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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' S t a t e C o l l e g e , 1963  M.S., San Diego S t a t e C o l l e g e , 1965  THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY  i n t h e Department of MATHEMATICS  We a c c e p t t h i s to the r e q u i r e d  t h e s i s as conforming standard  The U n i v e r s i t y o f B r i t i s h A p r i l , 1972  Columbia  In p r e s e n t i n g an the  thesis  in partial  f u l f i l m e n t of the  advanced degree at the  University  of B r i t i s h Columbia, I agree  Library  this  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  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 extensive for  s c h o l a r l y p u r p o s e s may  by h i s r e p r e s e n t a t i v e s .  be  thesis for financial  written  permission.  25,  gain  Mathematics  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a  Date A p r i l  g r a n t e d by  1972  the  Head o f my  Columbia  s h a l l not  be  for  that  study.  copying of t h i s  thesis  Department  I t i s understood that copying or  of t h i s  Department of  requirements  or  publication  allowed without  my  ii Chairman:  P r o f e s s o r Z. A. Melzak  ABSTRACT  A g l o b a l a n a l y t i c function i s uniquely by one o f i t s f u n c t i o n elements. i s i n t u r n completely  determined  the T a y l o r s e r i e s expansion  determined  T h i s f u n c t i o n element by the c o e f f i c i e n t s o f  about some p o i n t .  Therefore,  we s h o u l d be a b l e t o determine a l l o f the p r o p e r t i e s o f the f u n c t i o n from those o f the c o e f f i c i e n t s and by the formal p r o p e r t i e s o f T a y l o r s e r i e s .  D e t e c t i o n o f these  f u n c t i o n p r o p e r t i e s from those o f the c o e f f i c i e n t s i s the c e n t r a l problem o f the theory o f T a y l o r s e r i e s . a t e l y t h i s has proved and,  t o be an extremely  Unfortun-  difficult  problem  w i t h the e x c e p t i o n o f simple cases, very l i t t l e o f  the nature o f a f u n c t i o n i s known from the p r o p e r t i e s of  the c o e f f i c i e n t s . In Chapter  One o f t h i s t h e s i s , the c e n t r a l problem  of  the t h e o r y o f T a y l o r s e r i e s i s approached.  of  a f i x e d sequence o f c o e f f i c i e n t s , c e r t a i n rearrangements  of  the o r d e r o f the c o e f f i c i e n t s are a l s o c o n s i d e r e d .  Hopefully t h i s relaxation w i l l to  However, i n s t e a d  allow a d d i t i o n a l information  be d e t e c t e d . In p a r t i c u l a r , the P - p r e s e r v a t i o n s e t c o n s i s t s o f  those rearrangements o f the o r d e r o f the c o e f f i c i e n t s which p r e s e r v e a p r o p e r t y P.  This P-preservation s e t i s  iii maximal i n the sense t h a t each non-member 'maps' a f u n c t i o n w i t h p r o p e r t y P onto a f u n c t i o n without The  preservation sets  P.  (b^ten g r o u p s — a l w a y s semi-groups)  f o r s e v e r a l f u n c t i o n p r o p e r t i e s are  found.  T h i s concept o f p r o p e r t y p r e s e r v a t i o n a l s o examination of those  property  permits  p r o p e r t i e s which are i n v a r i a n t 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 s e r i e s i n t o s u b d i v i s i o n s d e t e r mined by subgroups o f S , the permutation group on  the  positive integers. In Chapter Two used t o sum  v a r i o u s summation p r o c e s s e s ,  the c o e f f i c i e n t s o f a power s e r i e s .  V-summation p r o c e s s e s ,  permutations which map  s e r i e s onto V^summable s e r i e s are d i s c u s s e d . o f Chapter One  i s continued,  v a t i o n s e t s are d i s c u s s e d .  V,  are  For  these  V-summable The  program  and the V-summability I t i s a l s o shown t h a t  preserthese  V-summability p r e s e r v i n g permutations leave the V-sum i n v a r 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 c o n s i s t i n g of a n a l y t i c f u n c t i o n p r o p e r t i e s on one and  the c o r r e s p o n d i n g  on the o t h e r s i d e . a beginning.  p r e s e r v a t i o n groups and  lattice side  semi-groups  This l a t t i c e c e r t a i n l y represents  Hopefully i t could lead to a d d i t i o n a l  i n s i g h t i n t o a n a l y t i c f u n c t i o n theory o r p o s s i b l y even l e a d to i n f o r m a t i o n about the subgroups o f S . 0)  only  iv  TABLE OF CONTENTS  Page Number Introduction  1  Chapter  5  One  P r e s e r v a t i o n Groups  5  Rational Functions  6  P o s i t i o n and Number o f Poles  10  A l g e b r a i c Functions  12  Radius o f Convergence  16  Entire  FUHULiona  Order o f E n t i r e F u n c t i o n s Type o f E n t i r e F u n c t i o n s  Chapter  23  28 32  35  Two  Summability  35  C o n d i t i o n a l l y Summable S e r i e s  36  Cesa.ro Means  41  P r e s e r v a t i o n Groups Conclusion Bibliography  =  55 56 60  V  ACKNOWLEDGEMENT  I Z.A. I  Melzak,  words this The  of  like  t o t h a n k my s u p e r v i s o r ,  f o rsuggesting  am e s p e c i a l l y  and of  would  grateful  the topic  Professor  of this  thesis.  f o r t h e encouragement,  o f wisdom he gave  me d u r i n g  advice,  the preparation  thesis. f i n a n c i a l support of the National  Canada and o f t h e U n i v e r s i t y  gratefully  acknowledged.  Research  Council  o f B r i t i s h Columbia i s  INTRODUCTION  The has  development o f the theory o f a n a l y t i c  f o l l o w e d two b a s i c  approaches:  the g l o b a l  functions Cauchy-  Riemann approach and W e i e r s t r a s s ' approach through power series  and f u n c t i o n  elements.  In t h i s t h e s i s we b e g i n  w i t h W e i e r s t r a s s ' approach. C o n s i d e r the sequence o f T a y l o r s e r i e s of the power s e r i e s  f(z)= E f z  origin for simplicity.  n  n  coefficients  expanded about the  (We s h a l l r e f e r t o the terms o f  t h i s sequence as the c o e f f i c i e n t s o f f o r simply as the coefficients.)  The c o e f f i c i e n t s o f f completely determine  a corresponding global  function.  We s h o u l d t h e r e f o r e be  able to d e t e c t a l l p r o p e r t i e s o f the f u n c t i o n  from those  of the c o e f f i c i e n t s and by the formal p r o p e r t i e s o f T a y l o r series.  D e t e c t i o n o f these f u n c t i o n  p r o p e r t i e s from those  of the c o e f f i c i e n t s i s the c e n t r a l problem o f the theory of T a y l o r  series.  It i s surprising function For  how l i t t l e o f the nature o f the  i s known from the p r o p e r t i e s o f the c o e f f i c i e n t s .  example, how can an a n a l y t i c  s e r i e s be r e p r e s e n t e d by a s i n g l e  function formula?  Hadamard g i v e s the r e l a t i v e p o s i t i o n  g i v e n by a T a y l o r A r e s u l t by  of p o l a r s i n g u l a r i t i e s  l y i n g w i t h i n the c i r c l e o f meromorphy.  Beyond Hadamard's  2  r e s u l t , not much i s known, y e t the p o s i t i o n o f s i n g u l a r p o i n t s i s o f paramount importance i n f u n c t i o n t h e o r y . (An e n t i r e f u n c t i o n i s c h a r a c t e r i z e d by the f a c t t h a t  00  i s i t s o n l y s i n g u l a r p o i n t , meromorphic f u n c t i o n s have o n l y p o l e s i n the f i n i t e complex plane, e t c . )  In Chapter One o f t h i s t h e s i s , the c e n t r a l problem of the  theory o f T a y l o r s e r i e s i s approached, but i n s t e a d  of a f i x e d sequence o f c o e f f i c i e n t s , c e r t a i n rearrangements of the o r d e r o f the c o e f f i c i e n t s are a l s o c o n s i d e r e d . H o p e f u l l y t h i s r e l a x a t i o n w i l l allow a d d i t i o n a l i n f o r m a t i o n t o be d e t e c t e d . J-J-  „  „  _ .  