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

Abstract

We consider the $α$-sine transform of the form $T_αf(y)=\int_0^\infty\vert\sin(xy)\vert^αf(x)dx$ for $α>-1$, where $f$ is an integrable function on $\mathbb{R}_+$. First, the inversion of this transform for $α>1$ 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 $α$-sine transform of a function $f$ admits a series representation for all $α>-1$, 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 $α$-sine transform. The same approach can be extended to a similar $α$-cosine transform on $\mathbb{R}_+$ for $α>-1$, and the two-dimensional spherical $α$-sine and cosine transforms for $α>-1$, $α\neq 0,2,4,\dots$. In an extensive numerical analysis, we consider a number of examples, and compare the inversion results of both methods presented.