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The Fourier Transforms of Smooth Measures on Hypersurfaces of Rn + 1

Published online by Cambridge University Press:  20 November 2018

Bernard Marshall
Bernard Marshall*
Affiliation:
McGill University, Montreal, Quebec
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The Fourier transform of the surface measure on the unit sphere in Rn + 1, as is well-known, equals the Bessel function

Its behaviour at infinity is described by an asymptotic expansion

The purpose of this paper is to obtain such an expression for surfaces Σ other than the unit sphere. If the surface Σ is a sufficiently smooth compact n-surface in Rn + 1 with strictly positive Gaussian curvature everywhere then with only minor changes in the main term, such an asymptotic expansion exists. This result was proved by E. Hlawka in [3]. A similar result concerned with the minimal smoothness of Σ was later obtained by C. Herz [2].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 38 , Issue 2 , 01 April 1986 , pp. 328 - 359
Copyright
Copyright © Canadian Mathematical Society 1986

References

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