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

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