Riesz bases from orthonormal bases by replacement

Given an orthonormal basis $ {\mathcal V}= {v_j} {j\in N}$inaseparableHilbertspace$H$andasetofunitvectors$ {\mathcal B}={w_j}{j\in N}$,weconsiderthesets$ {\mathcal B}_N$obtainedbyreplacingthevectors$v_1, …,, v_N$withvectors$w_1,, …,, w_N$\mathrm{.}Weshownecessaryandsufficientconditionsthatensurethatthesets$ {\mathcal B}_N$areRieszbasesof$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.