Embeddings of Müntz spaces: the Hilbertian case

Abstract

Given a strictly increasing sequence $Λ=(λ_n)$ of nonegative real numbers, with $\sum_{n=1}^\infty \frac{1}{λ_n}<\infty$, the Müntz spaces $M_Λ^p$ are defined as the closure in $L^p([0,1])$ of the monomials $x^{λ_n}$. We discuss properties of the embedding $M_Λ^p\subset L^p(μ)$, where $μ$ 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.