  em  It follows from (3.3)  that  [( « + 2)!/h °" *" ] [ F ( t i- h) - F ( t ) -  (3-8)  2  JLLl JEZ  : = 1  , (h /k!)F k  as h -> 0 and so C  ( k )  -I (t)J  1 ^( ~ D  = 0(1),  !)<*>  a n d  C  «  l V ( *  +  l )  (  t  )  are f i n i t e . Finally, C  c*  + 2)!/h  lim  DF ^  1  +  1  (t) F(t  oC + 1 h) - F ( t ) - ^ T " ( h / k i F  ( k )  (t)  k = 1  IwO i f the l i m i t on the r i g h t hand side exists. C  k  DF  By Theorem  exists p.p. i n E and i s equal to F,  «r 4- 1 THEOREM 3.1+.  3.2 x(t)  ( <* +• 2) Suppose that the r e a l part of series  i s bounded (C,*< ), (x « E, ) E | > 0, °< > - 1 ) .  Then the  (3.1)  11+  series (3.1)  Is summable (C, «*  ^ ),  f > 0, almost  everywhere In E to sum ^ ( ) ( t ) , where <^(t) - F ( t ) - 1 G(t) r  i s the function of Theorem 3.2 and r = Proof.  +- 2.  Marcinkiewicz and Zygmund have proved |^8,  Theorems 2 and 3J that i f the r e a l part of series (3.1) i s bounded (C,« ), (x £ E, |E| > 0, (3.1) p.p.  °< > - 1), then the series * *• f ),  i s bounded (C,«0 and summable (C,  ^ > 0,  i n E. Now, to f i x ideas take <=< = 1.  proof of Theorem 3.2, the series oo N>  - a s i n ( n t ) +- b cos(nt) n  oo  and -^T~ a cos(nt) +• b s i n ( n t )  n  n = 1  In view of the  n  n= 1  n^  n  n^  are uniformly convergent everywhere and hence are the Fourier series of F(t) and G(t) respectively. Theorem 3*2 that ( 3 ) ( ' ) F  in E.  t  a  n  d  G  (3)^  I t follows also from  exist almost everywhere  By a well known theorem (Cf. [l6]  , p. 257) the  Fourier series of F ( t ) and G(t) d i f f e r e n t i a t e d term by term three times are, almost everywhere In E, summable (C,k), k > 3 to the values ( 3 ) ( t ) F  Hence series (3.1)  a  n  d  G  ( 3 ) ^ ^ respectively.  i s summable (C, 1 + f ),  in E and also summable (0,*+) to ^ ( 3 ) ( t ) follows that the series (3.1) (C, 1 + <T) to sum ^ ( ) ( t ) . 3  <f > 0, p.p.  p.p. i n E . I t  i s almost everywhere summable  15  k.  r  The Expression of C o e f f i c i e n t s In Terms k + 2 ._ - Integral  The main r e s u l t of t h i s chapter may be formulated as  follows: THEOREM k.l.  I f k - 2m, l e t Q  = (2m + 2)1/ [(m + l ) i ]  2  and l e t (* ^) be the set c  (h.l)  - (2m + 2)rr , - 2m?r, ... , - 2 rr, 2 TT , . . . , (2m + 2)tr •  i f k = 2m f 1, l e t  r (2m + 3)1/ [(m *• 1)1 (m + 2 ) i ]  and l e t  (°< ) be the set (1+.2) - (2m + 2)-rr, - 2m-rr, ... , - 2n , 27r , ... , 2m"fT, (2m + 2)TT , (2m  W) TT .  Then i f series (3.1) i s summable (C,k) f o r a l l t to a function  CP(i)  - u(t) + i v ( t ) , \(P(t)\  series are given bv c  r  <  co, the c o e f f i c i e n t s of the  , U  /  5  V<t)  exp(- i p t ) d  For the proof several preliminary  k+ 2  t.  theorems must be  proved, the f i r s t one of which i s the analogue of a theorem by James [6, Theorem 5.*? p. 159J • 1  THEOREM k.2.  