Embeddings of M\"{u}ntz spaces: the Hilbertian case

TL;DR

This paper investigates the properties of embeddings of M"untz spaces into L^p spaces with respect to measures, focusing on the Hilbertian case p=2, and provides conditions for boundedness, compactness, and Schatten class membership.

Contribution

It offers new criteria for the boundedness, compactness, and Schatten class membership of embeddings of M"untz spaces into L^2 spaces, specifically in the Hilbertian setting.

Findings

Conditions for bounded embeddings of M"untz spaces

Criteria for compactness of embeddings

Characterization of Schatten class membership

Abstract

Given a strictly increasing sequence $\Lambda=(\lambda_n)$ of nonegative real numbers, with $\sum_{n=1}^\infty \frac{1}{\lambda_n}<\infty$, the M\"untz spaces $M_\Lambda^p$ are defined as the closure in $L^p([0,1])$ of the monomials $x^{\lambda_n}$. We discuss properties of the embedding $M_\Lambda^p\subset L^p(\mu)$, where $\mu$ is a finite positive Borel measure on the interval $[0,1]$. Most of the results are obtained for the Hilbertian case $p=2$, in which we give conditions for the embedding to be bounded, compact, or to belong to the Schatten--von Neumann ideals.