Mean convergence of generalized Walsh-Fourier series

Young, Wo Sang

doi:10.1090/s0002-9947-1976-0394022-8

Cited by 48 publications

(9 citation statements)

References 11 publications

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“…Not that Ψ 2 is the classical Walsh system. The basic properties of the generalized Walsh system of order have been obtained by Chrestenson, Fine, Watari, Young, Vilenkin, and others (see [11][12][13][14][15][16]). …”

Section: Definitionmentioning

confidence: 92%

Universality Properties of a Double Series by the Generalized Walsh System

Episkoposian

2013

International Journal of Analysis

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Section: Definitionmentioning

confidence: 92%

Universality Properties of a Double Series by the Generalized Walsh System

Episkoposian

2013

International Journal of Analysis

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“…We shall prove the weak type (1,1) result using Theorem 1.1 and the modified Calderón-Zygmund decomposition lemma obtained in [10].…”

Section: Proof Of Theorem 12mentioning

confidence: 94%

“…In the proof of the weak type (1,1) result, we modify the function b(t) in the Calderón-Zygmund decomposition to satisfy both ¡¡bdp = 0 and ¡¡ b(paf dp = 0, where ak depends on / and « . This modification of the Calderón-Zygmund decomposition was introduced in [10]. In the proof of the strong type (p, p) result, the sharp function is modified as follows: the mean value of / over / is replaced by a function fii.n = A+B<pZak (ak depending on / and n) that satisfies ¡{if-fi,n)dp = 0…”

Section: J{\s"f\>y] Jgmentioning

confidence: 99%

Weighted norm inequalities for Vilenkin-Fourier series

Young¹

1993

Trans. Amer. Math. Soc.

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Abstract. Let S"f be the nth partial sum of the Vilenkin-Fourier series of f € L1 . For 1 < p < oo , we characterize all weight functions w such that if / € IP{w), S"f converges to / in IP(w). We also determine all weight functions w suchthat {S"} is uniformly of weak type (1,1) with respect to IntroductionThe Vilenkin-Fourier series is a generalization of the Walsh-Fourier series. While the Walsh functions are characters of the countable direct product of groups of order 2, the Vilenkin system consists of characters of G = F|°^0 ZPi, a direct product of cyclic groups of order p¡, where p, is any integer > 2. In this paper, there is no restriction on the orders {/>,} .Let p be the Haar measure on G normalized by piG) = 1. We identify G with the unit interval (0, 1) in the following manner. Let «io = 1, mk = n,=o Pi' k -1.2,....We associate with each {x¡) £ G, 0 < x¡ < p¡, the point Y,™0Xjm~¿x £ (0, 1). If we disregard the countable set of ^,-rationals, this mapping is one-one, onto and measure-preserving. The characters of G axe the finite products of an orthonormal system {k(x) -exp(2nixk/pk), x = {xk} £ G, k = 0, 1, ... . To enumerate the finite products of {tpk} , we write each nonnegative integer « as a finite sum, « = YlT=oakmk , with 0 < ak < pk, and define Xn = nitLo ßk • ^ne functions {xn} are the characters of G, and they form a complete orthonormal system on G.For /el1, let . n -X Snfix)= / f(t)Yxj(x-t)dp(t), «=1,2,..., Jg j=0be the «th partial sum of the Vilenkin-Fourier series of /. In [10], we showed that for / £ Lp , 1 < p < oo, lim [ \S"fi-fi\pdp = 0.

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“…In [7] Schipp, F.Weisz studied scalar-valued tree martingales and proved some important inequalities on tree martingales. Using the results on tree martingales Schipp [5] and Young [8] proved that for an arbitrary Vilenkin system and for f ∈ L p (1 < p < ∞), the Vilenkin-Fourier series of f converges in L p norm to itself. In the case of tree martingales, the stopped martingale cannot be introduced and consequently the proofs of some basic inequalities are much more complicated than in the case of linear ordered martingales.…”

Section: Introductionmentioning

confidence: 99%