j. *  ^ J ^ ^ J - V.J-  w..  J. .  .XW..  can the o r d e r o f the c o e f f i c i e n t s o f t h i s f u n c t i o n be a l t e r e d so t h a t the new f u n c t i o n s t i l l  enjoys  p r o p e r t y P?  That i s , which elements o f S , the permutation  group on  the p o s i t i v e i n t e g e r s , map the f u n c t i o n f onto a new f u n c t i o n , f * , both e n j o y i n g p r o p e r t y P?  (The f u n c t i o n s  f and f * have the same s e t o f 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 o r d e r . from S  i s a c t u a l l y a map on t h i s o r d e r .  language w i l l be r e c t i f i e d  later.)  The  permutation  T h i s abuse o f  A permutation,  TT ,  p r e s e r v e s p r o p e r t y P i f f o r a l l f e n j o y i n g P, Tr(f) a l s o enjoys  P.  The s e t o f a l l permutations p r e s e r v i n g P i s  the P p r e s e r v a t i o n s e t .  Thus t o each p r o p e r t y P, t h e r e  3  corresponds a p r e s e r v a t i o n The  set  concept of p r o p e r t y  ( o f t e n a group).  preservation  allows  of those p r o p e r t i e s which are i n v a r i a n t under permutation groups.  examination  various  T h i s y i e l d s a d i v i s i o n of the  totality  of a l l power s e r i e s i n t o s u b d i v i s i o n s 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 n a l y t i c f u n c t i o n s by  a s s o c i a t e d Riemann s u r f a c e s .  Today  we  their  know t h a t each  Riemann s u r f a c e of an a l g e b r a i c f u n c t i o n i s t o p o l o g i c a l l y a sphere w i t h p handles, and is a single-valued The  Erlanger  considered  t h a t the a l g e b r a i c  function  f u n c t i o n of the p o i n t s on t h i s Programme i s another example.  groups of t r a n s f o r m a t i o n s  i n space.  surface. Klein  The  i n v a r i a n t t h e o r i e s of these groups each y i e l d a d e f i n i t e k i n d of geometry, and obtained  every p o s s i b l e geometry can  i n t h i s way.  p r o j e c t i v e geometry and  In Chapter Two used t o sum  K l e i n obtained  metric,  affine,  summation processes,  the c o e f f i c i e n t s of a power s e r i e s . permutations which map  s e r i e s onto V-summable s e r i e s are of Chapter One  and  a l s o topology through t h i s program.  various  V-summation p r o c e s s e s ,  be  i s continued,  v a t i o n s e t s are d i s c u s s e d .  and  considered.  V,  are  For  these  V-summable The  the V-summability  I t i s a l s o shown t h a t  program preserthese  V-summability p r e s e r v i n g permutations l e a v e the V-sum invariant. There are a l r e a d y two r e s u l t s i n t h i s f i e l d .  First,  the a b s o l u t e sum i s i n v a r i a n t under a l l rearrangements. Second,  Levi  [ 2 2 ] c o n s i d e r e d and found the p r e s e r v a t i o n  s e t f o r c o n d i t i o n a l summability. of L e v i ' s r e s u l t i s p r e s e n t e d .  A new, s h o r t e r p r o o f L e v i ' s theorem  g e n e r a l i z e d and proved f o r Cesaro  i s then  K-summability.  Many r e s u l t s i n r e l a t e d areas may be found i n the literature. Erdos  In a s o r t o f a n t i p o d a l r e s u l t , Bagemihl and  [ 4 ] c o n s i d e r e d the V-sum s e t (The s e t o f sums o f  a l l p o s s i b l e convergent rearrangements The sum s e t f o r a b s o l u t e convergence f o r c o n d i t i o n a l convergence continuum,  of a series.)  i s a single  point,  (not absolute) i t i s the  and f o r C- sums i t i s e i t h e r o f the form L  {a + bj} ,  j = 0, ± 1 , ± 2 , •••  o r i t i s g i v e n by Riemann s theorem. 1  5  PRESERVATION GROUPS  C o n s i d e r the f u n c t i o n oo  =  f(z)  I  f  n=Q  under rearrangements  n  a n a l y t i c at zero,  o f the o r d e r o f the c o e f f i c i e n t s .  Denote by S^ the permutation integers.  Z N  group on the p o s i t i v e  I f TT i s a permutation  b e l o n g i n g t o S , then  oo  f*(z). =  n I f •rr (n) n=0  i s a new f u n c t i o n o b t a i n e d from f by a rearrangement o f the c o e f f i c i e n t s o f f .  F o r convenience  of notation,  denote the f u n c t i o n f * ( z ) by T r [ f ( z ) ] o r simply by T r ( f ) .  The  permutation  TT p r e s e r v e s p r o p e r t y P i f , f o r a l l  f e n j o y i n g p r o p e r t y P, TT (f) a l s o enjoys p r o p e r t y P. set  o f a l l permutations  TT & S^ which p r e s e r v e p r o p e r t y P  i s c a l l e d the P - p r e s e r v a t i o n s e t .  Any p r e s e r v a t i o n s e t  i s n e c e s s a r i l y c l o s e d under composition at  l e a s t a semi-group.  The  and i s t h e r e f o r e  6  RATIONAL FUNCTIONS  The  following  are two well-known e q u i v a l e n t  of the s e t R o f r a t i o n a l f u n c t i o n s a)  descriptions  a n a l y t i c a t zero [26]:  R i s the s e t o f q u o t i e n t s P(z)/Q(z) o f p o l y n o m i a l s  w i t h complex c o e f f i c i e n t s and w i t h Q(0) ^ 0. b) regular  R i s the s e t o f power s e r i e s E f z n  n  that are  a t 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 . ~ i n+i j=u -  (n>n ) . 0  L  We s h a l l use the. f o l l o w i n g Theorem 1.1:  n  r e s u l t o f Szego  I f among the c o e f f i c i e n t s f  [12]:  of a Taylor  series  f ( z ) = £ f z , t h e r e are only a f i n i t e number o f d i f f e r e n t n  n  numbers, then e i t h e r f ( z ) = P ( z ) / ( l - z ) , where P(z) i s m  a p o l y n o m i a l and m a p o s i t i v e i n t e g e r , not  o r e l s e f ( z ) can  be c o n t i n u e d beyond the c i r c l e o f convergence o f Z f z . n  n  An  arithmetic  sequence i s a sequence •\n^)> l  common d i f f e r e n c e  p = n^. ^ - n^.  common d i f f e r e n c e  i s relevant,  sequence.  +  with a  When the value o f the  we use the term  p-arithmetic  7  Consider a permutation TT £.  .whose domain can be  w r i t t e n as a union o f a r i t h m e t i c sequences  i n such a way  t h a t the image under TT o f each o f these a r i t h m e t i c i s again an a r i t h m e t i c sequence.  Refer t o these  sequences  sequences  as t h e domain and range sequences, r e s p e c t i v e l y . For such a permutation, the union o f a r i t h m e t i c sequences  can be c o n s i d e r e d d i s j o i n t without any l o s s  of g e n e r a l i t y .  I f t h e r e were two d i s t i n c t , i n t e r s e c t i n g ,  domain a r i t h m e t i c sequences, d i f f e r e n c e s , say p and q.  they must have unequal common  The f i r s t sequence  w r i t t e n as the union o f q sequences  c o u l d be  with common d i f f e r e n c e  pq, the second as t h e union o f p (pq)-sequences. o r i g i n a l p, q-sequences (pq)-sequences.  These  The two  can then be r e p l a c e d w i t h the are d i s j o i n t s i n c e no two d i s t i n c t  (pq)-sequences i n t e r s e c t . I f TT can be c h a r a c t e r i z e d by the mapping o f a f i n i t e number o f domain sequences,  TT i s an a r i t h m e t i c permutation.  Since the domain o f TT i s s p l i t i n t o a f i n i t e number o f d i s j o i n t a r i t h m e t i c sequences,  let p  1  m u l t i p l e o f these common d i f f e r e n c e s .  be t h e l e a s t common I f p i s the minimum  of a l l p o s s i b l e p' , we say IT i s a p - a r i t h m e t i c  Theorem 1.2:  Arithmetic  permutation.  permutations p r e s e r v e r a t i o n a l  functions.  <Pj?oof S Let  f ( z ) = Ef z n  be a r a t i o n a l  11  f u n c t i o n and IT  be a p - a r i t h m e t i c permutation. Then, p-1 ir(f) = ir Ii=0  i  I  p-1 =  l - z  1  p  and z / ( l - z ) . 1  ] p  i  [TT( f o - 2 —  I  i=0 where f° [ z / ( l - z ) ]  -Z—  fo  l - z  ) ]  p  r e p r e s e n t s the Hadamard product o f f  The Hadamard product p r e s e r v e s  P  hence the products their coefficients  f° [ z / ( l - z ) ] 1  rationality,  are a l l r a t i o n a l  P  and  appear i n a p - a r i t h m e t i c sequence which  TT maps onto another a r i t h m e t i c  sequence.  