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

Given a strictly increasing sequence $\mathrm{\Lambda }=({\lambda }_n)$ of nonegative real numbers, with ${\sum }_{n=1}^{\mathrm{\infty }}\frac{1}{{\lambda }_n}<\mathrm{\infty }$, the M"untz spaces $M_{\mathrm{\Lambda }}^p$ are defined as the closure in $L^p([0,1])$ of the monomials $x^{{\lambda }_n}$. We discuss properties of the embedding $M_{\mathrm{\Lambda }}^p\subset L^p(\mu )$, 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.