On perturbing bases of complex exponentials in $L^{2}$ $(-\pi , \pi )$

Abstract:

A sequence of complex exponentials $\{ {e^{i{\lambda _n}t}}\}$ is said to be a Riesz basis for ${L^2}( - \pi ,\;\pi )$ if each function in the space has a unique representation $f = \Sigma {c_n}{e^{i{\lambda _n}t}}$, with $A\Sigma |{c_n}{|^2} \leqslant ||f|{|^2} \leqslant B\Sigma |{c_n}{|^2}$. It is known, for example, that if $|{\lambda _n} - n| \leqslant L < 1/4( - \infty < n < \infty )$, then $\{ {e^{i{\lambda _n}t}}\}$ is a Riesz basis. In this note we show that not only the orthonormal basis $\{ {e^{int}}\}$, but any Riesz basis of complex exponentials can be suitably perturbed.