We thus have i IT c  f ° - — p  z  )  =  g  k of  °  l-z*  1 - 5 8  where the non-zero  k  coefficients  of  fo[z /(l-z )] 1  p  and  cr  g o [ z /(l-z )]  are i d e n t i c a l  H  Since  f°[z /(l-z )] 1  p  satisfying N ^  j p(n+j)+i  i n v a l u e and o r d e r .  i s rational,  t h e r e are c^  9  Hence N :i  and  D q(n+])+k y  0  k k g o [z / ( 1 - z )]  Theorem 1.3:  i s rational.  I f the permutation TT p r e s e r v e s  rationality,  then rr must be a r i t h m e t i c .  Proof:  Assume TT i s not a r i t h m e t i c .  Then e i t h e r 1 o r 2  holds: 1.  Each i n t e g e r i n the domain o f TT belongs t o some  a r i t h m e t i c sequence whose image i s again an a r i t h m e t i c sequence, but rr cannot be c h a r a c t e r i z e d by the mapping o f a f i n i t e number o f domain sequences. p r e s e r v e r a t i o n a l i t y because E C j f 2.  n +  j  Then TT cannot  is a finite  property.  There i s an i n t e g e r b i n the domain o f TT such  t h a t the image o f each a r i t h m e t i c sequence  <an+b)> ,  a > 1, f a i l s t o be an a r i t h m e t i c sequence.  Then by Szego,  TT ( f ) cannot be r a t i o n a l  471 \ f (z)  =  2  v  z  where  an+b  10  POSITION AND  A permutation TT 6 for  NUMBER OF POLES  is called finite  a l l n l a r g e r than some N.  then f and TT (f) d i f f e r  i f TT (n) = n  I f TT i s a f i n i t e  only by a p o l y n o m i a l .  permutation  I f TT i s a  f i n i t e permutation, then the i n v e r s e permutation, TT ^ i s also f i n i t e .  Theorem 1 . 4 : if  f(z) =  The  R  f i n i t e permutation s e t i s a group.  I f F i s the group of f i n i t e permutations  Ef z n  11  i s a Taylor  and  s e r i e s , then  f and Tr(f) have the same s i n g u l a r p o i n t s on the c i r c l e of convergence for a l l f, =  Proof:  {  TT  £ S :  f and TT (f) have the same s i n g u l a r p o i n t s w i t h i n the c i r c l e of meromorphy for a l l f.  Each f i n i t e permutation o b v i o u s l y  s i n g u l a r p o i n t s on the c i r c l e moves an i n f i n i t e  preserves a l l  o f convergence.  Hence i f TT  number of c o e f f i c i e n t s , i t i s only  n e c e s s a r y t o e x h i b i t a power s e r i e s t h a t f and  }  f(.z) = E f z  Tr(f) do not have the same s i n g u l a r  n  n  so  points.  I f TT i s not an a r i t h m e t i c permutation, then i t maps a r a t i o n a l f u n c t i o n onto a f u n c t i o n w i t h a n a t u r a l  boundary  Hence l e t TT be a r i t h m e t i c w i t h p e r i o d p and l e t h  11  be (h  t h e maximum d i s t a n c e = s u p | TT ( n ) - n | ) . n  sequences  onto  Consider  Further,  q-arithmetic  TT m u s t  +  ;  x  k=Q  n  [  1  sequence  i s  moved  map p - a r i t h m e t i c  sequences.  f ( z ) = 1/(l-z^ ^) , q  where  any arithmetic  then  k  _ (ph+l)q z  ]  n^ < h .  Since  +  has of not  degree  z  (ph+i)  (q-i)  (ph+1) C3-1)  t h e denominator possessed  by f.  IT-Z  which  i s greater  than  the function  t h e degree  ir(f)  has poles  12  ALGEBRAIC FUNCTIONS  Definition:  The f u n c t i o n  f(z) represents  f u n c t i o n i f f s a t i s f i e s an a l g e b r a i c P(z,f) where P ( z , f ) number  be a l g e b r a i c .  n  n  m = 0, 1,  y L  ^_ n=0  represents  f  z  pn+m TTn+m ^  (p-1),  We d e f i n e  y  oo = 1. P  =  Z , p l-z^  _  C q  function.  = I  n=0  g z n  n  = f(.z) -  u f(03z: h  T h i s o p e r a t o r has the f o l l o w i n g  1.  g =0 ^n  2.  F o r each n there  g = P (to) f ^n n n  the power s e r i e s  an e r a s e r o p e r a t o r g ^ :  [f(z)]  h  Then f o r a l l p  m  p pn+m n ™  an a l g e b r a i c  g  where  = 0  i s a p o l y n o m i a l i n z and f over t h e complex  Let f(z) = E f z  all  Proof:  equation  field.  Lemma: and  an a l g e b r a i c  i f and only  and  1  i f f n  =0  i s a polynomial  P (co) = P , (co) . n ' n+p v  or P  properties:  n = -h(mod p ) ; ^ n  such t h a t  13  3.  I f f ( z ) i s a l g e b r a i c , then  g, [ f ( z ) ] np  i s also  algebraic.  These p r o p e r t i e s g  h p  are e a s i l y v e r i f i e d by expanding  [f(z)]:  g  [ f ( z ) ] - f ( z ) - a) f(wz) n  I (1 - w n=0  =  p  Now t o prove the lemma, c o n s i d e r gj the  where  h = 0, 1,  f i r s t operator to  function yields  (p-1) and  n + h  )f  z  n  the o p e r a t o r s h ^ m.  Apply  f ( z ) , the second o p e r a t o r t o the  j u s t o b t a i n e d , and so on.  Each a p p l i c a t i o n  an a l g e b r a i c f u n c t i o n w h i l e r e p l a c i n g  sequences o f c o e f f i c i e n t s w i t h z e r o s .  p-arithmetic  After  (p-1) a p p l i -  c a t i o n s we o b t a i n n-1  CO  { n  g. • } {f (z)} = • I  h=0  h  P  1  n  =o  P  P  n  +  m  . (u)f , z P  n  +  P  n  +  m  m  h^m  P(u)  I  f  , zP  n + m  which i s a l g e b r a i c ,  Denote by  .  n  the s e t o f permutations which map  14  a l g e b r a i c f u n c t i o n s onto a l g e b r a i c f u n c t i o n s .  We then  have the f o l l o w i n g theorem.  Theorem 1.5:  = Algebraic preservation set = Arithmetic  Proof:  permutation group.  P i c k any a r i t h m e t i c permutation TT w i t h p e r i o d p.  Then TT maps p - a r i t h m e t i c progressions, Let  progressions  onto  q-arithmetic  f o r some q.  f(z) =  T.f^z  be any a l g e b r a i c f u n c t i o n .  n  Then,  Trtftz)]  =  TT{  P7  1  I  I  f  k=0 n=0 p-1  <»  k=0  n=0  k=0 where the i n t e g e r the  'shift  a^  distance'.  z p  n  +  pn+k  K  ,, ^  n=0  n  q  K  i s dependent on k and This i s a f i n i t e  each a l g e b r a i c by the previous  by  an a l g e b r a i c  sum o f f u n c t i o n s ,  lemma.  I f TT i s not a r i t h m e t i c , then Szego provides  represents  (Theorem 1.1)  ( r a t i o n a l ) f u n c t i o n which, i s mapped  TT onto a f u n c t i o n with a n a t u r a l boundary.  15  The degree of ir(f) has an upper bound which i s a f u n c t i o n o f the o r i g i n a l degree of f and of the permut a t i o n TT. I f f s a t i s f i e s an i r r e d u c i b i l e p o l y n o m i a l i n z and f , P(z,f) = P ( z ) f  + P (z)f "  n  n  0  where each  + ••• + P ( z )  1  1  n  ^ ( z ) i s a p o l y n o m i a l i n z over the complex  numbers, then f i s a l g e b r a i c of degree n. the  We then have  following: 1.  I f a) i s complex, then  f ( u z ) i s a l g e b r a i c of  degree n. T-F  O  Vt  n c  a  nnoi f i tT O  i  n f orrar  f h o n  /,\  •£ ( rr \  - i c  a l g e b r a i c of degree n. 3.  The product  z^fCz)  i s a l g e b r a i c and of degree  at most n. 4.  I f g i s a l g e b r a i c and o f degree m, then  i s a l g e b r a i c and of degree at most 5.  If  g,  g,  (f)  i s a t most n  2  p  and  p  at most 6.  [18, p l 0 2 ] .  i s the e r a s e r o p e r a t o r , then the degree P  of  nm  f + g  _  1  [ I I g, ] (f) h=0 h^m  has degree  P  n  2  p - l,  The degree of  ' TT Cf)  r  2  i s a t most  (n  p-l  p - l p--l ) = n 2  4  16  RADIUS o f CONVERGENCE  Lemma;  L e t A = {TT :  n o n - a b e l i a n group.  IT Q.  and TT (n) ^ n}.  Then A i s a  We s h a l l r e f e r t o A as t h e asymptotic  group.  Proof:  I t i s o n l y necessary t o show c l o s u r e and t h e  e x i s t e n c e o f an i n v e r s e . If  TT. £ A, then t h e r e e x i s t s an  n(l  - e) ±  < 7r (n) < n ( l + e ) i  ±  and hence f o r the composition  Tr^[Tr2]  e. > 0  such t h a t  ,  i = 1, 2  we have  n ( l - e') < n ( l - e") < n ( l - e ) (1 - e ) 2  ±  <  T T (n) (1 - e x ) < •n [ T T (n) ]  <  1  2  2  and  Tr [Tf (n)] 1  2  < Tr (n)(l  = n ( l + e' ) .  2  + e )  < n ( l + e ) (1 + 2  e) ±  17 To show the i n v e r s e i s i n A p i c k e so t h a t  1 - e < 1/(1 +  and  1 + e > 1/(1 -  .  Then f o r the i n v e r s e o f T T ^ , we have  m(l - e) < m / ( l +  and  TT^ 6 A  whenever  1  Theorem 1.6:  1  (m)  < m/(1 - e ) < m ( l + e) 1  TT^ <£ A.  