The $ψ$-$ω$-Mellin transform

Abstract: This manuscript introduces a generalization of the Mellin integral transform within the framework of weighted fractional calculus with respect to an increasing function. The proposed transform is much more suitable for working with fractional operators that involve a weight and are defined with respect to a function. This work also explores the connections between the Laplace and Fourier integral transforms in the same context. To achieve this, a new formulation of the weighted Fourier integral transform with respect to a function is presented, along with a new version of the bilateral Laplace transform. We study some of the properties of these new operators, obtain an inversion formula and a convolution theorem, and also present a practical application as an illustrative example.