arXiv subjects
Publications and source records attributed to N. Areshidze.
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.
It is well-known the Lebesgue \cite{Lebesgue, Zygmund} test for trigonometric Fourier series. Taberski \cite{Taberski1, Taberski2} considered real-valued Lebesgue locally integrable functions $f$, such that \begin{equation*} \lim_{T \to \infty} \frac{1}{T} \int_{T}^{T+c} |f(t)| \, dt\ =0; \quad \lim_{T \to \infty} \frac{1}{T} \int_{-T-c}^{-T} |f(t)| \, dt \ =0 \end{equation*} for every fixed $c>0$. For this class of functions, he defined generalized Dirichlet's integrals. Besides, Taberski \cite{Taberski1,Taberski2} investigated problems of convergence and $(C,1)$-summability of these integrals. In this paper, the analogous of the Lebesgue test for the generalized Dirichlet's integrals is proved.
In this paper we improve and complement a result by Móricz and Siddiqi \cite{Mor}. In particular, we prove that their estimate of the Nörlund means with respect to the Vilenkin system holds also without their additional condition. Moreover, we prove a similar approximation result in Lebesgue spaces for any $1\leq p<\infty$.
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 $1\leq p<\infty$.