On a class of non-harmonic Fourier series

In the theory of non-harmonic Fourier series, one-quarter theorems deal with basis, frames and/or series expansion properties under some extreme conditions. In this paper we show the existence of a series representation analogous to the Fourier series for square integrable functions in terms of system of complex exponentials when the sequence of exponents is close to the "extreme case" sequence {n+sign(n)?}.