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Widths and rigidity

Yuri Malykhin

Abstract

We consider Kolmogorov widths of finite sets of functions. Any orthonormal system of $N$ functions is rigid in $L_2$, i.e. it cannot be well approximated by linear subspaces of dimension essentially smaller than $N$. This is not true for weaker metrics: it is known that in every $L_p$, $p<2$, the first $N$ Walsh functions can be $o(1)$-approximated by a linear space of dimension $o(N)$. We give some sufficient conditions for rigidity. We prove that independence of functions (in the probabilistic meaning) implies rigidity in $L_1$ and even in $L_0$ -- the metric that corresponds to convergence in measure. In the case of $L_p$, $1<p<2$, the condition is weaker: any $S_{p'}$-system is $L_p$-rigid. Also we obtain some positive results, e.g. that first $N$ trigonometric functions can be approximated by very-low-dimensional spaces in $L_0$, and by subspaces generated by $o(N)$ harmonics in $L_p$, $p<1$.

Widths and rigidity

Abstract

We consider Kolmogorov widths of finite sets of functions. Any orthonormal system of functions is rigid in , i.e. it cannot be well approximated by linear subspaces of dimension essentially smaller than . This is not true for weaker metrics: it is known that in every , , the first Walsh functions can be -approximated by a linear space of dimension . We give some sufficient conditions for rigidity. We prove that independence of functions (in the probabilistic meaning) implies rigidity in and even in -- the metric that corresponds to convergence in measure. In the case of , , the condition is weaker: any -system is -rigid. Also we obtain some positive results, e.g. that first trigonometric functions can be approximated by very-low-dimensional spaces in , and by subspaces generated by harmonics in , .
Paper Structure (33 sections, 33 theorems, 139 equations)

This paper contains 33 sections, 33 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Average widths and $L_1$-rigidity.
  3. Approximation in the space of measurable functions.
  4. $S_{p'}$-property.
  5. Positive approximation results.
  6. Organization of the paper.
  7. Acknowledgement.
  8. Preliminaries
  9. Metrics
  10. Widths
  11. Average widths.
  12. Notation.
  13. Properties.
  14. Average width in Hilbert space.
  15. Trigonometric width.
  16. ...and 18 more sections

Key Result

Theorem A

Let $w_1,w_2,\ldots$ be the Walsh system in the Paley numeration. For any $p\in[1,2)$ there exists $\delta=\delta(p)>0$ such that for sufficiently large $N$ the inequality holds

Theorems & Definitions (57)

  • Theorem A: Mal22
  • Theorem 1.1
  • Theorem 1.2
  • Theorem B
  • Theorem 1.3
  • Theorem 1.4
  • Theorem C: IvYu80
  • Theorem 1.5
  • proof
  • proof
  • ...and 47 more