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We revisit the Fourier transform on  space. We have known that the Fourier transform is well defined on  space. Consequently, some applications of the Fourier transform, especially on differential equations, is limited by the initial conditions that must be in . In this paper, we shall introduce generalization of the Fourier transform on  spaces with  which is called by weighted Fourier Transform, and discuss its properties. The invers transform  is studied and we also obtain a result which describe how the wighted Fourier transform is applied to solve generalization of the heat equation problem.

Fourier transform; L^p spaces; wighted Fourier transform; invers transform; heat equation