If F (  (a,b) then i t i s ? ° * k  a r a^ < a  2  < ... < a  2  n  k+  2  ) ( t ) exists and i s f i n i t e i n  - lntegrable - b.  If a  g  over (a-^jt) where < t < a  g +  ^, then  16  (••A)  (-i)  s+  *  k  /  2  F  (  k  +  2 ) ( t  )d  „ t =  k  2  k + 2. 1=1  i  i  where  X(t;a.)r J T " (t - a j / a , - a ) , r * 1  1  Proof.  r  1  r  The function  2  k Q(t) = F ( t ) -  X(t;a )F(a )  y  1  i  i s continuous and each Q ( ) ( t ) exists and i s f i n i t e i n n  [a,b] , 1 £ n < k + 2.  - 0, i = 1,2,3,  Furthermore, q(a ) t  ... , k *• 2, and * 2)  ^ ( k  ^ (k + 2 ) Q  ' (k * 2 ) ^ -  ( t )  F  ( t )  . 2).  Q ( k  A  ^ " °°» ^ ( k + Q  2)  ( t )  *  ( t )  »  *"°°>  =  in (a,b). Thus Q(t) i s both a major and a minor function of F  (k + 2 ) ^  o  v  e  ^ i^  r  a  #  J t  f  o  l  l  o  w  s  t h  at  F  ( + k  2  ) ^  i s  - integrable over ( a ^ t ) , and (h.k) follows from the d e f i n i t i o n of t h e ^ °  k  +  2  - integral.  If the series (3.1)  THEOREM *t.3.  for a l l t i n [a,b] to a function £P(t) where [a,b]  r  i  In  where a  Proof.  - u(t) + i v ( t ) ,  Is a f i n i t e , closed i n t e r v a l and  then (Pit) [a»bj  i s summable (C,k)  s  k  +  2  ,|^(t)l  < oo ,  - Integrable over (a^;t) f o r every t a^ t  Series (3.1)  ... < a = b. n  integrated term by term k •+• 2 times  17  converges uniformly to a continuous function By Theorem 3.1  H(t) + i K ( t ) .  H  ( )( ) t  a  n  d  K  r  ( ) ^ ) exist t  r  and are f i n i t e f o r a l l t i n [a,b] , 1 < r < k + 2, and H  (k + 2)(t)  =  u ( t )  ' ( k •*- 2 ) ( t ) K  theorem H ^ ^ 2 ) ^ *  2  ~  ^  u  a  n  =  d  K  v ( t )  *  B  y  t  (k + 2 ) ^  h  P  e  =  v  r e v i o u s  ^  a r e  Hence u(t) + i v ( t ) - C^(t) i s  - integrable.  ^>k -f 2 _ integrable over ( a ^ ; t ) . LEMMA. p > 1,  is.  Under the hypotheses of Theorem +  Proof.  Clftlexpf-lptl,  - Integrable over ( a ^ j t ) .  2  By hypothesis c exp(int) - ^ ( t ) ,  (C,k),  n  n= o for a l l t i n [a,b] .  It follows that  c e x p ( i ( n - p)t) - ^ ( t ) e x p ( - i p t ) , n  (C,k),  n - o for a l l t i n [a,b] , and therefore that  (**#5)  2  c e x p ( i ( n - p)t) - - c exp(- ipt) n  D  n= p - Cp _ ^expC- i t ) +  ciexp(- i ( l - p)t) CP(t)exp(By Theorem ^ . 3 ,  0(t)  i p t ) ' (9(t), is ^  for every t i n [a,b] . ?°  k  +  2  k  +  2  (C,k).  - integrable over ( a ^ t )  I t follows that (^(t)exp(- ipt) i s  - integrable since each of the other functions on  the right hand side of (*t.5) i s ^ °  k +  2  - integrable.  18  Theorem k.l can now be proved. Proof. i s even.  Consider f i r s t the expression f o r c  Q  when k  It follows from Theorem k,2 and the proof of  Theorem k.3 that, <- 1) [  th.6)  <- i)Ck  +  i  ( k+  2  )  /  2  i  2)/2 / * # -4^)  ^K(t) -  +  k  ( t  )  f  2  d  * CP(t)d  p  k  ^"^Xct;  (k + 2)/2  +  +  t=H(t) -  2  ^j.)K( «<I)J  * [(k" + 2)/2]  5  where C ^ ) i s "the set (^.1).  