Kolmogorov n-Widths of Function Classes Induced by a Non-Degenerate Differential Operator: A Convex Duality Approach

Patrick L. Combettes; Dinh Dũng

Kolmogorov n-Widths of Function Classes Induced by a Non-Degenerate Differential Operator: A Convex Duality Approach

Classical Analysis and ODEs 2014-12-22 v1 Analysis of PDEs Functional Analysis

Authors: Patrick L. Combettes , Dinh Dũng

Abstract

Let $P(D)$ be the differential operator induced by a polynomial $P$ , and let ${U^{[P]}_2}$ be the class of multivariate periodic functions $f$ such that $\|P(D)(f)\|_2\leq 1$ . The problem of computing the asymptotic order of the Kolmogorov $n$ -width $d_n({U^{[P]}_2},L_2)$ in the general case when ${U^{[P]}_2}$ is compactly embedded into $L_2$ has been open for a long time. In the present paper, we use convex analytical tools to solve it in the case when $P(D)$ is non-degenerate.

Keywords

Kolmogorov n-widths differential operator function classes convex duality asymptotic order

Cite

@article{arxiv.1412.6400,
  title  = {Kolmogorov n-Widths of Function Classes Induced by a Non-Degenerate Differential Operator: A Convex Duality Approach},
  author = {Patrick L. Combettes and Dinh Dũng},
  journal= {arXiv preprint arXiv:1412.6400},
  year   = {2014}
}

Kolmogorov n-Widths of Function Classes Induced by a Non-Degenerate Differential Operator: A Convex Duality Approach

Abstract

Keywords

Cite

Related papers