ℓ1-summability of higher-dimensional Fourier series

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

It is proved that the maximal operator of the ℓ1-Fejér means of a d-dimensional Fourier series is bounded from the periodic Hardy space Hp(Td) to Lp(Td) for all d/(d+1) <p ≤ ∞ and, consequently, is of weak type (1, 1). As a consequence we obtain that the ℓ1-Fejér means of a function f ∈ L1(Td) converge a.e. to f. Moreover, we prove that the ℓ1-Fejér means are uniformly bounded on the spaces Hp(Td) and so they converge in norm (d/(d+1) <p <∞). Similar results are shown for conjugate functions and for a general summability method, called θ-summability. Some special cases of the ℓ1-θ-summation are considered, such as the Weierstrass, Picard, Bessel, Fejér, de la Vallée Poussin, Rogosinski and Riesz summations.

Original languageEnglish
Pages (from-to)99-116
Number of pages18
JournalJournal of Approximation Theory
Volume163
Issue number2
DOIs
Publication statusPublished - Feb 2011

Fingerprint

Fourier series
Summability
High-dimensional
Summation
Converge
Conjugate functions
Maximal Operator
Friedrich Wilhelm Bessel
Hardy Space
Norm

Keywords

  • ℓ1-θ-summation
  • Fourier series
  • Hardy spaces
  • Interpolation
  • P-atom

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics
  • Mathematics(all)
  • Numerical Analysis

Cite this

ℓ1-summability of higher-dimensional Fourier series. / Weisz, F.

In: Journal of Approximation Theory, Vol. 163, No. 2, 02.2011, p. 99-116.

Research output: Contribution to journalArticle

@article{3a8708ebccd349bcaf506ae134575370,
title = "ℓ1-summability of higher-dimensional Fourier series",
abstract = "It is proved that the maximal operator of the ℓ1-Fej{\'e}r means of a d-dimensional Fourier series is bounded from the periodic Hardy space Hp(Td) to Lp(Td) for all d/(d+1) d) converge a.e. to f. Moreover, we prove that the ℓ1-Fej{\'e}r means are uniformly bounded on the spaces Hp(Td) and so they converge in norm (d/(d+1)",
keywords = "ℓ1-θ-summation, Fourier series, Hardy spaces, Interpolation, P-atom",
author = "F. Weisz",
year = "2011",
month = "2",
doi = "10.1016/j.jat.2010.07.011",
language = "English",
volume = "163",
pages = "99--116",
journal = "Journal of Approximation Theory",
issn = "0021-9045",
publisher = "Academic Press Inc.",
number = "2",

}

TY - JOUR

T1 - ℓ1-summability of higher-dimensional Fourier series

AU - Weisz, F.

PY - 2011/2

Y1 - 2011/2

N2 - It is proved that the maximal operator of the ℓ1-Fejér means of a d-dimensional Fourier series is bounded from the periodic Hardy space Hp(Td) to Lp(Td) for all d/(d+1) d) converge a.e. to f. Moreover, we prove that the ℓ1-Fejér means are uniformly bounded on the spaces Hp(Td) and so they converge in norm (d/(d+1)

AB - It is proved that the maximal operator of the ℓ1-Fejér means of a d-dimensional Fourier series is bounded from the periodic Hardy space Hp(Td) to Lp(Td) for all d/(d+1) d) converge a.e. to f. Moreover, we prove that the ℓ1-Fejér means are uniformly bounded on the spaces Hp(Td) and so they converge in norm (d/(d+1)

KW - ℓ1-θ-summation

KW - Fourier series

KW - Hardy spaces

KW - Interpolation

KW - P-atom

UR - http://www.scopus.com/inward/record.url?scp=78651366292&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78651366292&partnerID=8YFLogxK

U2 - 10.1016/j.jat.2010.07.011

DO - 10.1016/j.jat.2010.07.011

M3 - Article

AN - SCOPUS:78651366292

VL - 163

SP - 99

EP - 116

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

SN - 0021-9045

IS - 2

ER -