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.