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Harmonic analysis of scalar and vector fields in n

Published online by Cambridge University Press:  24 October 2008

J. Denmead Smith
J. Denmead Smith
Affiliation:
Department of Mathematics, University of Reading
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Abstract

It is shown that a real scalar function in n which is of class Cn and either has zero mean on all spheres of unit radius, or has zero mean in all balls of unit radius admits a unique expansion in terms of eigenfunctions of the Laplacian operator. In a similar manner, a suitably smooth vector-valued function in n which has zero flux through all spheres cf unit radius is shown to be decomposable as the sum of a solenoidal part and a series of conservative parts that are eigenfunctions of the Laplacian. Applications are given, including some in complex analysis.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 72 , Issue 3 , November 1972 , pp. 403 - 416
Copyright
Copyright © Cambridge Philosophical Society 1972

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References

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