Abstract In this study, a weighted generalization of the Fourier transform using a weight function w with respect to a function g is defined and some fundamental properties are examined. Then, the existence of the weighted Fourier transform is proven, and the transforms of some example functions are calculated to facilitate understanding of the transform, and then the inverse of the weighted Fourier transform is given. Furthermore, a new weighted convolution operation is defined, and a weighted convolution theorem is proven. Finally, the weighted Fourier transform of the weighted derivative and the fractional weighted derivative of a function f with a weighted w respect to another function g is investigated. Additionally, the certain theorems are verified with examples, and our study is enriched with numerical graphs.
Sermutlu et al. (Mon,) studied this question.