k  t  k +2  y" X(t;^,)H(^ 2  1^1  = Q(t)  1  iR(t),  + 1»  The function H(t) + iK(t)may  be written i n the form ^ P ( t ) + B^Ct) + i H ( t ) , where S P ( t ) = 2  c t  k  Q  * /(k+2)' 2  period 2 T T .  2  I f t = 0, the r i g h t hand side of (*+.6) becomes  ^ p (0)  0*.7)  and B^Ct) and H ( t ) are periodic with  - ^^X(o;  ^ ^ P c ^ )  + [%((» i i  H (O)] 2  k +• 2 A(0; ^  -  [^(o) t i H ( o ) ] . 2  This reduces to  C  + 2 T T (- o q )  k Q  (k + + 2)1 2TT  I I  i = 1  (, 1) " • ""  (cf. James, £7] » p. 106).  Oc +2)/2 • n n 'k  (2rr) ^ k  C o  2  19  Hence  <2*r)  k  +  2  y  ( *  )  The expression f o r c  when k i s odd can be  Q  obtained i n a similar manner since (-1)  (if.8)  +  +  ^(t)d ' t  2  ?  I*  (- 1) Y  k  * ^  +  / *  1  ^(t)d  k  +  2*  =  (t) + H ^ t ) + i H ( t ) , • 2  (k + l ) / 2  s  r  "  £k + l)/2]  + 1»  •where (* ^) i s the set (*f.2). To obtain the formulae f o r c ,p > 1 , rewrite p  equation C+.5)  i n the form  A.9)  a_exp(Irt)=  r =0  ©(t),  (C,k)  r  where r - n - p, n = p, p +' 1 , ... , and a„ = a r n-p 7  7  The constant term i n (*+.9) i s a Tk~rc P  2TT  7  Q  = Cp, and f o r k even,  k + 2 iT- P1 (-  1)  L(k +  2)/2]  - k  +  2  A  g  20  (- i ) since  (  k  +  2  ^  ^(t)exp(- ipt)d  2  k  +  2  t,  s-  0 exp(- i m t ) d  k  ±  2  t  = 0.  (^ ^) It follows that c = ^  6?(t)exp(- ipt)d  kT~2  P  y  k  t.  •»•  2  For k odd the argument i s s i m i l a r . Remark.  The proof of Theorem M-.l.is simpler than  the proof of the analogous r e s u l t by James JV» pp. 105 106]  since the r e s u l t s on the formal m u l t i p l i c a t i o n of  trigonometric series are not needed.  -  21  5»  The Expression of C o e f f i c i e n t s i n Terms of the C <* +- l  p  ~ Integral  Suppose that the series (1.2) i s bounded  THEOREM.  * - 0 , 1, 2, ... jLn (- TT, 7r ) and  (C,<* ), f o r a f i x e d l e t r = «* +- 2.  I f F(t) and G(t) are the functions of  Theorem 3.2, the c o e f f i c i e n t s of series (1.2) are given by TT  (5.1)  a = _L_ f F , ( t ) c o s ( n t ) d t (r) n TT J  =  v  -rr  G. ( t ) s i n ( n t ) d t ,  (r) N  /  (5.2)  b - _L n ^  / F , . (t)sin(nt)dt = J (r) -n (t)cos(nt)dt, ^ (r)  TT  where the integrals are C ^ ^  - integrals.  To f i x ideas take <* - 2.  Proof.  It follows from  Theorem 3«2, Theorem 3.3, and the d e f i n i t i o n of the C  N  P -  i n t e g r a l that F (t) - F (--r) = F S  S  0 £ s < 3. for  1 f P  (t) - F  ( s )  (--)  = v / /  (  s  +  1  )  ( x ) d x ,  Hence, using the property of Integration  by parts  the C P - i n t e g r a l £2] ,  C^PF C  ( s )  n  ( 1 + )  F  ^ C  (t)cos(nt)dt (  2  )  (  t  )  n  2  c  o  s  (  n  t  )  d  t  2  P / ^ F  ( 3 )  (t)nsin(nt)dt =  " " 0^/^ <1>Ct>n slnCJa-t)« = c  F  3  22 77 CP / 0  since  F(t)n cos(nt)dt - a^tr lf  n - 1  ,  [a cos(nt) •*- b s i n ( n t ) J /n n  Q  convergent Fourier series of F(t)., follow s i m i l a r l y .  i s the  uniformly  The other r e s u l t s  23 6.  