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

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.