Riesz bases from orthonormal bases by replacement

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.