The C  k  +  ±  ?  - integral and  the ^°  k  +  2  It has been shown by James [6] i n t e g r a b i l i t y implies ¥° indefinite V  *  k  2  k  +  - integral  that C  - i n t e g r a b i l i t y and  2  i  r  - i n t e g r a l , the next one  i n t e g r a l , and CP D  that  indefinite  an i n d e f i n i t e C-  A relationship  the d e f i n i t e C : k  since, when series (1.2)  the  - fold  -jP -  so on, the outermost integral being an  - Integral.  integral and  -jP -  +  - integral i s equal to an (r + 1)  integral in which the innermost i n t e g r a l i s an CP  k  indefinite  between the d e f i n i t e <T  f  k  ]_P - integral suggests i t s e l f  4  i s summable (C,k), the theorem in  Chapter 5 and Theorem *+.l give analogous expressions for c o e f f i c i e n t s c , one  i n terms of the  n  other i n terms of the C THEOREM. and C_P (6.1)  +  - integral,  the the  -^P - i n t e g r a l .  Suppose that f ( t ) i s periodic with period 2 TT  , - ( r - 2)TT,  when r i s even, and  (6.2)  k  - integrable over (- T ,*T). - r<  -  2  the  - (r +• 1)  TT  Let ( *^)  , - 2T,  ...  be the  2-rr,  ...  set  , (r + 2)-TT,  set  , _ (r - 1) -rr,  2-w,  2-rr ,  ...  ,  (r - l)-rr , (r +• l ) * r , when r i s odd. F (x) r  If = CP  F (x) = C P k  k  /"x / — r  f(t)dt, 7/  F k  +  1  (t)dt,  0 i k £ r -  1,  2k  F(x) -  F(x),  o  then I? - 1 (2  /  )  it  r  f(t)d  r  +  t - CP r  • ( o ^ )  Proof.  Consider the case when r i s odd.  for r even i s similar.) (- 1) f  (6.3)  2  ( r  1-  1  )  /  (The proof  I t i s known ^6, p. 168^ that ^  2  +  r  +  f°  1  f(t)d  p  4  • t -  A * ) t  F(0) -  r •*• 1 i ^ 1  x  A (0;  ^  ) F ( < 1  . ) .  1  It i s easy to v e r i f y that F ( X ) = C P /  f(t)dtr j c _ / f ( t ) d t r IT 2 77 TT  X  x  +G (x)=  where G (x) i s periodic with period 2-n. 2  F{x) = ^ Mx  T  +  /  V(r •l)lj  p  *  XyU -v- G ( x ) , p  Y  < L  I t follows that  + G (x) + (a polynomial i n x Q  2""".  of degree r ) , where G (x) i s periodic with period Q  The r i g h t hand side of (6.3) thus reduces to ( - l )  fc-  (  1  r  +  ,  1  )  /  > ( 2  T  )  r  «t)a  the required r e s u l t .  -  1  / ! 7 -  1  « * . P- 18).  t - _i_ c p /  This y i e l d s  f(t)at,  25  7«  Uniqueness Theorems f o r Multiple Trigonometric Series  Since the triangular method of summation defined in Chapter 2 reduces the summation of a multiple trigonometric series to the Cesaro summation of a single series,  i t is  possible to use the r e s u l t s of the preceding chapters to obtain uniqueness theorems f o r multiple trigonometric series. THEOREM 7.1. a function CPjx^  I f the series (1.1) i s summable (T,k) to x , ... , x , t) = g  n  f o r a l l values (x^ -*• t , x  2  #?(x,t),  +• t , ... , x  Q  |6?