Classical Mathematical Physics [Dynamical Systems and Field by W. Thirring

By W. Thirring

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P q In the proof of this theorem we require the following lemma. 1. For 1 < p < » , - + —=* 1, the Walsh series CO is the Walsh Fourier series of a function / € L g €L if and only if for every with Walsh Fourier coefficients b, , oo is (C , 1) summable PROOF. 1) 0 fc =o {f(x)-s (x)) °khk' to 1 lim [ g(x) n-*oo 0 dx = 0 . 2) Since lim I I / - S || = 0 , 1 < p < oo by v i r t u e of a w e l l known theorem of n->oo n p P a l e y ( s e e f o r example Wade [ 1 4 ] ) we g e t ( 4 . 2 ) i n view of t h e r e l a t i o n : 1 f g{x) 1 Moreover, (f(x)-s (x)) dx L Wf-Sn\\L 0 i f \f(x)g(x) For t h e s u f f i c i e n c y p a r t , l e t g€ L \ dx= k=o L °]

T a h a \a2\ <\/2(l-3) , \a3\ < 2(1-3) and . 1). It was shown in [7] that the sharp coefficient bounds for a2 , α 3 are |α2Ι < 2(1 -β) , \a3\ < (l-B)(3-2ß) . It would be of interest to know what are the sharp bounds for the coeffi­ cients a2 , α 3 in the class S (3) . 4. The Class C (3) . 3) i s the function defined in ( 2 . 4 ) . THEOREM 4 . 1 . Let oo f(z) belong to J ■ = z + ^ an zn , n=2 C (3) . Then \a2\ Mor eover, for the class 0(0)

By t h e same r e a s o n i n g , i t f o l l o w s I, <ω , An~l/1) that 0(1) uniformly in x + t €. [a , o] . Now, for x + yE [b ,°o) and x + t € [a 9 c] due to the boundedness of [0,oo) and / r ) € C [ 0 , b ) we have v (η·\ <κ y-t) where s>r and 6 is such that 0 <6

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