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 M\"{u}ntz space $\overline{M^2_{\Lambda}}(\mathbb{D})$ 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.