(x,t)) < co ,  •+• t ) where  0 5 x^, x , ... , x , t < 2 "f then the c o e f f i c i e n t s of 2  Q  the series are given by  ^(x,t)e"  where m  1  n  + ... + m  n  = p,  i p t  d  k + 2  t d x ... dx , x  Ik and ( «• .) are defined as i n 1 —:  Theorem ^.1, and the n outer integrals are Riemann integrals. Proof.  For s i m p l i c i t y i n w r i t i n g , the proof w i l l be  given f o r n = 2. By hypothesis, ^  + ]>Z C (x ,x )e P i  p  p  1  2  for a l l t and 0 < x  1 ?  x  2  t  = ^(  X ; L  ,x ,t), 2  (C,k),  < 2-rr, I t follows from Theorem * K l  n  26  C (x ,x ) = 1  P  2  >  \  (7.2)  c  k  0 < x^,x  i  m  x  l  1  1  1  2  0 2  0  2  2  NOW l e t m and n be any two positive integers  such that m + n = p. e"  i(m,x, +• m x )  e  m  / 6^(x ,x ,t)e- i P * ^ , t ,  < 2TT.  2  m  Multiplying both sides of (7.2) by  and integrating with respect to x^ over (0,2'"')  yields (2T)(c  )e ^P ' - m . • i  m,p  2 t c mn e  (7.3) 2  'o  -V  l n x  2  m ) x  2 —  -  i» ( 2 7 r  Then, multiplying both sides of (7.3) by e"  l n x  2 and  integrating with respect to x~ over (0,2 t t ) gives (7.W  . 2TT  (2^) = 2-"- i(mx 1 + e (2 7)\ 2  r a n  0 0 the required r e s u l t .  n X  2  }  /  P  k  0 C^ (x ,x ,t)e- P d )  i  1  '(^)  2  t  k v 2  t  d x ^ ,  27  A more general r e s u l t i s obtainable i n a similar way i n virtue of the theorem of Chapter 5. THEOREM 7.2.  I f series  X  e m  c  i ( m  > \ x  7  i s bounded (T,k) f o r a l l values (x.^ + t , ... ,x + t ) , n  0 < x , x , ... »x ,t i 2tr, and i f r - k + 2, then the 1  2  n  series C (x ,x , ... ,x ) i p t p l 2 n e 7  7  7  (ip)  r  converges uniformly f o r a l l t and each (x) to a continuous function  ^ ( t ) , and the c o e f f i c i e n t s of the series are  given by  'n  1' 2 TT  2 n integrals  (SO"' 0  71  e" ' '0  2r" > ^ /^/ (t)e" P dtdx  1(m  2  x  integrated  (7.6)  (for n = 2)  t  +  - Integral.  The convergence property of the  The theorem of Chapter 5 gives C ( x , . x ) p 1 2 \ i(m,x, * m x„) /_ c e l l 22 ,——— _ m-iJ-' .jij + m s p cm1 ' 2 2T°k.  0  l f>(r)(t)j"1Ptdt» P  ^0  ...  '0  series i s obvious.  m  1  T  where the inner i n t e g r a l i s a C^ Proof,  : L  ( r )  28  and (7.5) follows from (7.6) i n the same way that (7.1) (7.2).  follows from  A uniqueness theorem which i s stronger than Theorem 7.1 may be obtained i n a very interesting manner. It i s stronger i n the sense that the set of n - tuples (x^,x , ... , X ) i n the hypothesis i s reduced to a 2  q  countable set. THEOREM 7.3.  I f f o r a countable set A of n - tuples  10' 20> n - 1 0' ni*' io' 20> n -1 and 1 = 1, 2, 3, ... , series (1.1) i s summable (T,k) to ( x  0  X  , x  x  f o r a l l values ( x  2,3,*..,  1 Q  f  + t,x  2 0  o  f  r  l  x  e  d  x  x  «• t , ... , x  , x  n i  + t), 1=1,  0 < t < 2 t t , then the series vanishes ident-  ically. Proof.  Only the case n = 3 w i l l be considered.  