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Kolmogorov widths of a Sobolev class with restrictions on the derivatives in different metrics

A. A. Vasil'eva

Abstract

In this paper, we obtain order estimates for the Kolmogorov widths of periodic Sobolev classes with restictions on derivatives of order $r_j$ with respect to $j$th variable in metrics $L_{p_j}$ ($1\le j\le d$).

Kolmogorov widths of a Sobolev class with restrictions on the derivatives in different metrics

Abstract

In this paper, we obtain order estimates for the Kolmogorov widths of periodic Sobolev classes with restictions on derivatives of order with respect to th variable in metrics ().
Paper Structure (6 sections, 20 theorems, 122 equations)

This paper contains 6 sections, 20 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Auxiliary assertions
  3. On estimates of the widths of an intersection of finite-dimensional balls
  4. Proof of Theorem \ref{['main']}
  5. Proof of Theorem \ref{['main1']}
  6. Proof of Theorems \ref{['not_emb']} and \ref{['main2']}

Key Result

Theorem 1

Let $d\in \mathbb{N}$, $d\geqslant 2$, $1<q<\infty$, $1<p_j<\infty$, $r_j>0$, $j=1, \, \dots, \, d$, and let $\frac{\langle \overline{r}\rangle}{d}+ \frac{1}{q} - \frac{\langle \overline{r}\rangle}{\langle \overline{p}\circ\overline{r}\rangle} > 0$. Suppose that

Theorems & Definitions (34)

  • Definition 1
  • Theorem 1
  • Remark 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • ...and 24 more