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.

Convergence of Marcinkiewicz Means of Integrable Functions with Respect to Two-Dimensional Vilenkin Systems

2004 ◽  
Vol 11 (3) ◽  
pp. 467-478
Author(s):  
György Gát
Keyword(s):  
Maximal Operator ◽  
Weak Type ◽  
Integrable Function ◽  
Two Dimensional ◽  
Vilenkin Groups ◽  
Almost Everywhere ◽  
Integrable Functions ◽  
Marcinkiewicz Means

Abstract We prove that the maximal operator of the Marcinkiewicz mean of integrable two-variable functions is of weak type (1, 1) on bounded two-dimensional Vilenkin groups. Moreover, for any integrable function 𝑓 the Marcinkiewicz mean σ 𝑛𝑓 converges to 𝑓 almost everywhere.

Uniform estimates for families of singular integrals and double Fourier series

1986 ◽  
Vol 41 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Elena Prestini
Keyword(s):  
Fourier Series ◽  
Hilbert Transform ◽  
Singular Integrals ◽  
Double Fourier Series ◽  
Variable Coefficients ◽  
Partial Sums ◽  
One Dimensional ◽  
Uniform Estimates ◽  
Almost Everywhere ◽  
The One

AbstractIt is an open problem to establish whether or not the partial sums operator SNN2f(x, y) of the Fourier series of f ∈ Lp, 1 < p < 2, converges to the function almost everywhere as N → ∞. The purpose of this paper is to identify the operator that, in this problem of a.e. convergence of Fourier series, plays the central role that the maximal Hilbert transform plays in the one-dimensional case. This operator appears to be a singular integral with variable coefficients which is a variant of the maximal double Hilbert transform.

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 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.

scholarly journals A note on weighted maximal inequalities

1997 ◽  
Vol 40 (1) ◽  
pp. 193-205
Author(s):  
Qinsheng Lai
Keyword(s):  
Maximal Operator ◽  
Weak Type ◽  
Strong Type ◽  
Maximal Inequalities ◽  
Littlewood Maximal Operator

In this paper, we obtain some characterizations for the weighted weak type (1, q) inequality to hold for the Hardy-Littlewood maximal operator in the case 0<q<1; prove that there is no nontrivial weight satisfying one-weight weak type (p, q) inequalities when 0<p≠q< ∞, and discuss the equivalence between the weak type (p, q) inequality and the strong type (p, q) inequality when p≠q.

scholarly journals BEST CONSTANTS IN THE WEAK-TYPE ESTIMATES FOR UNCENTERED MAXIMAL OPERATORS

2012 ◽  
Vol 54 (3) ◽  
pp. 655-663
Author(s):  
ADAM OSȨKOWSKI
Keyword(s):  
Maximal Operator ◽  
Borel Measure ◽  
Weak Type ◽  
Integrable Function ◽  
Maximal Operators ◽  
Weak Type Estimates ◽  
Locally Integrable Function ◽  
Image Position ◽  
Strong Maximal Operator ◽  
General Product

AbstractLet μ be a Borel measure on ℝ. The paper contains the proofs of the estimates and Here A is a subset of ℝ, f is a μ-locally integrable function, μ is the uncentred maximal operator with respect to μ and cp,q, and Cp,q are finite constants depending only on the parameters indicated. In the case when μ is the Lebesgue measure, the optimal choices for cp,q and Cp,q are determined. As an application, we present some related tight bounds for the strong maximal operator on ℝn with respect to a general product measure.

scholarly journals Convergence in Measure of Logarithmic Means of Quadratical Partial Sums of Double Walsh-Kaczmarz-Fourier Series

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Ushangi Goginava ◽  
Károly Nagy
Keyword(s):  
Fourier Series ◽  
Orlicz Space ◽  
Baire Category ◽  
Partial Sums ◽  
Convergence In Measure ◽  
Logarithmic Means

The main aim of this paper is to prove that the logarithmic means of quadratical partial sums of the double Walsh-Kaczmarz series does not improve the convergence in measure. In other words, we prove that for any Orlicz space, which is not a subspace ofL log+ L(I2), the set of the functions the logarithmic means of quadratical partial sums of the double Walsh-Kaczmarz series of which converge in measure is of first Baire category.

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