The Asymptotic group, A  fTT £ S : L  TT p r e s e r v e s radius} c  CO  of  Proof• Let  TT"  <  f (z) = If z n  n  convergence.  where l i m I f 1  n  1  |  n  = R . 1  F i r s t we  show t h a t i f TT i s asymptotic, then TT p r e s e r v e s r a d i u s o f convergence. We break the p r o o f i n t o two cases depending a l l the c o e f f i c i e n t s  Case I .  on whether  are e v e n t u a l l y l e s s than one o r n o t .  | f | <_ 1  f o r a l l s u f f i c i e n t l y l a r g e n.  T h i s case f o l l o w s from one s t r i n g o f 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 f o r the two not so obvious  inequalities  18 are g i v e n  i n a and b below.  -.1 _  e )  l/(l-e)  f  —(a)  1  n  ,1/nd-e) 1  m  |  —  1  n  ,1/Tr(n) 1  and  ra|f |  < HH|f |  1A(n)  n  1 / n { 1 + e )  n  Now  to v e r i f y a and b:  a)  given  lim|f |  <  = R  1 / / n  n  ( b )  _ 1  (R  _ 1  + ejVd+e)  i f and o n l y  if e >0 and N, there e x i s t s <% n > N such t h a t  f j  or  1  /  n  > (R  _ 1  -1  _  - e)  equivalently  f n  ,l/n(l- ) £  >  (R  £ )  l/(l- ) e  and b)  given  e > 0, there e x i s t s  N such t h a t n > N  implies  f j  1  /  < CR"  n  1  + e)  or e q u i v a l e n t l y j  f  |l/n(l+e)  <  (R  -1  +  e )  l/U+e)  19 Hence  limIf 1  l-*-/ ^ ) = R 1  n  1  Case I I :  11  f o r case I .  1  F o r each p o s i t i v e i n t e g e r N, t h e r e  n > N such t h a t  ]f | > 1.  i s an  T h i s proof f o l l o w s t h e same  o u t l i n e as case I .  -1 _  ,1/U+e) , ' —(a) ra  v  ,1/nd+e)  f  n  1  ^  <  —  1  1  n  ,1/TrCn) 1  and  Tim | f | 1  Now a)  n  < IT5|f — n  1 / l T ( n )  1  1  lim | f | given  = R  1 / / N  |  1  /  N  (  1  1  "  E  )  <,..  ( R "  —(b)  1  + e)  1  /  n  (  1  +  £  > (R- - e) 1  > ( R - -  )  1  s )  1  /  (  1  +  £  )  and b)  given  e > 0, there  e x i s t s N such t h a t n > N  implies If  or  equivalently  |  n•  1  /  N  /  (  1  -  £  e x i s t s .; n > N such t h a t  equivalently  | f J  1  i f and o n l y i f  1  e > 0 and N, t h e r e  IfJ /"  or  1  <  ( R  _  1  +e)  )  .  20  |l/n(l-e)  K  { R  -1  +  e )  l/U-e>  Hence sz i l/fr (n) lT~-—i im f \ ' = R„-l n  and t h e r e f o r e asymptotic permutations p r e s e r v e the r a d i u s of convergence. F i n a l l y , we  show t h a t the s e t o f r a d i u s o f convergence  p r e s e r v i n g permutations i s a subset o f the a s y m p t o t i c group. Suppose IT p r e s e r v e s r a d i u s o f convergence. lim  n  Tr(n)/n =  We want t o show  1.  Suppose not.  Then  l i m Tr(n)/n = a > 1,  Consider oo  z  n=0  2  n  where R  Now, a)  -1  = lim  _-n11/n _ 1 1  1  from the d e f i n i t i o n of  2  l i m Tr(n)/n = a, we have  g i v e n e > 0, there e x i s t s an N such t h a t n > N  implies < (a + e)  or  u(n) < n(a  +e)  21  and b)  g i v e n e > 0 and N, there e x i s t s  * n > N such  that IT  (n)  >  n  (a - e)  or  Tr(n) > n ( a - e)  Using these two e x p r e s s i o n s to e v a l u a t e we  lira] 2 |-'-/' ( ) n  IT  n  obtain  n£| -n|l/ir(n) 2  =  2  ±  ^  n  5  |  ~n j 1/n (a+s)  2  =  m  2  -l/(a+e)  =  -l/(a+e)  and i3H|2- | n  >  1 / i r ( n )  Ti¥| - | n  ( b )  1 / n ( a  2  -  e )  =  _ 2  1  /  (  a  -  e  .  )  Thus HE| -n|lA(n) 2  =  2  ~ l / a  which has the same r a d i u s of convergence o n l y when  Remark: Ef z n  n  The above counterexample i l l u s t r a t e s  a power s e r i e s  w i t h r a d i u s o f convergence 2, mapped onto a  power s e r i e s w i t h r a d i u s o f convergence 2  1 / / a  .  a = 1.  new  It i s  i n t e r e s t i n g t o note t h a t the r a d i u s : o f convergence cannot be moved a c r o s s  |z| = 1 .  Suppose there was  and a f u n c t i o n f where R(f) > 1 w h i l e  a permutation TT  R[Tr(f)] < 1.  Then  22  f(l) is  = Ef  i s absolutely  absolutely  c o n v e r g e n t and h e n c e  convergent which i s absurd.  7r[f(l)]  =f(l)  23  ENTIRE FUNCTIONS  We have been f o r m u l a t i n g  a dual l a t t i c e composed o f  f u n c t i o n p r o p e r t i e s and p r e s e r v a t i o n groups. e n t r i e s enable p r e d i c t i o n o f i n f o r m a t i o n new e n t r i e s . preservation  As an example, c o n s i d e r  The p r e s e n t  about p o s s i b l e  the e n t i r e f u n c t i o n s  set.  For an e n t i r e f u n c t i o n , Cauchy's n - t h r o o t equals  zero:  The f u n c t i o n  R  Obviously  _ 1  f(z) = Ef^z  = Tim" I f 1  n  1  !  1  /  n  11  test  i s entirei f  = 0.  the s e t o f permutations p r e s e r v i n g e n t i r e  f u n c t i o n s w i l l c o n t a i n the s e t p r e s e r v i n g  r a d i u s o f conver-  gence . With any f u n c t i o n , Cauchy's n - t h r o o t t e s t a unique  R  1  which i s non-negative.  associates  Now g i v e n any  e n t i r e f u n c t i o n f , permuting i t s c o e f f i c i e n t s can o n l y make the  R  1  associated with  f larger.  From t h i s i t  would appear the o n l y r e s t r i c t i o n on a member o f S , g  the e n t i r e f u n c t i o n p r e s e r v a t i o n s e t , i s t h a t there i s some upper bound on t h e forward t r a v e l o f the permutations.  Following  t h i s reasoning,  TT c o u l d have unbounded  t r a v e l i n the r e v e r s e d i r e c t i o n .  But these are n o t  symmetrical t r a v e l c o n d i t i o n s . t h a t the o n l y be  set S  w i l l not  e  T h i s tends to i n d i c a t e  contain  a l l inverses  s h a l l follow t h i s o u t l i n e , proving  members of S  are c h a r a c t e r i z e d by  g  [TT (n) = 0 (n) .. or simply contain  inverses  Theorem 1.7:  and  If S  entire functions,  Proof:  First,  TT = 0 ( n ) ] , and  that S  i s consequently not  i s the  g  then  e  {TT <£" S  :  I  ^  71  S  the  does  g  preserving  TT (nl = 0 (n)_ 1 .  : W  TT = 0 (n) } C  S  . Q  Let f be  any  R"  let  n  1  = U S i|f i |  1 / / n  given  f ( z ) = Z f z , where n  n  = l i m | fn |  1 / n  n  TT = 0 (n)  Now, lim|f |  entire function,  lequivalently l e t e  > 0,  n  = 0, t h e r e  let  e  1 / n  CTT) = £„. 0'  e x i s t s an  = 0  TT Cn) ;  < nk ^  I  1 / n  such t h a t  k £^  =  e  equivalently |f 1 n 1  1  1 / n k  ~ < e  J  Since  implies If  not  a group.  set of permutations =  s  that  an upper bound  —————  or  will  a semi-group.  We  and  and  for n > N •e  n >  25 Rewording: I1 f  n  j  1  /  /  n  k  1  Given  e > 0, t h e r e e x i s t s an N  < e whenever  n > N  e  .  Hence  such t h a t  £  l i m If 1  n  1  1  1 / / n k  = o.  F i n a l l y , using t h i s l i m i t i n addition to  ^ |f 1  ^ 1/TT (n) „ i^. ,1/nk | < |f | / n 1  n  /  1  T  W  x  1  1  for  n K  1  1  |f | < 1 n 1  yields T . i£ i 1/TT (n) . lim f ' = 0. ' n 7  v  1  Second,  S e  —  Suppose not. TT f- 0 ( n ) .  IT = 0 (n) } .  C J ^ G S : 1  co  * u e S  Then there must be some  where  G  T h i s means, f o r each N and k, there e x i s t s an  n > N such t h a t Let N = k.  TT (n) > nk. Then t h e r e e x i s t s TT(n ) > n k k  k  n, > k such t h a t k > n g[n ) k  k  where g(n^) i s a p o s i t i v e i n c r e a s i n g f u n c t i o n of n , k  n f(.z) = Ef z n n and A i s a c o n s t a n t . Here Now  consider  R"  1  = lim|f |  Next, f o r '  n  1 / n  . where  , -A-n'g(n) f = e n  = lim|e * 9 > | =  TT (f) = Ef z n  A  T t ( n )  ( n  0  (Hence e n t i r e )  R  1  = lim  -A*n«g(n)  > lim  The s e t  a semi-group w i t h  Proof:  Closure  - s - i — -A-n*g(n) = lim e ^  l/n«g(n)  1/TT (n) ^ ' >  -A  TT <£ S . e  This contradicts Theorem 1.8:  n  l/iT(n) '  = {TT £ S^:  TT (n) = 0 (n) }  forms  identity.  f o l l o w s immediately from the d e f i n i t i o n  of S , hence i t i s a semi-group. g  The s e t S  e  does not form a group.  The f o l l o w i n g i s  an example o f a permutation which does not p r e s e r v e e n t i r e t y , y e t i t s i n v e r s e does. Let  and d e f i n e TT as f o l l o w s : For a, b ^ 1 0 For  n = 10 , l e t  x  and  a < b, r e q u i r e t h a t  TT (n) = m  where m i s p i c k e d t o s a t i s f y  1/m 10 ! X  TT (a) < Tr(b).  