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Double Walsh series with coefficients
of bounded variation of higher order


Authors: Chang-Pao Chen and Ching-Tang Wu
Journal: Trans. Amer. Math. Soc. 350 (1998), 395-417
MSC (1991): Primary 42C10
DOI: https://doi.org/10.1090/S0002-9947-98-01899-6
MathSciNet review: 1407697
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $D_{j}^{k}(x)$ denote the Cesàro sums of order $k$ of the Walsh functions. The estimates of $D_{j}^{k}(x)$ given by Fine back in 1949 are extended to the case $k>2$. As a corollary, the following properties are established for the rectangular partial sums of those double Walsh series whose coefficients satisfy conditions of bounded variation of order $(p,0), (0,p)$, and $(p,p)$ for some $p\ge 1$: (a) regular convergence; (b) uniform convergence; (c) $L^{r}$-integrability and $L^{r}$-metric convergence for $0<r<1/p$; and (d) Parseval's formula. Extensions to those with coefficients of generalized bounded variation are also derived.


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Additional Information

Chang-Pao Chen
Affiliation: Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 30043, Republic of China
Email: cpchen@math.nthu.edu.tw

Ching-Tang Wu
Affiliation: Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 30043, Republic of China

DOI: https://doi.org/10.1090/S0002-9947-98-01899-6
Received by editor(s): May 17, 1995
Received by editor(s) in revised form: July 30, 1996
Additional Notes: The first author’s research is supported by National Science Council, Taipei, R.O.C. under Grant #NSC 84-2121-M-007-026.
Article copyright: © Copyright 1998 American Mathematical Society

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