# Stability theorems for exponential bases in $L^2$

Let $D$ be a domain of $\R^d$; we say that $L^2(D)$ has an exponential basis if there exists

sequence of functions ${\mathcal B}=\{ e^{2\pi i \langle s_m x\rangle}\}_{ m \in Z^d}$,

with $s_m\in\R^d$, with the following property: every function in $L^2(D)$ can be written in

a unique way as $\sum_{m\in\Z^d} c_m e^{ 2\pi i \langle s_m, x\rangle} $, with $c_m \in \C$.

For example, $\{ e^{2\pi i mx}\} _{m \in Z}$ is an exponential basis of $L^2(0, 1 )$.

Exponential bases are very useful in the application, especially when they are orthogonal; however,

the existence or non-existence of exponential bases is proved only on very special domains of $\R^d$.

In particular, it is not known whether the unit ball in $\R^2$ has an exponential basis or not.

An important property of exponential bases is their stability. That is, if $\{ e^{2\pi i \langle s_m,

x\rangle}\}_{ m \in Z^d}$ is an exponential basis of $L^2(D)$ and $\Delta=\{\delta_m\}_{ m \in Z^d} $ is

a sequence of sufficiently small real number, then also $\{ e^{2\pi i \langle s_m+\delta_m,

x\rangle}\}_{ m \in Z^d}$ is an exponential basis of $L^2(D)$. In this talk I will discuss the existence

and stability of exponential bases on special 2-dimensional domains called trapezoids.

I will also generalize a celebrate theorem by M. Kadec and obtain stability bounds for exponential bases on domains of $\R^d$.

The result that I will present in my talk are part of joint projects with my students A. Kumar and S. Pathak.

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- Last modified: 5/05/2014