1 -  x  T h i s i s e q u i v a l e n t t o p i c k i n g m l a r g e r than  27  ra >  giW  lo  >  0.  log(l-i) F i r s t , observe t h a t TT (f) i s not e n t i r e  R  1  = limIf 1  However, Tr (m) < m -1  -1  TT  I  l/ir(n)  lim  n  1  1/m  1 10  because  =  1.  !  (EL S : e  f o r those m which are the images o f powers  of 10 and TT-^Cn)  < n + p(n)  where  of 10 which i s l e s s than n.  p(n)  i s the h i g h e s t power  28  ORDER OF ENTIRE FUNCTIONS  Definition:  The e n t i r e f u n c t i o n  — l o g [log M(r)] lim — — -—— log r  f (z) i s o f o r d e r p i f  0 < p <  p,  where M(r) denotes the maximum modulus o f f ( z ) f o r  |z| = r .  A c o n s t a n t has order 0, by c o n v e n t i o n . Asymptotic Theorem 1.9:. {Permutation} Group  The function  Lemma:  rollowing  P of e n t i r e  11  r  e  s  e  r  v  e  s  order i functions ' t  n  e  lemma g i v e s the order of an e n t i r e  as a f u n c t i o n  The e n t i r e  ITT <~ S "to*  o f the sequence o f c o e f f i c i e n t s [ 7 ] .  function  f ( z ) = Ef z , n ' 11  f + 0, i s o f n '  f i n i t e o r d e r i f and o n l y i f  -r-.— n• l o g (n)  y  is finite;  = lim  1 ~—log f n  and then the o r d e r p o f f ( z ) i s equal t o y.  For the proof o f Theorem 1.9, we show i n 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  group p r e s e r v e the o r d e r o f e n t i r e  functions.  permutation  29  Let  f(z) = Ef^z  o r d e r p.  be an e n t i r e f u n c t i o n of  11  Denote the o r d e r of Tr(f) by p , where TT i s i n  the asymptotic permutation group.  p  finite  = lim  TT  (n) • Log ]og 1 log f n  TT  (n) (n)  K  Then  n (1+e) ° l o g [n (1+e) ]  j-^  log n  s i n c e TT i s asymptotic t o n. Now,  e > 0  given  1 1+e  ,  and f o r a l l s u f f i c i e n t l y  log(1+e) (1+e)-log(n)  l a r g e n,  ,  Equivalently, l o g [ n ( l + e ) ] < (1+e) • log.(n) and hence  P i r  1  TiTa n d + e ) l o g [ n ( l + e ) ] log n  Since  Tr(n) > n ( l - e )  £  U  +  e  )  2 .  m  n-log(n) < log n  = YTrn  71 ( n )  1  ?  g  ( n )  log n  ]  ( 1 + £  f o r l a r g e n, t h e r e i s a s i m i l a r  lower bound:  TT  p  > Tim  n  (1-e) l o g [n (1-e) ] log "n  30 Now,  for  1 1-e  e > 0,  , l o g (1-e) ^ , + —(1-e)?—• l o g ;—(n) > 1  _  TT  T  f o r a l l l a r g e n.  Equivalently, log[n(l-e)] and  >  (l-e)-log(n)  thus  ^ 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) log log n n This yields  p  = p. TT  Finally,  i f e i t h e r f o r TT ( f ) h a s u n b o u n d e d  t h e n t h e y b o t h have unbounded  order.  S e c o n d , we show t h a t a l l o r d e r p r e s e r v i n g are i n the asymptotic  order,  permutations  group.  Consider f Cz)  y L  £  n  _n-log(n)  The f u n c t i o n f ( z ) i s a n e n t i r e  R  •1  function:  -r-.— i -n«log(n) . 1/n —. -log(n) i _ = limje ^ | ' = limje ^ | = 0.  The o r d e r o f f ( z ) i s f i n i t e :  31  P  If  p  TT  ITH —  =  n , 1  °g  log|e  ( n )  TT i s asymptotic, then  = H i lJn)Iog[Tr(n)] n*log(n) — <  = ITS {<a+e)  +  , . =i ,  n , l o g ( n )  m  I  l i m [IT (n)/n] = a > 1,  and  n(a+e)log[n(a+e)3 n«log(n)  IS±£il£2l5±£l}  =  ( a  +  e )  .  Also = Xlm  ( ) 9 tit (n) ] n-log(n) n  1  l Q  >  —  h ( a - e ) l o g [n ( a - e ) ] n«log(n)  log(n)  Thus, i f TT i s not c o n t a i n e d i n the asymptotic permut a t i o n group, then TT does not p r e s e r v e o r d e r .  32  TYPE OF ENTIRE FUNCTIONS  Definition:  The e n t i r e f u n c t i o n  f(z) of p o s i t i v e order  p i s o f type x i f Hm  { r - l o g [M(r) ] } = T , _ p  (0 < T £ »)•  r;->-oo  where M(r) denotes  Theorem 1.10:  The  the maximum modulus o f f ( z ) f o r |z| = r .  {TT € S : co 1  following  characterization  ., . " P r e s e r v e s , the type j of e n t i r e functions J  theorem from Boas  =  { LJ  p  ^  e  Asymptotic J £ m  t  a  t  i  o  n J  }  group  [7] provides a  o f the type o f an e n t i r e f u n c t i o n i n  terms o f the c o e f f i c i e n t s .  Theorem 1.11:  If  v = lim(n[f  |/ ) , p  n  0 < v < °°, then the  n-M»  function  f ( z ) = Xf z  and o n l y i f v = exp. of growth  i s of order p If  and type T i f  v = 0 o r °°, f ( z ) i s , r e s p e c t i v e l y ,  (p,0) o r o f growth not l e s s than  (p,°°); and  conversely.  Proof o f Theorem 1.10:  I f TT i s a member o f the asymptotic  group, then f o r e > 0, n ( l - e) < Tr(n) < n ( l + e)  forsufficiently  l a r g e n,  We ality.  may  assume  ]f | < 1 without  any l o s s  of  gener-  Hence,  n(l-e)|f  I P /  n  1  1  *  1  - ^  <  T R  (  N  )  |  F  1  |P/nU-e)  n  1  p  /  K  FF(N)  ,  F  1  | P/TT  n  (n)  1  and  ir(n) | f  n  |  p  A  (  n  < Tr(n)|f |  )  n  (  1  +  £  < n (1+e) | f  )  n  R  |  p  /  n  (  1  +  e  )  Thus  T •  lim n->°° and  the type  If doesn't  \  i  jr  i  P/TT (n) w /  f \ n  of the entire  preserve  the order  I t would  generating  set  ,  (n)  _ | p/n = n f n I  H /  1  function  1  f (z)  i s preserved  TT i s n o t a m e m b e r o f t h e a s y m p t o t i c  Remark:  far.  TT  set f o rthe preservation  of a l l shifts,  where  permutation  which  sequences.  Although  functions,  permutation  these  i n the proofs they  basic  groups mentioned  group  a p-shift i s a  interchanges  t h e n TT  functions.  b e i n t e r e s t i n g i f t h e r e w a s some  The a r i t h m e t i c  mentioned  of entire  group,  b y TT.  i s generated  thus by t h e  p-arithmetic  two a d j a c e n t  p-arithmetic  p - s h i f t s were n o t s p e c i f i c a l l y  i n v o l v i n g r a t i o n a l and  n e v e r t h e l e s s were  o f fundamental  algebraic importance.  34  The  t r a v e l p e r m i t t e d by the asymptotic group and the  bounded forward t r a v e l semi-group i s so random and the two s e t s are so l a r g e t h a t i s doesn't  appear t h e r e w i l l  be a s i g n i f i c a n t l y s m a l l e r subset which would them.  generate  35 CHAPTER  Let  Ef  denote  We  shall  be  distinguished  Ef  is  refer  to  TWO  a  series  several by  V-summable  an  to  SUMMABILITY  of  summation  attached  S  complex  or  valued  processes;  i n i t i a l .  We  terms.  these  say  w i l l  the  series  symbolically oo  v-Tf  =  s.  n=0 A a  permutation  series,  Ef  permutations A  TT  , (n)  where  w  of  the  of  the  permutation  series TT i s  a  positive  Ef  again  member  of  S  V-summable  type  of  TT p r e s e r v e s  consisting In  of  this  those  Ef  V,  n  we  V-summabilitv  b i l i t y group  and,  absolute  also  group  further, leaves  i f  set  i t  the  a  Ef  sum  , .  each  preservation  set  V-summability.  convergence;, c o n d i t i o n a l  found  is  With  preserving  summability is  of  Tr(n;  V-summable.  permutations  chapter,  preservation  is  associate  convergence, and Cesa.ro The  the  -  whenever  summability,  ,  integers.  =  is  to  means  shown  for  are considered.  each  that  the  type  of  summa-  preservation  invariant. oo  The.preservation  group  for  the  absolute  sum,  £  n=l of  a  series  Ef  is  n  S  to  .  |f  |,  36  CONDITIONALLY  The N lim  series  i  =  exists  n  u  n  Levi  We  borrow  result. order  and i s  of  Levi's  A new p r o o f ,  Cesaro  Let  IT b e  transforms series  TT  a  summation,  and proved  convergence  terminology  capable i s  of  the  result  preservation  i n  order  to  group.  state  generalization to  his  higher  presented.  a permutation  the i n f i n i t e  If  For  stated  the conditional some  i f  f i n i t e .  [22] o r i g i n a l l y  which gave  SERIES  c o n d i t i o n a l l y corweraervt ~ — —  n  Y f  N+c*  is  If  SUMMABLE  of  the positive  serie.s  into  If.  integers  the  which  rearranged  , .. (n)  given  positive  f  a  jumping-out  f  a  jumping-in  term  integer i f  n,  c a l l  m <_ n  and  TT (m)  >  n,  m > n  and  TT (m)  <_ n .  