Embeddings of Müntz spaces: The Hilbertian case

Abstract:

Given a strictly increasing sequence $\Lambda =(\lambda _n)$ of nonnegative real numbers, with $\sum _{n=1}^\infty \frac {1}{\lambda _n}<\infty$, the Müntz 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.