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The Fourier Transforms of Smooth Measures on Hypersurfaces of Rn + 1
Published online by Cambridge University Press: 20 November 2018
Extract
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].
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- Copyright © Canadian Mathematical Society 1986
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