Chirp transforms and Chirp series

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 eiϕ(t)θ(ω), the Chirp transform is an unitary isometry from L2(R,dϕ) onto L2(R,dθ) and it can be explicitly defined in terms of generalized Hermite polynomials. The corresponding Chirp series take einθ(t) as a basis which in some sense is dual to the theory of non-harmonic Fourier series which take eiλnt 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.