and  This  terminology  n-th  p a r t i a l A  block  set of of  term  arises  sum o f  naturally  the o r i g i n a l  consecutive  jumping-out the s e t has  (b)  the term the set  only  i n i s  terms  of  one  considers  terms  just  jumping-out  the  the rearranged  the series  jumping-out  the series  not a  when and of  (jumping-in)  (a)  following  i f  i s  series.  called  a  i f (jumping-in) preceeding  terms,  and  (jumping-in)  just  term.  37  The is  total  denoted  k(n),  the  TT a n d  is  Theorem  number  by  k(n).  value  2.1  leaves  the  We Schur  and  of  A  shall  blocks,  that  each  depends the  i f  terms  for  the  a l l  for  any  in  on  the  of  the  series  use  the  following  the  TT p r e s e r v e s  Such  a  n,  a  function  permutation  series.  associated  n.  given  TT d e t e r m i n e s  solely  permutation  only  bounded  sum  [12,  Theorem  which  (Levi):  uniformly  these  Notice  independent  convergence i f is  of  of  conditional  function,  k(n),  permutation  also  unaltered.  theorem  by  Kojima  and  p385].  2.2:  The  necessary  and  s u f f i c i e n t  conditions  that  oo  z' n  =  Y  a  ,z,  should  Tl / Kl  vergent  be  convergent  whenever  z,  Kl  is  con-  Kl  are oo  (a)  i.e.  I  |  k=l (b)  lim  a  the  l i m i t  (c)  I  vI  a  n  , n,k  =  a, k  of  a  , n ,k  =  A  a  of  n;  exists  when  tends  to  a  k  is  l i m i t  fixed  a when  '  Moreover,  z'  independently  M  ~  '  k=l  (d)  <  =  i f  z^. ->  z  we  lim  z' n  az  +  n->-oo  =  have  i)  -j k=l  a, ( z , k  k  -  z)  and  n  n  -> °°;  -»- °°.  38  If a matrix  (a , ) s a t i s f i e s niK c i n Theorem 2.2, c a l l i t r e g u l a r .  Proof o f Theorem 2.1:  k = £f  Let  denote the n-th p a r t i  1  00  sum of the s e r i e s  c o n d i t i o n s a, b, and  T f k n 1  and assume t h a t  lim z = z n n->-°°  exists  and i s f i n i t e . Call  S the C summation m a t r i x where S i s an i n f i n i t e —o  dimensional m a t r i x w i t h u n i t e n t r i e s on and below the main d i a g o n a l and zeros above the main d i a g o n a l :  10  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 m a t r i x i n which n  the n-th row e n t r y i s f , then the matrix n 1  S(f  n  ) = (z ) n  i s a l s o an i n f i n i t e column m a t r i x whose e n t r i e s are the n-th p a r t i a l We  sums  z  n can now w r i t e the f o l l o w i n g :  1 0 0 •  \  -110 0-1 0  s  - 1  1 0-1  ../  (z )  - (f )  (z )  = P(f )  k  PS  _ 1  k  SPS (z ) k  (SPS  )  k-th column o f  ,  k  - SP(f )  _ 1  Let  k  k  .  denote the e n t r y i n the n-th  row,  SPS . - 1  R e f e r r i n g t o Theorem 2 . 2 , l e t  a^ ^ =  (SPS  v  so t h a t  (a  We  s h a l l show t h a t I k=l  -1 =  S  P  S  (a , ) n f K.  i s regular.  oo.  oo  (a)  n,k>  |a n  J ,  K  = .1 k=l  |(SPS ) -1  n  ,  K  | < ~  M  uniformly,  The m a t r i x SP Is o b t a i n e d from the m a t r i x S by a rearrangement of the columns  of the matrix  There i s a 1 i n row n of SPS  S.  c o r r e s p o n d i n g t o each  1  1,0 p a i r i n the n-th row of SP. There i s a -1 i n row n o f SPS  1  c o r r e s p o n d i n g t o each  40 0, 1 p a i r i n the n-th row  of  SP.  Thus each b l o c k of c o n s e c u t i v e l ' s i n the n-th of  SP i s denoted by e n t r i e s 1 and -1 i n SPS  f i r s t b l o c k i n any  row begins  o n l y a 1 w i t h t h a t block.) to  the jumping-out Now,  ,I  Ela  i n the f i r s t  1  .  row  ( I f the  column, a s s o c i a t e  These b l o c k s of l ' s correspond  (jumping-in)  terms.  i s u n i f o r m l y bounded i f and o n l y i f  the number of b l o c k s i n the n-th row,  k(n), i s uniformly  bounded. (b) zero f o r  The  terms i n the k-th column of SPS  n > N, hence '  lim a N  (c)  The  , = a, = n, k k  From  are a l l  0.  ->OD  e n t r i e s i n the f i r s t  a f t e r some p o i n t .  1  column of SP are a l l 1  (a) above we  have  CO  I  k=l  a.n ,k  = A = 1 n  for  =  (d)  From  (b) and  1.  (c) , each a, = 0  = l i m z' = z n n-5-°°  and the sum  n > N  of the s e r i e s i s u n a l t e r e d .  and  a =  1.  41  CESARO MEANS  The r e s u l t s of the p r e c e e d i n g s e c t i o n 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  by Hardy  suggested  [16] .  Write A° = f + f, + ••• + f n 0 1 n n  and  k -k 1 _.k 1 k 1 = A + A, + ••• + A n 0 1 n —  A  Definition:  A  I f k. i s the l e a s t p o s i t i v e  i n t e g e r so t h a t  l i m k!n A = A n n->-°<> k  exists  and  is finite,  t h e n we  k  say t h a t  J-  n=0 (C ,k) to sum A and  f  i s summable n  write •I n=0  f  = A  (C,k).  I f S i s the summation m a t r i x , then 1 0 0 '210 S  2  =  1 0 0 •* • »  13 1 0  3 2 1  6 3 1  4 3 2  10 6 3  \  42 and  i n g e n e r a l , the term a t the i n t e r s e c t i o n o f the n - t h k+1 row and j - t h column of S is /• k + l - j [  where is  b  ( ? ^-T^} (n-j)  j  + K ^\ (•(n-j); -f I (n-j) i  _ n , j "  i s to be i n t e r p r e t e d  ll  as zero i f (n-j)  negative. Further, 10  0 0  10  /-- 2 1 0 0 -2  0 0  3 10  1-2 1 0  0  3-3 1 0  .-3  0 1-2 1  -1 3-3 1  0 0 1-2  0-1  3-3  \: and  \ /  /  i n g e n e r a l , the term a t the i n t e r s e c t i o n o f the n - t h  row and the j - t h column o f S ^ ^ ^ is k +  -(k+1)  where  (^-j)  negative or i f  t  o  b  e  }  • =  i  n t e r  P  C-l)  r e t  ed  n + j  (  k + 1  >  as zero i f  (n-j) > k+1.  L e t D be the d i a g o n a l  matrix  /  O  V2  D  1/3  \  o  1/4  /  (n-j) i s  43 We can now w r i t e the column m a t r i x which has the Cesaro  (C,k)  n - t h p a r t i a l sum,  k!n  A ,  i n the n - t h  row: k„k+l  k! D S  The s e r i e s  Ef^  < i»  •  f  i s Cesaro  (C,k) summable i f t h e k k+1  sequence  o f terms i n the column m a t r i x  converge.  Assuming  this  sequence  klD S  (f^)  does converge, which  permutation m a t r i c e s , P, i n s u r e t h a t  k! D S "'"P (f ^) k  k+  also  converges? Theorem 2.3: if  A permutation TT p r e s e r v e s Cesaro k-summability  and o n l y i f  I  i=l  i  k  l  I  j=l  C-l)  i + TT (.j)  fn-j+ky ( n-j '  k+1 >| = . 0 (n ). . Cj ) - i k  TT  Such a permutation a l s o l e a v e s the Cesaro k-sum o f the series  Remark:  unaltered.  (An i n t u i t i v e  approach t o the C ^ - p r e s e r y a t i o n groups)  I f a s e r i e s converges C Q , then the d i f f e r e n c e between partial  sums, N'  N n  44  must approach zero as (jumping-in)  N -> °°.  The b l o c k s o f jumping-out  terms have t h i s form and, t h e r e f o r e , a l s o  approach zero. S i n c e the p a r t i a l  sums o f the rearranged  series  differ  from the p a r t i a l sums of the o r i g i n a l s e r i e s by a bounded number o f b l o c k s , the two p a r t i a l sums must have the same limit. Bounded b l o c k t r a v e l may a l s o be d e s c r i b e d by r e q u i r i n g the t o t a l number o f unordered w i t h i n the p a r t i a l sums t o be bounded. block begins  pairs of c o e f f i c i e n t s (Note t h a t each  and terminates w i t h an unordered  S i n c e terms o f the s e r i e s order b i n o m i a l w e i g h t i n g , the n - t h p a r t i a l  pair.)  E f . are g i v e n a k - t h  (^^j^),  when t h e y appear i n  sum, no simple c h a r a c t e r i z a t i o n  appears p o s s i b l e .  However, a few comments might convey some  i n s i g h t i n t o these  permutations.  For C^~summability, the f i r s t c o n d i t i o n f o r r e g u l a r i t y 2  of  DS PS  —2  -1  D  r e q u i r e s t h e a b s o l u t e row sum to be u n i f o r m l y  bounded: 1  |a  n k  | < -M  uniformly.  Now,  • L  I|a ,| = i n,k' n 1  L  I n,k i|b  1  1  .|, '  45  where t h e b , are zero f o r each o r d e r e d 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 f o r each unordered triple  (slight oversimplification).  Thus, we r e q u i r e  the expected v a l u e (or p o s i t i o n ) o f the unordered  triples  to be u n i f o r m l y bounded:  E(X) = |b | < M n I i n,k L  1  v  uniformly.  