Inversion of α-sine and α-cosine transforms on ℝ
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We consider the α-sine transform of the form T_α f(y)=∫_0^∞|sin(xy)|^α f(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.
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