Inversion of α-sine and α-cosine transforms on R. (27th July 2021)

• Record Type: Journal Article
• Authors: Hoang, Ly Viet; Spodarev, Evgeny
• Is Part Of: Inverse problems. Volume 37:Number 8(2021)
• Publication Date: 2021-07-27
• Subjects: Fourier analysis -- integral transform -- sine transform -- cosine transform -- spherical cosine transform -- hypergeometric function -- stable process; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/ac1327
• Languages: English
• ISSNs: 0266-5611

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.