Riesz bases from orthonormal bases by replacement

TL;DR

This paper investigates conditions under which modified orthonormal bases in a Hilbert space form Riesz bases after replacing some vectors, providing criteria, estimates, and applications to exponential bases in Euclidean domains.

Contribution

It establishes necessary and sufficient conditions for Riesz basis formation from orthonormal bases with replacements, including estimates of frame constants and applications to exponential bases.

Findings

Conditions for Riesz bases after vector replacements

Estimates of frame constants for the modified bases

Applications to exponential bases on domains in R^d

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.