IJPAM: Volume 108, No. 3 (2016)

THE FUNCTIONAL EQUIVALENCE OF STABLE AND
UNSTABLE INVERSIONS OF THE MELLIN TRANSFORM

William G. Hawkins
College of Optical Sciences
University of Arizona
Tucson, AZ 44122, USA


Abstract. We use the term ``stable" to mean numerical stability, that the formal solution of an integral equation can be well-approximated using standard methods, including but not limited to hypergeometric functions or numerical integration. Thus, valid trial solutions may be found by symbolic integration or computerized numerical integration. We specifically analyze a class of forward problems that can be expressed as Meijer G-Functions. In Euclidean spaces $\mathbb{R}^n$, the conditions under which an unstable inverse of the Mellin Transform has an equivalent stable inverse are established. These inverses are formally and analytically equivalent, so that any closed form solution of one is also a solution of the other. For this reason, it is a functional equivalence. Mathematically, at least, this non-uniqueness is benign, much like the indeterminacy of a square root or phase. But the mathematical formalism and feasibility of numerical modeling of these inverses may be radically different. As such, we have shown that a linear inverse may not be unique. The inverse harmonic Radon Transform is an example of this difficulty. We demonstrate that, under certain easily satisfied conditions, a stable inversion does exist.

More importantly, a stable low pass inversion is an accurate picture of the underlying physics if the forward problem exists and is a Hermitian operator. We demonstrate these results with the $n$-dimensional Radon Harmonic Transform. We also provide the reader with brief introductions to ultra-harmonic functions, the Funk-Hecke Theorem, and the methods of the Mellin Transform applied to generalized hypergeometric functions.

Received: June 16, 2016

AMS Subject Classification: 45Q05, 44A12, 47A52, 35R30, 33C20, 33C55, 42A38

Key Words and Phrases: Mellin transforms, Slater's Theorem, Funk-Hecke Theorem, $n$-dimensional radon transform, harmonic analysis, ill-posed problems, Meijer G-functions, Fox H-functions, Fredholm integral equations of the first kind, ultra-spherical polynomials

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DOI: 10.12732/ijpam.v108i3.17 How to cite this paper?

Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2016
Volume: 108
Issue: 3
Pages: 671 - 707


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CC BY This work is licensed under the Creative Commons Attribution International License (CC BY).