On the Stability of Multivariate Trigonometric Systems

Abstract

Kadec's 1/4-theorem says that if {λn: n∈Z} is a sequence of real numbers for which |λn−n|≤L<14, then {eiλnω: n∈Z} forms a Riesz basis for L2[−π,π]. S. Favier, R. Zalik, C. Chui, and X. Shi extended this result to the multivariate case. But their results lead to very small stability bounds. In this paper, we give an optimal stability bound for the multivariate trigonometric systems. Moreover, for the case of Fourier frames in L2[−π,π]d, we also give the stability bounds.