For C^-summability, we roughly r e q u i r e the expected value o f the unordered  (k+2)-tuples t o be u n i f o r m l y  bounded.  Proof o f Theorem 2.3: We seek c o n d i t i o n s on P such t h a t (f.) i s C. -summable i f and o n l y i f P('f.) i s C.-summable. 1 k * l k E q u i v a l e n t l y , we seek c o n d i t i o n s on P such t h a t the sequence o f terms forming t h e column o f k!D S* "'" (f ) k  C+  converges i f and o n l y i f t h e sequence o f terms forming the column o f  k!D S k  k + 1  PCf.) = Ik!D S k  k + 1  Jk!D kSk+l  pS  0  k k+l = ICa,.)Jk!D"S 0  converges.  46  Lemma:  The n - t h row sum o f  S  PS  'D  K  i s asymptotic  k  to n .  Proof:  We proceed by i n d u c t i o n over the Cesaro number k .  Assume  S PS k  D~  - k  ( k - 1 )  = (b. . ) D " ~ ( k  has n - t h row  1 )  if 3  sum asymptotic t o n  k _ 1  ;  then  S  k + 1  PS~  ( k + 1 )  D  _ k  has n - t h  k _ 1  ^  may be  k_  row  sum asymptotic t o n . The  sum o f the n - t h row o f  S PS k  _ k  D~^  r e w r i t t e n as .k-1, k-1 ) x b . ~ n v n, 1 x v  We wish t o show .k  n  y x i y  y iii  b  . -  y  b. ._,,] ~ ' n  i i y b. . - y b. . _ j j = l 3,x. j 4 i D^+l H  co  3 =1  i=l  n j=l  co  i=l  3 , : L  J'^-  1  a> i=l  3  i=2  1  n I  =1  {*>, ! + D  ,  x  I  i=2  b. 11  '  i  I i  k  -  Ck-l) ]} k  47  I  [b  1=1 ?  -'' ]  - k j  n I b ^ 33=1  k  _  1  k , n  k-£  -  o(  +  b  i=2  3  .{o( '  + °(j - )] k  2  n . „ I .(j - ) j=l  1  )  + I  1  J r  n , , - k [ j*' j=l  . , k.  i^  b  i=2  J  [b  o(n  I  + (-k)  k  +  2  k-1 ,n . k  _  1  )  k n  Now, matrix  (a)  r e t u r n i n g to the p r o o f o f Theorem 2.3, the (a^  )  I  |a  i s regular:  .| < M  independently o f n.  Ca .) = k ! D S n, i  C o n s i d e r the n-th row of  k  I4- |a n, l • | = Kk 4-I ^ 1  K  1  I i  k 4n l  k  l  II  4' j  v  (S  k  +  1  1  )  II ( S 4-  k + 1  v  . ( s "  n,j '  J  (  k  )  1  )  PS  (  .(PS-  n,T  +  k + 1  )  ' TT  ,.,  .  k + 1  C k + 1 )  (i) , l  ) " D  )  k  48 - i y - i iv ~ k I 4  f_u  k  n  < n  1  1  1  i + T T  C:)  t ~i \ n-j  <  a.  . = 0  n  {  X)  +k  k  +  ,  1  ;  fO(n )] k  k  [Mn ] k  k  = M .  (b)  CO  l i m a. •  . = a. = 0 i  i , i  £  a  ± >  .  = A.  since  i , i  for  ~ 1.  3=1  F i n a l l y , the sum (d)  Since  i s unaltered:  a = 1  z' = l i m z n n->-°°  and  = az +  a  k  =  0,  T a, Cz .-, k k k=l L  n  - z) = z  i > I.  Theorem 2.4:  If  c^,  k-summability  p r e s e r v a t i o n semi-group, C (^C k  Proof:  (k > 1) .  k = 0, 1, 2, •••  Ci = 1, 2, 3,  k + i  denotes t h e Cesa.ro then •••).  We e x h i b i t a permutation  iT  k  & C^ - C k+i'  Let k  n+1  m  if n = 2 n-1  k  n  , m integral, m  if n = 2 7i  The permutation  k  +1,  otherwise. i n t e r c h a n g e s two s u i t a b l y  thin  subsequences and l e a v e s a l l o t h e r terms unchanged.  First, in C  k  TT £' C k  The permutation  k  Tr  k  i f and o n l y i f  D S k  i s r e g u l a r , where P  k  k + 1  Pk S -  ( k + 1  v  V  k  i s the permutation m a t r i x a s s o c i a t e d  w i t h TT . k  I t i s o n l y necessary t o show t h a t  I  |a< .|< M 1=1 ' k)  n  where  i s contained  a  Ck) n,D  r L  independent  3  JcJc+r k  ^-(k+l) -k D }  .  of n  50  if  b  C k )  . = (sk+Vs-(k+1))  .  then a  (k) 1 . = — n,D n  k  , (k) .k b .j n,j  Proof: b  . = cs n,_j C k )  =  k + 1  P-s-  ( k + 1 )  )  . nj r  c-D  i  i+j  ( ;i:r ) n  k  i=l  t !h k  J  k+1 = since i-j  ^^j)  0  =  (-D  i i-j=0 f  o  r  a  1  i + j  c -^:: ) n  k  c :h k  ,  J  values of  1  i - j except f o r  = 0, 1, 2, •••, (k+1). Letting  t = i-j  k+1 = TT I - t=0 t  K  yields  C-D ! [(.n-j)+k-t'][Cn-j)+l-f] } ( 1  k + 1  t  )  51 (-D {( J: ) t  K  k  1  Cn-j)  + Jkk+l^f) (n-j)  k  k _ 1  ' t=0 + 0((n-j) - }} k  1 k K  .  .. k  k  , 1 (n-D) r  r "  n  t  t=0  (n-j)  k+1 I |(k+l-2t') ( - D " ^ ) + t=0  k _ 1  + 0((n-j)  The f i r s t  / t /k+l> , (-D ( ) •+  L  1  *  +  2  1  k _ 2  1  )}  summation i n the l a s t e x p r e s s i o n equals k+1  the b i n o m i a l expansion  of  (1-1)  = 0.  summation can be r e w r i t t e n as a f i r s t in t'.  When the permutation  permutation,  t  1  = t  degree p o l y n o m i a l 1  equals the i d e n t i t y  and the summation equals zero.  o b t a i n the permutation differ  matrix P  The second  P" = P^.,  e x a c t l y two o f t h e t  from t and hence the summation cannot  R e t u r n i n g t o t h e p r o o f o f Theorem 2.4; t  OO  and o n l y i f  U  j=l  n  '  \  \ |a ;|< M j=l '  3  uniformly.  J  k j=l  n  n  '  3=1 3= 2  3  , J  To 1  equal z e r o . ir^. &  i f  +  52 2k+l 2  ,  v  ..k-l.k  j=l  K  3=2 The l a s t  two e x p r e s s i o n s f o l l o w from the lemma.  For n ^ , the summation w i l l be maximum when  I j=l  |a  (k)  n  J  ,  | < **±i I (2 " ^ k ^ k s=l  2k+l  k P  -  2  )(2  r P ^ , J c k-1 „ k { ,=1 I (2 ) ^ 2^ 1  P  JV  .k  k S  p + 1  x  2  =  P ~ l  ,i  oi  2k+l | Y „k  2  k  k  P  2  i  S+ l  k  j  +  ( 2 f c + 1 )  s=l  p  2  + (2k+l)  k  1  J  = 4k + 2 .  ••4  k S  s + 1  )  n = j,  k  } + (2k+l)  hence  53  We next show ^r-v— l  i  m  n->co  TT^ ^  i (k+i) | JJ n,j I j = i '->  i  l  i  m  n  =  l  - I T i n  2  +  lim P  k  (2  2  J  <2*Y  p  C2~  )  k  k+i  P  (2^ )  k  k + i  j  -  k P  ,k  -  2~  k  P  -  2  k+i  -  J  k  = lim P  p  (2^ )  13  k + i  2  k  )  S  k  +  ( 2  i  p  1 v  k+i )  k  S  )  k  +  ^k^.k+i  (2  )  J  i  lc  P  J  s=l  (2  lim  b  C2  (2" k  v  S  I  k+i  )  i,(k+i)|.k+i .1j = lJ n , j 13  J  P  P  1  (n-j)  )  k  ^ L -  r  b  . ,k  p  o  J n,j  I  1 C  f  m  J=2  OT  > lim P  i  1 k  +  I  lim k+i n n"  >_ l i m n  k  Y-r— 1 -k+T n n  CO  v  a  i  c  -P-1 k+i  k  p-l k  k  +  ).  i  k  P  L  (2^ )  i  54 k, i p  lim P  Two  cases  First,  (2  )  remain; i 9^ i + i  c  c  f  o  r  L  1  1 ]  Define 2  TT-,  P  + 1  >P  (n) = <  it  i  i s almost  Second,  0  n  £ C  0  n  P n = 2^  if  n = 2  for integral +  P  p,  1,  otherwise.  n Then  if  - C. r  i > 1. —  for  T h i s p r o o f i s omitted; ^  i d e n t i c a l to the p r e v i o u s p r o o f .  C  Q  r/_ C\  i _> 1.  for  Define P"2p TTQ  +  1  if  n =  if  n = 2p +  2p,  (n) = L2p  1.  Here TTQ i n t e r c h a n g e s even and odd terms, at most one k(n)  jumping-in  and one  There i.s  jumping-out term, hence  i s u n i f o r m l y bounded and TTQ £ C Q .  s i m i l a r to the p r e v i o u s cases shows  An  TT ^ c^  argument for  i >_ 1,  PRESERVATION GROUPS  Number o f p o l e s on the c i r c l e  Finite'permutations  of convergence Number o f poles w i t h i n the  Finite  permutations  c i r c l e o f meromorphy Rational Algebraic  functions functions  Radius o f convergence  Arithmetic  permutations  Arithmetic  permutations  Asymptotic  permutations  L V—  Type o f e n t i r e Entire  functions  A A U  permutations  Bounded forward t r a v e l  functions  Polynomials Sum of a b s o l u t e l y  Asymptotic  W  convergent  All  permutations  All  permutations  series Sum o f c o n d i t i o n a l l y  convergent  Bounded b l o c k movement  series Sum o f Cesaro C, summable k series  Bounded  (k+2)^tuple  movement  56 CONCLUSION  T h i s t h e s i s has  b a r e l y touched the p o s s i b l e problems  i n v o l v e d w i t h rearrangements of c o e f f i c i e n t s and v a t i o n groups. tensions  are the obvious  ex-  t o the dual l a t t i c e of f u n c t i o n p r o p e r t i e s  and  preservation to be  A d d i t i o n a l l y , there  preser-  groups.  There are other  types of summation  considered. The  l a t t i c e of p r e s e r v a t i o n  i n t r i g u i n g property ments.  groups p o i n t s out  of the c h a r a c t e r  This property  of the  an  rearrange-  appears t o be of a fundamental  nature. Each permutation group which preserved property  was  described  by a t r a v e l c o n d i t i o n whereas  those permutations which p r e s e r v e d were d e s c r i b e d that pairs  by  a function  different  smoothness c o n d i t i o n s  (k-tuples) were kept  i n the  summabilities sense  together.  