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 e i ϕ ( t ) θ ( ω ) , the Chirp transform is an unitary isometry from L 2 ( R , d ϕ ) onto L 2 ( R , d θ ) and it can be explicitly defined in terms of generalized Hermite polynomials. The corresponding Chirp series take e i n θ ( t ) as a basis which in some sense is dual to the theory of non-harmonic Fourier series which take e i λ 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. Qianʹs 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.