Strong convergence theorems for two-parameter Walsh-Fourier and trigonometric-Fourier series

Ferenc Weisz

Studia Mathematica (1996)

  • Volume: 117, Issue: 2, page 173-194
  • ISSN: 0039-3223

Abstract

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The martingale Hardy space H p ( [ 0 , 1 ) 2 ) and the classical Hardy space H p ( 2 ) are introduced. We prove that certain means of the partial sums of the two-parameter Walsh-Fourier and trigonometric-Fourier series are uniformly bounded operators from H p to L p (0 < p ≤ 1). As a consequence we obtain strong convergence theorems for the partial sums. The classical Hardy-Littlewood inequality is extended to two-parameter Walsh-Fourier and trigonometric-Fourier coefficients. The dual inequalities are also verified and a Marcinkiewicz-Zygmund type inequality is obtained for BMO spaces.

How to cite

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Weisz, Ferenc. "Strong convergence theorems for two-parameter Walsh-Fourier and trigonometric-Fourier series." Studia Mathematica 117.2 (1996): 173-194. <http://eudml.org/doc/216250>.

@article{Weisz1996,
abstract = {The martingale Hardy space $H_p([0,1)^2)$ and the classical Hardy space $H_p(^2)$ are introduced. We prove that certain means of the partial sums of the two-parameter Walsh-Fourier and trigonometric-Fourier series are uniformly bounded operators from $H_p$ to $L_p$ (0 < p ≤ 1). As a consequence we obtain strong convergence theorems for the partial sums. The classical Hardy-Littlewood inequality is extended to two-parameter Walsh-Fourier and trigonometric-Fourier coefficients. The dual inequalities are also verified and a Marcinkiewicz-Zygmund type inequality is obtained for BMO spaces.},
author = {Weisz, Ferenc},
journal = {Studia Mathematica},
keywords = {martingale and classical Hardy spaces; p-atom; atomic decomposition; Walsh functions; Hardy-Littlewood inequality; Walsh-Fourier series; trigonometric-Fourier series; martingale Hardy space; classical Hardy space; strong convergence; Marcinkiewicz-Zygmund type inequality; BMO spaces},
language = {eng},
number = {2},
pages = {173-194},
title = {Strong convergence theorems for two-parameter Walsh-Fourier and trigonometric-Fourier series},
url = {http://eudml.org/doc/216250},
volume = {117},
year = {1996},
}

TY - JOUR
AU - Weisz, Ferenc
TI - Strong convergence theorems for two-parameter Walsh-Fourier and trigonometric-Fourier series
JO - Studia Mathematica
PY - 1996
VL - 117
IS - 2
SP - 173
EP - 194
AB - The martingale Hardy space $H_p([0,1)^2)$ and the classical Hardy space $H_p(^2)$ are introduced. We prove that certain means of the partial sums of the two-parameter Walsh-Fourier and trigonometric-Fourier series are uniformly bounded operators from $H_p$ to $L_p$ (0 < p ≤ 1). As a consequence we obtain strong convergence theorems for the partial sums. The classical Hardy-Littlewood inequality is extended to two-parameter Walsh-Fourier and trigonometric-Fourier coefficients. The dual inequalities are also verified and a Marcinkiewicz-Zygmund type inequality is obtained for BMO spaces.
LA - eng
KW - martingale and classical Hardy spaces; p-atom; atomic decomposition; Walsh functions; Hardy-Littlewood inequality; Walsh-Fourier series; trigonometric-Fourier series; martingale Hardy space; classical Hardy space; strong convergence; Marcinkiewicz-Zygmund type inequality; BMO spaces
UR - http://eudml.org/doc/216250
ER -

References

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