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Approximation by matrix transform means with respect to the Walsh system in Lebesgue spaces

István Blahota, Dóra Nagy

Abstract

In this paper, we improve, complement and generalize (from Nör\-lund to matrix transform means) a result of Móricz and Siddiqi \cite{MS} and some statements of Areshidze and Tephnadze \cite{AT}, and (from $T$ (weighted) to matrix transform means) Anakidze, Areshidze, Persson and Tephnadze \cite{AAPT}.

Approximation by matrix transform means with respect to the Walsh system in Lebesgue spaces

Abstract

In this paper, we improve, complement and generalize (from Nör\-lund to matrix transform means) a result of Móricz and Siddiqi \cite{MS} and some statements of Areshidze and Tephnadze \cite{AT}, and (from (weighted) to matrix transform means) Anakidze, Areshidze, Persson and Tephnadze \cite{AAPT}.

Paper Structure

This paper contains 4 sections, 16 theorems, 62 equations.

Table of Contents

  1. Notation and definitions
  2. Historical overview
  3. Auxiliary results
  4. On the rate of the approximation

Key Result

Theorem 1

Let $f\in L_{p}(G),\ 1\leq p\leq\infty$ and let $\{q_{k}: k\in\mathbb{N}\}$ be a sequence of nonnegative numbers such that If ${q_{k}}$ is non-decreasing, then while if ${q_{k}}$ is non-increasing, then

Theorems & Definitions (20)

  • Theorem 1: Móricz and Siddiqi MS, Theorem 1
  • Remark 1
  • Theorem 2: Blahota and K. Nagy BN1, Theorem 1
  • Theorem 3: Areshidze and Tephnadze AT, Theorem 1
  • Lemma 1: Paley's lemma SWSP, p. 7.
  • Lemma 2
  • Lemma 3: Gát G1, Corollary 6
  • Lemma 4: Fine Fine, Lemma 2
  • Lemma 5: Yano Y1
  • Lemma 6: Toledo Tol
  • ...and 10 more