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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