The Functional Equivalence of Stable and Unstable Inversions of the Mellin Transform

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