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

Patrick L. Combettes; Dinh Dũng

paper

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

arXiv:1412.6400

Abstract

Let $P (D)$ be the differential operator induced by a polynomial $P$ , and let $U_{2}^{[P]}$ 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 $d_{n} (U_{2}^{[P]}, L_{2})$ in the general case when $U_{2}^{[P]}$ 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.