There are s e v e r a l p o s s i b l e approaches toward an u n d e r l y i n g  c h a r a c t e r i z a t i o n of these permutations.  Possibly generating l i k e l y t h a t the 'types'  obtaining  s e t s c o u l d do t h i s .  rearrangement  'type'  i s basic,  of rearrangements might be t r a v e l  smoothness c o n d i t i o n s , and  I t seems more Three  conditions,  density preserving  conditions.  57  The  permutations c o n s i d e r e d so f a r c o u l d a l l be  r e p r e s e n t e d by  a 0-1  doubly s t o c h a s t i c  types of permutations generated by (but  not  0-1)  m a t r i c e s might be  t r a n s i t i o n m a t r i c e s might be of s e r i e s  c o u l d be  or Laurent  stochastic  considered.  Also,  of i n t e r e s t . Other types  series.  p r o p e r t i e s have been c o n s i d e r e d .  t o l e r a n c e allowed i n the  while s t i l l p r o p e r t y was Are  doubly  o r d e r of the  guaranteeing p r e s e r v a t i o n of a discussed.  coefficients particular  A l o g i c a l q u e s t i o n i s then:  t h e r e p r o p e r t i e s of power s e r i e s which are  of the  ordering?  coefficients clusions series  be  (and  That i s , s p e c i f y i n g possibly  drawn about the  composed of a l l the  There are First,  Other  r e a r r a n g e d , f o r example, D i r i c h l e t  Several function The  matrix.  two  the  s o l e l y the  m u l t i p l i c i t y ) , can  p r o p e r t i e s of any  Szego s t a t e s  that  along t h i s i f the  form a set of f i n i t e c a r d i n a l i t y , then the represents a function  which i s e i t h e r  depending on whether the sequence or not.  coefficients  Second, i f the  bounded away from the  second and  set  con-  set? line.  coefficients  power  series  rational.or form a  of  power  c o e f f i c i e n t s from t h i s  well-known r e s u l t s  a theorem by  independent  singular,  periodic  set of c o e f f i c i e n t s t h i r d quadrants by  is rays  58  emanating from the o r i g i n , then the power s e r i e s w i t h r a d i u s o f convergence R has a s i n g u l a r i t y a t A t h i r d r e s u l t may be o b t a i n e d Weyl's c r i t e r i o n  z = R.  as a c o r o l l a r y o f  f o r 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 o f a power s e r i e s are not u n i f o r m l y t r i b u t e d on the u n i t c i r c l e ,  dis-  then some Hadamard power o f  f, f w i l l be s i n g u l a r a t  ( k )  z  = I(f ) z k  n  z = R.  n  ,  Unfortunately  there  i s no  guarantee t h a t f w i l l be s i n g u l a r . F u n c t i o n s w i t h n a t u r a l boundaries problem.  suggest another  A power serie.s i s s i n g u l a r i f i t s c i r c l e o f  convergence i s a n a t u r a l boundary.  Are t h e r e ,  then,  functions  which are s i n g u l a r and a l l o f whose permutations are a l s o singular? and  We r e c a l l t h a t almost a l l decimals are i r r a t i o n a l  t h a t almost a l l f u n c t i o n s  are nowhere d i f f e r e n t i a b l e .  Perhaps almost a l l power s e r i e s are s i n g u l a r . C o n s i d e r the s e t c o n s i s t i n g o f the c o e f f i c i e n t s o f such a s i n g u l a r f u n c t i o n . the e x i s t e n c e set condition.  The author attempted t o show  o f s i n g u l a r f u n c t i o n s by s p e c i f y i n g some I t was c o n j e c t u r e d  i e n t s were s u i t a b l y s c a t t e r e d ,  t h a t i f the s e t o f c o e f f i c -  say dense i n some r e g i o n ,  then any f u n c t i o n formed w i t h c o e f f i c i e n t s from t h i s s e t must be s i n g u l a r .  An i n t e r e s t i n g counterexample w i t h  c o e f f i c i e n t s dense i n the plane and only  two  poles  59 is  the  d e r i v a t i v e of  f(z)  =  Je n=0  i a n  (cos  ^ 2^ \ 1-e  - In where a and any  rational  e  bn)z  1  i(a+b)  i(a+b)n  b are p i c k e d k.  n  +  so  z  , 1-e e  i(a-b) z  i(a-b)n- \  that  (a+b)  ^ k(a-b)  for  60 BIBLIOGRAPHY  Agnew, R. P., On Rearrangements of S e r i e s , American Mathematical S o c i e t y Bulletin,46(1940). A h l f o r s , Lars V., McGraw H i l l ,  Complex A n a l y s i s , Second E d i t i o n , 1966.  Armstrong, Convergence Radius of R e g u l a r l y Monotonic Functions, Duke Math J o u r n a l , V o l . 37 No. 1, March 1970. Bagemihl and Erdos, P., Rearrangements o f C-, -summable Series, A c t a Mathmatica, V o l . 92 (1954)7 B i e b e r b a c h , Ludwig, V e r l a g , 1955.  Analytische Fortsetzung,  Springer-  B l i s s , G i l b e r t Ames, A l g e b r a i c F u n c t i o n s , American Mathematical S o c i e t y Colloquium P u b l i c a t i o n s V o l . XVI, New York, 1933. Boas, R. P., 1954.  E n t i r e Functions,  Academic P r e s s ,  Inc.,  Carlson, F r i t z , Uber P o t e n z r e i h e n mit g a n z z a h l i g e n Koeffizienten, Mathematische Z e i t s c h r i f t , V o l . IX, October 1919. Cartan, H e n r i , Elementary Theory of A n a l y t i c Functions of one or s e v e r a l Complex V a r i a b l e s , A d d i s o n Wesley P u b l i s h i n g Co., Inc., Palo A l t o , 1963. C a s s e l s , J.W.S., An Approximation,  I n t r o d u c t i o n to Diophantine Cambridge U n i v e r s i t y P r e s s ,  1957.  Davenport, H., Erdos, P., and LeVeque, W.J., On Weyl's C r i t e r i o n f o r Uniform D i s t r i b u t i o n , M i c h i g a n Mathematical J o u r n a l , V o l . 10 C1963). Dienes, P., The T a y l o r S e r i e s , New York, 1957.  Dover P u b l i c a t i o n s , Inc.  F i n e , N.J. and Schweigert, G.E., On the Group of Homeomorphisms of an A r c , Annals of Mathematics, V o l . 62 No. 2, September 1955.  61 [14]  Ford, Walter B., S t u d i e s on D i v e r g e n t S e r i e s and Summability, C h e l s e a P u b l i s h i n g Co., New York, 1960.  [15]  Guha, U.C., On L e v i ' s Theorem on Rearrangements of Convergent S e r i e s , Indian J o u r n a l o f Mathematics, V o l . 6, 1967.  [16]  Hardy, G.H., Divergent Series, P r e s s , 1949.  [17]  Hausdorff, F e l i x , Zur V e r t e i l u n g der f o r t s e t z b a r e n P o t e n z r e i h e n , Matheroatische Z e i t s c h r i f t . V o l . 4, 1919.  [18]  H i l l e , Einar, A n a l y t i c F u n c t i o n Theory, V o l s . I & I I , Ginn and Co., 19 62.  [19]  Klein, Felix, Elementary Mathematics from an Advanced S t a n d p o i n t — A r i t h m e t i c , Algebra, A n a l y s i s , Dover P u b l i c a t i o n s , New York.  [20]  Klein, Felix, Elementary Mathematics from an Advanced Standpoint—Geometry, The M a c M i l l a n 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  Oxford at the Clarendon  U ± U X X  W  X.  J.11 L  J-i._l.J-  U  London, 1'92 8. [22]  L e v i , F.W., Rearrangements o f Convergent S e r i e s , Duke Mathematical J o u r n a l , V o l . 13, 19 46.  [23]  Lorentz, Z e l l e r , S e r i e s Rearrangements S e t s , A c t a Mathematica, 1958.  [24]  Mahler, K., On the T a y l o r C o e f f i c i e n t s of R a t i o n a l F u n c t i o n s , Proceedings of the Cambridge P h i l o s o p h i c a l S o c i e t y , V o l . 52, P a r t 1, 1956.  [25]  Melzak, Z.A., A Countable I n t e r p o l a t i o n Problem, Proceedings of the American Mathematical S o c i e t y , V o l . I I , No. 2, A p r i l 1960.  [26]  Melzak, Z.A., Power S e r i e s R e p r e s e n t i n g C e r t a i n R a t i o n a l F u n c t i o n s , Canadian J o u r n a l o f Mathematics, V o l . 12, 1960.  [27]  Moore, C h a r l e s N., Summable S e r i e s and Convergence Factors, American Mathematical S o c i e t y C o l l o q u i u m P u b l i c a t i o n s , V o l . XXII, 1938.  and A n a l y t i c  62 [28]  Niven, Iven, I r r a t i o n a l Numbers, Mathematical A s s o c i a t i o n of. America, Carus Mathematical Monographs, No. 11, 1956.  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