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Approximation of generalized Wiener classes of functions of several variables in different metrics

Andrii Shidlich

Abstract

The paper presents new and known results on estimates of important linear and nonlinear approximation characteristics of generalized Wiener classes of functions of several variables in different metrics.

Approximation of generalized Wiener classes of functions of several variables in different metrics

Abstract

The paper presents new and known results on estimates of important linear and nonlinear approximation characteristics of generalized Wiener classes of functions of several variables in different metrics.
Paper Structure (21 sections, 21 theorems, 274 equations)

This paper contains 21 sections, 21 theorems, 274 equations.

Table of Contents

  1. Introduction
  2. Approximative characteristics
  3. Approximative characteristics of the classes ${\mathcal{F}}_{q}^{\Psi}$ in the spaces ${\mathcal{S}}^p$
  4. Exact values of best approximations and basis widths
  5. Exact values of best approximations and basis widths
  6. Exact values of best $n$-term approximation
  7. Order estimates of best $n$-term approximations and basis widths of the classes ${\cal F}_{q,r}^{\psi}$ in the spaces ${\mathcal{S}}^p$
  8. Approximation characteristics of the classes ${\cal F}_{q,r}^{\psi}$ in the spaces $L_p$
  9. Proof of theorems
  10. Proof of Theorem \ref{['Th:ApChFS01']}
  11. Auxiliary estimates
  12. Proof of Theorem \ref{['Th:_OE_Sp1']}
  13. Proof of Theorem \ref{['Th:_OE_Sp2']}
  14. Proof of Theorem \ref{['Th:_OE_Sp3']}
  15. Proof of Theorem \ref{['Th:_OE_Lp1']}
  16. ...and 6 more sections

Key Result

Theorem 1

Let $m\in {\mathbb N}$ and $\Psi=\{\Psi_k\}_{k \in {\mathbb Z}^d}$ be a sequence of complex numbers, $\Psi_k\not=0$, such that there exists a non-increasing rearrangement $\bar{\Psi}= \{\bar{\Psi}_j\}_{j=1}^\infty$ of the system $\{|\Psi_k|\}_{k \in {\mathbb Z}^d}$. (i) If $0<q<p=\infty$ or $p=q=\in and Moreover, for any collection $\gamma_m^*\in\Gamma_m$ satisfying Eq:ExV_BaW08, relation Eq:ExV_

Theorems & Definitions (28)

  • Theorem 1
  • Lemma 1
  • Remark 1
  • Theorem 2
  • Remark 2
  • Theorem 3
  • Proposition 1
  • Theorem 4
  • Remark 3
  • Remark 4
  • ...and 18 more