Inversion of α-sine and α-cosine transforms on $R$

We consider the α-sine transform of the form $T_{\alpha }f\left(y\right)={\int }_0^{\infty }\left|sin\right(xy\left){|}^{\alpha }f\right(x)dx$ for α > −1, where f is an integrable function on $R_{+}$. 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 $R_{+}$ for α > −1, and the two-dimensional spherical α-sine and cosine transforms for α > −1, α ≠ 0, 2, 4, …. In an extensive numerical analysis, we consider a number of examples, and compare the inversion results of both methods presented. (paper)