Analogues of Finite Blaschke Products as Inner Functions

TL;DR

This paper generalizes finite Blaschke products within various reproducing kernel Hilbert spaces, explores their relation to Shapiro--Shields functions, and characterizes entire inner functions on weighted Hardy spaces.

Contribution

It introduces a broad framework for inner functions as analogues of finite Blaschke products and connects them to shift-invariant subspaces and classical functions.

Findings

Only monomials are entire inner functions on weighted Hardy spaces.

Establishes a link between generalized inner functions and Shapiro--Shields functions.

Provides a unified perspective on inner functions in different RKHS settings.

Abstract

We give a generalization of the notion of finite Blaschke products from the perspective of generalized inner functions in various reproducing kernel Hilbert spaces. Further, we study precisely how these functions relate to the so-called Shapiro--Shields functions and shift-invariant subspaces generated by polynomials. Applying our results, we show that the only entire inner functions on weighted Hardy spaces over the unit disk are multiples of monomials, extending recent work of Cobos and Seco.