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

Abstract

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