Wall rational functions and Khrushchev's formula for orthogonal rational functions

TL;DR

This paper explores the properties of Wall rational functions and their role in Khrushchev's formula for orthogonal rational functions, using the Nevalinna-Pick algorithm to analyze their convergence and approximation characteristics.

Contribution

It introduces a novel application of the Nevalinna-Pick algorithm to derive Khrushchev's formula for orthogonal rational functions and studies the properties of Wall rational functions.

Findings

Nevalinna-Pick algorithm creates homeomorphisms between measures, functions, and sequences.

Continued fraction expansion of Schur functions identified with rational approximants.

Convergence analysis of Wall approximants in indeterminate cases provided.

Abstract

We prove that the Nevalinna-Pick algorithm provides different homeomorphisms between certain topological spaces of measures, analytic functions and sequences of complex numbers. This algorithm also yields a continued fraction expansion of every Schur function, whose approximants are identified. The approximants are quotients of rational functions which can be understood as the rational analogs of the Wall polynomials. The properties of these Wall rational functions and the corresponding approximants are studied. The above results permit us to obtain a Khrushchev's formula for orthogonal rational functions. An introduction to the convergence of the Wall approximants in the indeterminate case is also presented.