Fourier transforms via heat kernel

ISSN： 1065-2469

Source： Integral Transforms and Special Functions, Vol.15, Iss.4, 2004-08, pp. : 295-302

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Abstract

Making use of the heat kernel method which represents various generalized functions as initial values of the solutions of the heat equation, we interpret the Fourier transform as the initial value of the temperature transform T defined by (Tu) (x, t) = lang u, e^-^ix^-^t^||² rang and give a new proof of the Fourier inversion formula and invariance of the spaces of tempered distributions, Fourier hyperfunctions and generalized functions of Gelfand-Shilov type S under the Fourier transformation.

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