Transplantation formulae and hadamard's method of descent

Abstract

We define a partial Radon transform mapping functions on ℝn+l to functions on ℝn which intertwines the Laplace operator on the two spaces. As a consequence, transplantation formulae relating the radial eigenfunctions of the Laplacian on Euclidean spaces of different dimensions are obtained. Our formulae provide a geometric interpretation of integral formulae for Bessel functions of Abel type, which are found useful in potential theory. The formulae portray a view of Hadamard's method of descent within the realm of harmonic analysis, allowing the transplant of local problems from even dimensions to odd dimensions and unifying the techniques of several authors.

Document Type

Article

DOI

https://doi.org/10.1017/S0013091505000933

Keywords

Bessel functions, Fourier transform, Method of descent

Publication Date

6-1-2007

Journal Title

Proceedings of the Edinburgh Mathematical Society

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