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 ${\sum }_{n=1}^{\infty }{\left|⟨f,r_n⟩\right|}^2<\infty$, where $⟨f,g⟩={\int }_0^{1}f(t)\cdot \overline{g(t)}dt$.