Data-driven optimal approximation on Hardy spaces in simply connected domains

TL;DR

This paper develops a data-driven method for optimal function approximation in Hardy spaces within simply connected domains, linking complex analysis with control theory and numerical methods, and demonstrating its effectiveness through numerical experiments.

Contribution

It introduces a novel data-driven algorithm for optimal approximation in Hardy spaces, connecting complex analysis with discrete-time system interpolation and numerical methods.

Findings

The algorithm computes locally optimal approximants effectively.

Connections established between Hardy space approximation and classical numerical methods.

Numerical experiments validate the proposed approach.

Abstract

We consider optimal interpolation of functions analytic in simply connected domains in the complex plane. By choosing a specific structure for the approximant, we show that the resulting first order optimality conditions can be interpreted as optimal $\mathcal{H}_2$ interpolation conditions for discrete-time dynamical systems. Connections to the implicit Euler method, the midpoint method, and backward differentiation methods are also established. A data-driven algorithm is developed to compute a (locally) optimal approximant. Our method is tested on three numerical experiments.