New Convolutions for Quadratic-Phase Fourier Integral Operators and their Applications
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We obtain new convolutions for quadratic-phase Fourier integral operators (which include, as subcases, e.g., the fractional Fourier transform and the linear canonical transform). The structure of these convolutions is based on properties of the mentioned integral operators and takes profit of weight-functions associated with some amplitude and Gaussian functions. Therefore, the fundamental properties of that quadratic-phase Fourier integral operators are also studied (including a Riemann–Lebesgue type lemma, invertibility results, a Plancherel type theorem and a Parseval type identity). As applications, we obtain new Young type inequalities, the asymptotic behaviour of some oscillatory integrals, and the solvability of convolution integral equations.
KeywordsConvolution Young inequality oscillatory integral convolution integral equation fractional Fourier transform linear canonical transform
Mathematics subject classificationPrimary 44A35 Secondary 42A38 42A85 43A32 44A20 45E10
The first-named author was supported in part by FCT–Portuguese Foundation for Science and Technology through the Center for Research and Development in Mathematics and Applications (CIDMA) at Universidade de Aveiro, within the Project UID/MAT/04106/2013. The two last named-authors were partially supported by the Viet Nam National Foundation for Science and Technology Development (NAFOSTED).
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