The system $\{ e^{i\lambda_n t}\}_{n\in\mathbb Z}$ is a Riesz basis for $L^2(-\pi,\pi)$, for $\lambda_n\in\mathbb R$ and under Kadec's condition; see:
M.I. Kadec, The exact value of the Paley-Wiener constant, Soviet Math. Dokl., 5 (1964), 559-561.
This result has many application, fro example, in sampling theory of bandlimited functions. The Besicovitch almost periodic functions in $B^2$ (the Besicovitch space) have an expansion as $$\sum a_ne^{i\lambda_n t}$$ with $\sum a_n^2$ finite and $\lambda_n$ real. Also these functions have some application (in signal theory, for example).
My question is in the title of this post. Do you know any application (in certain space) of $\{ e^{i\lambda_n t}\}_{n\in\mathbb Z}$ for $\lambda_n\in\mathbb C$? Answers, references (book, papers etc) are welcome.
Thanks in advance.