Hardy Subspaces with Sparse Fourier Spectrum and Muntz Space

Abstract

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