Weak type inequality for logarithmic means of Walsh-Kaczmarz-Fourier series

Ushangi Goginava; Károly Nagy

doi:10.14321/realanalexch.35.2.0445

Weak type inequality for logarithmic means of Walsh-Kaczmarz-Fourier series

Ushangi Goginava, Károly Nagy

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

The main aim of this paper is to prove that the Nörlund logarithmic means t_n^κ f of one-dimensional Walsh-Kaczmarz-Fourier series is weak type (1,1), and this fact implies that t_n^κf converges in measure on I for every function f ∈ L(I) and t_n,m^κ f converges in measure on I² for every function f ∈ Lln⁺ L(I²): Moreover, the maximal Orlich space such that Nörlund logarithmic means of two-dimensional Walsh-Kaczmarz-Fourier series for the functions from this space converge in two-dimensional measure is found.

Original language	English
Pages (from-to)	445-462
Number of pages	18
Journal	Real Analysis Exchange
Volume	35
Issue number	2
DOIs	https://doi.org/10.14321/realanalexch.35.2.0445
Publication status	Published - 2010
Externally published	Yes

Keywords

Convergence in measure
Double Walsh-Kaczmarz-Fourier series
Orlicz space

ASJC Scopus subject areas

Analysis
Geometry and Topology

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10.14321/realanalexch.35.2.0445

Cite this

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title = "Weak type inequality for logarithmic means of Walsh-Kaczmarz-Fourier series",

abstract = "The main aim of this paper is to prove that the N{\"o}rlund logarithmic means tnκ f of one-dimensional Walsh-Kaczmarz-Fourier series is weak type (1,1), and this fact implies that tnκf converges in measure on I for every function f ∈ L(I) and tn,mκ f converges in measure on I2 for every function f ∈ Lln+ L(I2): Moreover, the maximal Orlich space such that N{\"o}rlund logarithmic means of two-dimensional Walsh-Kaczmarz-Fourier series for the functions from this space converge in two-dimensional measure is found.",

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author = "Ushangi Goginava and K{\'a}roly Nagy",

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doi = "10.14321/realanalexch.35.2.0445",

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T1 - Weak type inequality for logarithmic means of Walsh-Kaczmarz-Fourier series

AU - Goginava, Ushangi

AU - Nagy, Károly

PY - 2010

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N2 - The main aim of this paper is to prove that the Nörlund logarithmic means tnκ f of one-dimensional Walsh-Kaczmarz-Fourier series is weak type (1,1), and this fact implies that tnκf converges in measure on I for every function f ∈ L(I) and tn,mκ f converges in measure on I2 for every function f ∈ Lln+ L(I2): Moreover, the maximal Orlich space such that Nörlund logarithmic means of two-dimensional Walsh-Kaczmarz-Fourier series for the functions from this space converge in two-dimensional measure is found.

AB - The main aim of this paper is to prove that the Nörlund logarithmic means tnκ f of one-dimensional Walsh-Kaczmarz-Fourier series is weak type (1,1), and this fact implies that tnκf converges in measure on I for every function f ∈ L(I) and tn,mκ f converges in measure on I2 for every function f ∈ Lln+ L(I2): Moreover, the maximal Orlich space such that Nörlund logarithmic means of two-dimensional Walsh-Kaczmarz-Fourier series for the functions from this space converge in two-dimensional measure is found.

KW - Convergence in measure

KW - Double Walsh-Kaczmarz-Fourier series

KW - Orlicz space

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JO - Real Analysis Exchange

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Weak type inequality for logarithmic means of Walsh-Kaczmarz-Fourier series

Abstract

Keywords

ASJC Scopus subject areas

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