arXiv subjects
Publications and source records attributed to G. Tephnadze.
In this paper we present and prove some new results concerning approximation properties of $T$ means with respect to the Vilenkin system in Lebesgue spaces and Lipschitz classes for any $1\leq p<\infty$. As applications, we obtain extension of some known approximation inequalities.
The famous Carleson-Hunt theorem has been in focus of interest for a long time. This theorem concerns convergence almost everywhere of Fourier series of $f\in L_p$ functions for $1<p\leq \infty.$ Kolmogorov constructed a function $f\in L_1$ such that the partial sums of Fourier series diverge everywhere. On the other hand, we have boundedness result for Féjer means for all $1\leq p\leq \infty$. Similar results are proved for the partial sums and Féjer means of Vilenkin-Fourier series. But also here it appears the questions what happens on any subset $E$ of measure zero, can we even have a function which diverge there? We contribute with a new result concerning this question and prove by the concrete construction that for any set $E$ of measure zero there exists a function $f\in L_p(G_m) (1\leq p<\infty)$ such that the Féjer means with respect to Vilenkin systems diverge on this set, which follows similar result for the partial sums. The key is to use new constructions of Vilenkin polynomials, which was introduced in \cite{PTW2}. In fact, the theorem we prove follows from the general result of \cite{Kar}, but we provide an alternative approach and the constructed function in our proof has a simple explicit representation.
In this paper we improve and complement a result by Móricz and Siddiqi \cite{Mor}. In particular, we prove that their inequality of the Nörlund means with respect to the Walsh system holds also without their additional condition. Moreover, we prove some new approximation results and inequalities in Lebesgue spaces for any $1\leq p<\infty$.
In this paper, we investigate very general approximation kernels with special properties, called an approximate identity, and prove almost everywhere and norm convergence of these general methods, which consists of a class of summability methods and provide norm and a.e. convergence of these summability methods with respect to the trigonometric system. Investigations of these summations can be used to obtain norm convergence of Fejér means with respect to the Vilenkin system also, but these methods are not useful to study a.e. convergence in this case, because of some special properties of the kernels of Fejér means. Despite these different properties we give alternative methods to prove almost everywhere convergence of Fejér means with respect to the Vilenkin systems.
We prove that there exists a martingale $f\in H_p $ such that the subsequence $\{L_{2^n}f \}$ of Nörlund logarithmic means with respect to the Walsh system are not bounded in the Lebesgue space $weak-L_p $ for $0<p<1 $. Moreover, we prove that for any $f\in L_p(G),$ $p\geq 1, $ $L_{2^n}f$ converge to $f$ at any Lebesgue point $x$. Some new related inequalities are derived.
In this paper we discuss and prove some new strong convergence theorems for partial sums and Fejér means with respect to the Vilenkin system.
In this paper we derive a new strong convergence theorem of Riesz logarithmic means of the one-dimensional Vilenkin-Fourier (Walsh-Fourier) series. The corresponding inequality is pointed out and it is also proved that the inequality is in a sense sharp, at least for the case with Walsh-Fourier series.
In this paper we investigate convergence and strong summability of the two-dimensional Vilenkin-Fourier series in the martingale Hardy spaces.
In this paper we characterize subsequences of Fejér means with respect to Vilenkin systems, which are bounded from the Hardy space $H_{p}$ to the Lebesgue space $L_{p},$ for all $0<p<1/2.$ The result is in a sense sharp.
It is important in many applications to be able to extend the (outer) unit normal vector field from a hypersurface to its neighborhood in such a way that the result is a unit gradient field. The aim of the paper is to provide an elementary proof of the existence and uniqueness of such an extension.
In this paper we derive characterizations of boundedness of the subsequences of partial sums with respect to Vilenkin system on the martingale Hardy spaces when $ 0<p<1 $. Moreover, we find necessary and sufficient conditions for the modulus of continuity of $f\in H_{p}$ martingales, which provide convergence of subsequences of partial sums on the martingale Hardy spaces. It is also proved that these results are the best possible in a special sense. As applications, both some well-known and new results are pointed out.
In this paper we prove and discuss a new divergence result of Nörlund logarithmic means with respect to Vilenkin system in Hardy space $H_1. $
The restricted maximal operators of partial sums with respect to bounded Vilenkin systems are investigated. We derive the maximal subspace of positive numbers, for which this operator is bounded from the Hardy space $%H_{p}$ to the Lebesgue space $L_{p}$ for all $0<p\leq 1.$ We also prove that the result is sharp in a particular sense.
We prove that certain means of the quadratical partial sums of the two-dimensional Vilenkin-Fourier series are uniformly bounded operators from the Hardy space $H_{p}$ to the space $L_{p}$ for $0<p\leq 1.$ We also prove that the sequence in the denominator cannot be improved.
As main result we prove strong convergence theorems for Cesáro means $% \left(C,α\right) $ on the Hardy spaces $H_{1/\left(1+α\right) } $% , where $0<α<1.$
In this paper we prove and discuss some new $\left( H_{p},L_{p}\right)$-type inequalities of weighted maximal operators of Vilenkin-Nörlund means with non-increasing coefficients. These results are the best possible in a special sense. As applications, both some well-known and new results are pointed out in the theory of strong convergence of Vilenkin-Nörlund means with non-increasing coefficients.
As main result we prove strong convergence theorems of Vilenkin-Fejér means when $0<p\leq 1/2.$
In this paper we prove and discuss some new $\left(H_{p},L_{p}\right)$ type inequalities of maximal operators of Vilenkin-Nörlund means with non-decreasing coefficients. We also apply these inequalities to prove strong convergence theorems of such Vilenkin-Nörlund means. These inequalities are the best possible in a special sense. As applications, both some well-known and new results are pointed out.