Hardy Subspaces with Sparse Fourier Spectrum and Muntz Space

Let $\mathrm{\Lambda }=\{{\lambda }_n{\}}_{n=1}^{\mathrm{\infty }}\subset \mathbb{N}$ with ${\lambda }_n$ strictly increasing and such that ${\sum }_{n=1}^{\mathrm{\infty }}{\lambda }_n^{-1}<\mathrm{\infty }$. We show that a Hardy subspace $H^2(\mathbb{D},\mathrm{\Lambda })$ consisting of functions with sparse Fourier spectrum $\mathrm{\Lambda }$ coincides with a Müntz space $\overline{M_{\mathrm{\Lambda }}^2}(\mathbb{D})$ characterized by square-summability of coefficients relative to a biorthogonal family. As consequences, we obtain a new characterization of the Hardy norm in $H^2(\mathbb{D},\mathrm{\Lambda })$ and an integral representation formula for the Fourier coefficients. The proof uses the biorthogonal representation developed in a previous work of the author.