A generalization of Hardy's operator and an asymptotic Muntz-Szasz Theorem

TL;DR

This paper extends Hardy's operator to a broader class of bounded operators on L^2, and establishes an asymptotic Muntz-Szasz theorem characterizing limits of monomials with growing exponents.

Contribution

It introduces a generalization of Hardy's operator and proves an asymptotic Muntz-Szasz theorem for monomials with increasing exponents.

Findings

Operators preserving monomial structure leave functions vanishing on [0,s] invariant.

Characterization of limits of monomials with exponents between n and 2n.

Extension of classical theorems to asymptotic regimes.

Abstract

The Hardy operator has all the monomial functions as eigenvectors. We study bounded operators on L^2 that take monomial functions to multiples of other monomials, with a shifted exponent. We prove that they all leave the space of functions vanishing on [0,s] invariant. We prove an asymptotic Muntz-Szasz theorem, characterizing the set of functions that are limits of linear combinations of monomials with exponents between n and 2n.