Momentum transforms and Laplacians in fractional spaces

TL;DR

This paper introduces a new class of unitary transformations and Laplacians in fractional and fractal spaces, extending Fourier analysis and fractional geometries to more general measures and infinite-dimensional settings.

Contribution

It generalizes Fourier transforms to fractional spaces, defines new Laplacians, and extends fractional geometries to broader measures and infinite dimensions.

Findings

Defined a class of unitary transforms between fractional position and momentum spaces

Diagonalized new Laplacian operators using these transforms

Extended fractional space concepts to infinite-dimensional and more general measures

Abstract

We define an infinite class of unitary transformations between position and momentum fractional spaces, thus generalizing the Fourier transform to a special class of fractal geometries. Each transform diagonalizes a unique Laplacian operator. We also introduce a new version of fractional spaces, where coordinates and momenta span the whole real line. In one topological dimension, these results are extended to more general measures.