On measures which generate the scalar product in a space of rational functions

TL;DR

This paper characterizes measures on the unit circle that induce a scalar product on a space of rational functions equivalent to the one generated by the Lebesgue measure, linking measure properties to rational function spaces.

Contribution

It provides a description of measures on the unit circle that generate a scalar product matching that of the Lebesgue measure within a rational function space.

Findings

Identifies measures that generate the same scalar product as Lebesgue measure on rational functions.

Connects measure properties with the structure of rational function spaces.

Provides conditions for measures to induce specific scalar products.

Abstract

Let $z_1,z_2,\,\ldots\,,z_n$ be pairwise different points of the unit disc and $\mathscr{L}(z_1,z_2,\,\ldots\,z_n)$ be the linear space generated by the rational fractions $\frac{1}{t-z_1} , \frac{1}{t-z_2} , \cdots\ , \frac{1}{t-z_n}\cdot$ Every non-negative measure $\sigma$ on the unit circle $\mathbb{T}$ generates the scalar product \[\langle\,f\,,\,g\,\rangle_{\!_{L^2_\sigma}} =\int\limits_{\mathbb{T}}f(t)\,\bar{g(t)}\,\sigma(dt), \quad \forall\,f,g\,\in\,L^2_\sigma.\] The measures $\sigma$ are described which satisfy the condition \[\langle\,f\,,\,g\,\rangle_{\!_{L^2_\sigma}}= \langle\,f\,,\,g\,\rangle_{\!_{L^2_m}},\quad \forall\,f,g\in\mathscr{L}(z_1,z_2,\,\ldots\,z_n),\] where $m$ is the normalized Lebesgue measure on $\mathbb{T}$.