In view of the proof of Theorem 7.1, i t i s clear that  (7.7)  Y L °»»-" * * " ° " m * m + m = p 1 2 3 ' 1 2 3 for fixed 20 » > 3 , ... . This gives e  x  , x  a  n  d  i  =  1  1(  1  10  2 X 2  4  3 X 3 l )  =  2  1 0  y  fm ,m c  ,1 m^ + m^ * m^ = • p « • * • * ~ m. + m. =• o L * x  e  m  J  1 ( m  i 10 X  +  m  2 20 X  )  e  im.x_ 3 3i  29  5  m  m-^ + m + m^  =  P  2  r / c l m  l  , m  2  , D 1  e  3  i m  3 3i  e , m  =  x  ipx  3 i  e  - i(  m i  ,  +  m) 2  2 *P ~ l ~ 2 m  m  1 'm^,m,p - m^ - nig  e  - i(m + m ) 1  2  2  m  l  m  2  =  i = 1, 2, 3» ... •  I  (7.8)  ^  It follows that the equation  O  1' 2  1  , P  2  has a countable number of roots, 21 32 33 x  the l e f t hand side of (7.8)  ,x  iX  m  J  l *  m  2 * 3 m  =  p <  ra  i s a polynomial of degree p  p o s s i b i l i t y i s that c' _ _ _ _ m2,m ,p- m^ - m and hence that c 1' 2»^3 2  l * 2  ... . But  9  and cannot have a countable number of zeros. r  m  - iu e  2  The only  r cl = 0, m^,m,m ' such that 1 3 m  2  m 2  3  30  BIBLIOGRAPHY  1.  S. Bochner, "Summation of multiple Fourier series by spherical means", Trans. Amer. Math. Soc. v o l . *t0  (1936) pp. 175 - 207. s 2.  J . C. B u r k i l l , "The Cesaro-Perron scale of integration", Proc. London Math. Soc. (2) v o l . 39 (1935) pp. 5^1  -  552. 3.  , "The expression  of trigonometrical  series i n Fourier form", J . London Math. Soc. v o l .  11 (1936) pp. hZ - **8. k,  Min-Teh Cheng, "Uniqueness of multiple  trigonometric  series", Ann. of Math. v o l . 52 (1950) pp. h03  -  *4-l6.  5.  G. H. Hardy, "Divergent s e r i e s " , Oxford, 19*+9.  6.  R. D. James, "Generalized n - th primitives", Trans. Amer. Math. Soc. v o l . 76 (195 *) pp. 1**9 1  7.  176.  , "Summable trigonometric s e r i e s " , P a c i f i c J . Math. v o l . 6 (1956) pp. 99 - 110.  8.  J . Marcinkiewicz and A. Zygmund, "On the d i f f e r e n t i a b i l i t y of functions and summability of trigonometrical s e r i e s " , Fund. Math. v o l . 26 (1936) pp. 1 - 1*3.  9. 10.  S. Saks, "Theory of the i n t e g r a l " , Warsaw, 1937. W. L. C. Sargent, "On the Cesaro derivates of a function", Proc. London Math. Soc. (2) v o l . *+0  (1936) pp. 235 - 25^.  31 11.  W. L. C. Sargent (cont.), "A descriptive d e f i n i t i o n of Cesaro-Perron integrals", Proc. London Math. Soc.  (2) v o l . k? (19^2) pp. 212 - 2*+7. 12.  , "On generalized derivatives and Cesaro-Denjoy  integrals", Proc. London Math. Soc.  (2) v o l . 52 (1951) pp. 365 - 376. 13.  V. L. Shapiro, "An extension of r e s u l t s i n the uniqueness theory of double trigonometric s e r i e s " , Duke Math. J . v o l . 20 (1935) pp. 359 - 365.  lh.  , "A note on the uniqueness of double trigonometric s e r i e s " , Proc. Amer. Math. Soc. v o l .  (1953) pp. 692 - 695. 15. - S. Verblunsky, "On the theory of trigonometric series ¥11", Fund. Math. v o l . 23 (193*0 pp. 193 -  236. 16.  A. Zygmund, "Trigonometrical s e r i e s " , Warsaw, 1935.  `

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