Nonharmonic Fourier series

Remarks

When $A = \pi$ and $\lambda_n = n$, this is an ordinary Fourier series, and the $e_n / \sqrt{2\pi}$ form an orthonormal basis. In general the $e_n$ are not orthogonal, and the theory asks which sequences $\Lambda$ make $(e_n)$ complete, a Riesz basis or a frame of $L^2(-A, A)$. When $(e_n)$ is a Riesz basis, every $f \in L^2(-A, A)$ has a unique expansion $f = \sum c_n e_n$, and $\sum \abs{c_n}^2$ is comparable to $\norm{f}^2_{L^2}$.

Example: let $A = \pi$ and $\lambda_n = n + \delta_n$ with real $\delta_n$ and $\sup_n \abs{\delta_n} < 1/4$. By Kadec's $1/4$ theorem, $(e^{i\lambda_n t})$ is a Riesz basis of $L^2(-\pi, \pi)$, so every function in $L^2(-\pi, \pi)$ is the sum of a unique nonharmonic Fourier series with these frequencies. The constant $1/4$ cannot be replaced by a larger number.

Non-example: let $A = \pi$ and $\lambda_n = 2n$. The functions $e^{2int}$ all have period $\pi$, so they cannot represent $f = 1_{(0, \pi)}$, which is not $\pi$-periodic on $(-\pi, \pi)$. So the closed span of $(e_n)$ is not all of $L^2(-\pi, \pi)$ and the system is not complete.