Almost Everywhere Convergence of (𝐶, α)-Means of Quadratical Partial Sums of Double Vilenkin–Fourier Series

2006 ◽  
Vol 13 (3) ◽  
pp. 447-462
Author(s):  
György Gát ◽  
Ushangi Goginava
Keyword(s):  
Fourier Series ◽  
Maximal Operator ◽  
Weak Type ◽  
Partial Sums ◽  
Almost Everywhere Convergence ◽  
Almost Everywhere

Abstract We prove that the maximal operator of the (𝐶, α)-means of quadratical partial sums of double Vilenkin–Fourier series is of weak type (1,1). Moreover, the (𝐶, α)-means of a function 𝑓 ∈ 𝐿1 converge a.e. to 𝑓 as 𝑛 → ∞.

scholarly journals Almost everywhere convergence of Cesàro means of two variable Walsh – Fourier series with varying parameteres

2021 ◽  
Vol 73 (3) ◽  
pp. 291-307
Author(s):  
A. A. Abu Joudeh ◽  
G. G´at
Keyword(s):  
Fourier Series ◽  
Maximal Operator ◽  
Weak Type ◽  
Partial Sums ◽  
Cesàro Means ◽  
Almost Everywhere Convergence ◽  
Almost Everywhere ◽  
Integrable Functions ◽  
Walsh Group ◽  
Cesáro Means

UDC 517.5 We prove that the maximal operator of some means of cubical partial sums of two variable Walsh – Fourier series of integrable functions is of weak type . Moreover, the -means of the function converge a.e. to for , where is the Walsh group for some sequences .

scholarly journals Almost everywhere convergence of a subsequence of the logarithmic means of Vilenkin-Fourier series

2008 ◽  
Vol 21 (3) ◽  
pp. 275-289
Author(s):  
György Gát ◽  
Károly Nagy
Keyword(s):  
Fourier Series ◽  
Maximal Operator ◽  
Weak Type ◽  
Integrable Function ◽  
Partial Sums ◽  
One Dimensional ◽  
Almost Everywhere ◽  
Logarithmic Means ◽  
The One

The main aim of this paper is to prove that the maximal operator of a subsequence of the (one-dimensional) logarithmic means of Vilenkin-Fourier series is of weak type (1, 1). Moreover, we prove that the maximal operator of the logarithmic means of quadratical partial sums of double Vilenkin-Fourier series is of weak type (1, 1), provided that the supremum in the maximal operator is taken over special indices. The set of Vilenkin polynomials is dense in L1, so by the well-known density argument the logarithmic means t2n(f) converge a.e. to f for all integrable function f. .

On the (𝐶, α)-Means of Quadratic Partial Sums of Double Walsh–Kaczmarz–Fourier Series

2009 ◽  
Vol 16 (3) ◽  
pp. 489-506
Author(s):  
Gyäorgy Gát ◽  
Károly Nagy
Keyword(s):  
Fourier Series ◽  
Weak Type ◽  
Integrable Function ◽  
Partial Sums ◽  
Almost Everywhere

Abstract The main aim of this paper is to prove that the (𝐶, α)-means of quadratic partial sums of double Walsh–Kaczmarz–Fourier series are of weak type (1, 1) and of type (𝑝, 𝑝) for all 1 < 𝑝 ≤ ∞ (0 < α < 1). Moreover, these (𝐶, α)-means converge to 𝑓 almost everywhere for any integrable function 𝑓.

scholarly journals Generalized localization for spherical partial sums of multiple Fourier series

2019 ◽  
Vol 489 (1) ◽  
pp. 7-10
Author(s):  
R. R. Ashurov
Keyword(s):  
Fourier Series ◽  
Multiple Fourier Series ◽  
Partial Sums ◽  
Localization Principle ◽  
Open Set ◽  
Almost Everywhere

In this paper the generalized localization principle for the spherical partial sums of the multiple Fourier series in the L2-class is proved, that is, if f L2 (ТN) and f = 0 on an open set ТN then it is shown that the spherical partial sums of this function converge to zero almost - ​everywhere on . It has been previously known that the generalized localization is not valid in Lp (TN) when 1 p 2. Thus the problem of generalized localization for the spherical partial sums is completely solved in Lp (TN), p 1: if p 2 then we have the generalized localization and if p 2, then the generalized localization fails.

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