Riesz bases from orthonormal bases by replacement

Abstract

Given an orthonormal basis $\mathcal{V}=\{v_j{\}}_{j\in N}$ in a separable Hilbert space $H$ and a set of unit vectors $\mathcal{B}=\{w_j{\}}_{j\in N}$, we consider the sets ${\mathcal{B}}_N$ obtained by replacing the vectors $v_1,...,v_N$ with vectors $w_1,...,w_N$. We show necessary and sufficient conditions that ensure that the sets ${\mathcal{B}}_N$ are Riesz bases of $H$ and we estimate the frame constants of the ${\mathcal{B}}_N$. Then, we prove conditions that ensure that $\mathcal{B}$ is a Riesz basis. Applications to the construction of exponential bases on domains of $R^d$ are also presented.