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

Abstract

Abstract We consider the α-sine transform of the form T α f ( y ) = ∫ 0 ∞ | sin ( x y ) | α f ( x ) d x 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.