Constrained extremal problems in the Hardy space H2 and Carleman's formulas

TL;DR

This paper investigates constrained approximation problems in the Hardy space H2, establishing existence, uniqueness, and dual formulations, and extends the analysis to a finite-dimensional polynomial setting.

Contribution

It introduces new results on constrained extremal problems in H2, including dual formulations and computational methods, with a finite-dimensional polynomial extension.

Findings

Existence and uniqueness of solutions are proven.

A dual formulation via a critical point equation is derived.

A polynomial version of the problem is analyzed.

Abstract

We study some approximation problems on a strict subset of the circle by analytic functions of the Hardy space H2 of the unit disk (in C), whose modulus satisfy a pointwise constraint on the complentary part of the circle. Existence and uniqueness results, as well as pointwise saturation of the constraint, are established. We also derive a critical point equation which gives rise to a dual formulation of the problem. We further compute directional derivatives for this functional as a computational means to approach the issue. We then consider a finite-dimensional polynomial version of the bounded extremal problem.