Chirp transforms and Chirp series

Abstract

Motivated by the recent work on the non-harmonic Fourier atoms initiated by T. Qian and the non-harmonic Fourier series which originated from the celebrated work of Paley and Wiener, we introduce an integral version of the non-harmonic Fourier series, called Chirp transform. As an integral transform with kernel $e^{i\varphi (t)\theta (\omega )}$, the Chirp transform is an unitary isometry from $L^2(\mathbb{R},d\varphi )$ onto $L^2(\mathbb{R},d\theta )$ and it can be explicitly defined in terms of generalized Hermite polynomials. The corresponding Chirp series take $e^{in\theta (t)}$ as a basis which in some sense is dual to the theory of non-harmonic Fourier series which take $e^{i{\lambda }_{n}t}$ as a basis. The Chirp version of the Shannon sampling theorem and the Poisson summation formula are also considered by dealing with sampling points which may non-equally distributed. Since the Chirp transform interchanges weighted derivatives into multiplications, it plays a role in solving certain differential equations with variable coefficients. In addition, we extend T. Qian's theorem on the characterization of a measure to be a linear combination of a number of harmonic measures on the unit disc with positive integer coefficients to that with positive rational coefficients.