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.
Categories: math.FA