On Strong Markushevich bases $\{t^{λ_n}\}_{n=1}^{\infty}$ in their closed span in $L^2 (0, 1)$ and characterizing a subspace of $H^2 (\mathbb{D})$

Abstract

Let $Λ=\{λ_n\}_{n=1}^{\infty}$ be a strictly increasing sequence of positive real numbers such that $\sum_{n=1}^{\infty}\frac{1}{λ_n}<\infty$ and $\inf(λ_{n+1}-λ_n)>0$. We investigate properties of the closed span of the system $\{t^{λ_n}\}_{n=1}^{\infty}$ in $L^2 (0,1)$, denoted by $\overline{M_Λ}$, and of the unique biorthogonal family $\{r_n (t)\}_{n=1}^{\infty}$ to the system $\{t^{λ_n}\}_{n=1}^{\infty}$ in $\overline{M_Λ}$. We show that the system $\{t^{λ_n}\}_{n=1}^{\infty}$ is a strong Markushevich basis in $\overline{M_Λ}$ and we obtain a series representation for functions in $\overline{M_Λ}$. We also construct a general class of operators on $\overline{M_Λ}$ that admit spectral synthesis. In particular, for all $ρ\in (0,1)$ the operator $T_ρ(f)=f(ρx)$ on $\overline{M_Λ}$ admits spectral synthesis. In addition, we characterize a certain subspace of the classical Hardy space $H^2 (\mathbb{D})$. Under the extra assumption that $Λ\subset\mathbb{N}$, let $H^2(\mathbb{D}, Λ)$ consist of functions $f$ in $H^2(\mathbb{D})$ so that the Fourier coefficients $c_n$ of the boundary function $f(e^{iθ})$ vanish for all $n\notin Λ$. We prove that $f\in H^2(\mathbb{D}, Λ)$ if and only if $f\in\overline{M_Λ}$ and $\sum_{n=1}^{\infty}\left| \langle f, r_n\rangle \right|^2<\infty$, where $\langle f, g\rangle= \int_{0}^{1} f(t)\cdot \overline{g(t)}\, dt$.