Double Walsh series with coefficients of bounded variation of higher order
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- by Chang-Pao Chen and Ching-Tang Wu PDF
- Trans. Amer. Math. Soc. 350 (1998), 395-417 Request permission
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.References
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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
- MR Author ID: 1161278
- 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.
- © Copyright 1998 American Mathematical Society
- 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