It is well-known that $\{e^{i n t}\}\_{n\in\mathbb Z}$ is an orthonormal basis for $L^2(-\pi,\pi)$. A theorem by Kadec (Kadec $1/4$ theorem) studies the perturbed exponential system:



> 
> If $\{\lambda\_n\}$ is a sequence of real numbers for which
>  $$|\lambda\_n-n|\leqq L<\frac{1}{4}, \ \ n=0, \pm 1, \pm 2, \dots$$
>  then $\{e^{i \lambda\_n t}\}\_{n\in\mathbb Z}$ satisfies the
>  Paley-Wiener criterion and so forms a Riesz basis for $L^2(-\pi,\pi)$.
> 
> 
> 


Any Hilbert space $\overline{\operatorname{span}}\left(\phi\_k\right)$ can be endowed with different equivalent norms, i.e. if $\|\cdot\|$ is a norm of $\overline{\operatorname{span}}\left(\phi\_k\right)$, then $\|\cdot\|'$ obeying
$$c\_1\|\Psi\|\leq \|\Psi\|'\leq c\_2\|\Psi\|, \ \ \forall \Psi\in \overline{\operatorname{span}}\left(\phi\_k\right); \ \ c\_1,c\_2>0$$
is also a norm of $\overline{\operatorname{span}}\left(\phi\_k\right)$. If a basis set is a Riesz system with respect to $\|\cdot\|$, one can always choose a second equivalent norm $\|\cdot\|'$ such that the basis set becomes orthonormal with respect to the appertaining scalar product $(\cdot,\cdot)'$. If A is the Schmidt matrix, which orthonormalizes a Riesz system, then
$$\|\Psi\|'=\|A\Psi\|, \ \ \forall \Psi\in \overline{\operatorname{span}}\left(\phi\_k\right).$$
For an application that I'm studying, I'd like to calculate $A$, which orthonormalizes Riesz system $\{e^{i \lambda\_n t}\}\_{n\in\mathbb Z}$ (under Kadec assumption). But I do not know where to start. Any suggestions please? Bibliography references and answers are welcome.



Transform Riesz basis $\{e^{i \lambda_n t}\}_{n\in\mathbb Z}$ to an orthogonal basis
In the conditions of Kadec's result , you can write $A=(I-S)^{-1}=\sum\_{m=0}^\infty S^m,$ where $S(f)(x)=\sum\_{n=-\infty}^\infty \hat f(n) (e^{inx}-e^{i\lambda\_n x})$ verifies $\Vert S\Vert<1.$ Here $\{ \hat f (n)\} $ are the Fourier coefficients of $f$. 


Then $A(e^{i\lambda\_n x})= e^{in x}$ and so $A$ orthogonalizes the Riesz basis. The fact that $\Vert S\Vert<1$ can be found for example in Benzinger, Nonharmonic Fourier series ans Spectral Theory, 1987, TAMS.


