arXiv subjects
Publications and source records attributed to L. -E. Persson.
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.
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 prove and discuss a new divergence result of Nörlund logarithmic means with respect to Vilenkin system in Hardy space $H_1. $