Inversion of $\alpha$-sine and $\alpha$-cosine transforms on $\mathbb{R}$

TL;DR

This paper develops new methods for inverting the $oldsymbol{ extalpha}$-sine and cosine transforms on $oldsymbol{ extbf{R}}$, providing series representations, explicit formulas, and numerical techniques for function reconstruction.

Contribution

It introduces a series representation for the $oldsymbol{ extalpha}$-sine transform involving Fourier transforms and hypergeometric coefficients, and extends inversion techniques to related transforms.

Findings

Derived explicit series representations for the $oldsymbol{ extalpha}$-sine transform.

Constructed a linear system for approximating the Fourier transform from the $oldsymbol{ extalpha}$-sine transform.

Validated the methods through extensive numerical analysis and comparisons.

Abstract

We consider the $\alpha$-sine transform of the form $T_\alpha f(y)=\int_0^\infty\vert\sin(xy)\vert^\alpha f(x)dx$ for $\alpha>-1$, where $f$ is an integrable function on $\mathbb{R}_+$. First, the inversion of this transform for $\alpha>1$ is discussed in the context of a more general family of integral transforms on the space of weighted, square-integrable functions on the positive real line. In an alternative approach, we show that the $\alpha$-sine transform of a function $f$ admits a series representation for all $\alpha>-1$, which involves the Fourier transform of $f$ and coefficients which can all be explicitly computed with the Gauss hypergeometric theorem. Based on this series representation we construct a system of linear equations whose solution is an approximation of the Fourier transform of $f$ at equidistant points. Sampling theory and Fourier inversion allow us to compute an estimate of $f$ from its $\alpha$-sine transform. The same approach can be extended to a similar $\alpha$-cosine transform on $\mathbb{R}_+$ for $\alpha>-1$, and the two-dimensional spherical $\alpha$-sine and cosine transforms for $\alpha>-1$, $\alpha\neq 0,2,4,\dots$. In an extensive numerical analysis, we consider a number of examples, and compare the inversion results of both methods presented.