Chirp transforms and chirp series. (English) Zbl 1200.42026

The theory of non harmonic Fourier series was originated by Paley and Wiener, and more recently, in a series of papers, many questions were studied. The authors introduce an integral version of the non-harmonic Fourier series, called Chirp transform. It becomes from the Fourier integral transform, if one replaces the kernel by \(e^{i\phi(t)\theta(\omega)}\), for certain functions \(\phi\) and \(\theta\), which is called Chirp atom. They consider the Chirp series, which is a discrete version of the Chirp transform associated to the Chirp atoms. They also establish the Chirp version of the Shannon sampling theorem and the Poisson summation formula. Finally they show that Chirp transform can be applied to some certain differential equations with non-constant coefficients.


42C20 Other transformations of harmonic type
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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