Nonharmonic Gabor Expansions

Abstract

We consider Gabor systems generated by a Gaussian function and prove certain classical results of Paley and Wiener on nonharmonic Fourier series of complex exponentials for the Gabor expansion‎. ‎In particular, we prove a version of Plancherel-Po lya theorem for entire functions with finite order of growth and use the Hadamard factorization theorem to study regularity‎, ‎exactness and deficiency of Gabor systems.