Constrained $L^2$-approximation by polynomials on subsets of the circle

TL;DR

This paper investigates polynomial approximation on subsets of the circle with pointwise bounds, showing convergence to a bounded extremal problem solution and demonstrating practical application with real hyperfrequency filter data.

Contribution

It introduces a new approximation method constrained by pointwise bounds on the complement, linking it to extremal problems in system identification.

Findings

Convergence of polynomial solutions to a bounded extremal problem as degree increases

Application to real hyperfrequency filter data

Provides a numerical example demonstrating practical relevance

Abstract

We study best approximation to a given function, in the least square sense on a subset of the unit circle, by polynomials of given degree which are pointwise bounded on the complementary subset. We show that the solution to this problem, as the degree goes large, converges to the solution of a bounded extremal problem for analytic functions which is instrumental in system identification. We provide a numerical example on real data from a hyperfrequency filter.