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Some orthogonal and complete sets of Bessel functions associated with the vibrating plate

Published online by Cambridge University Press:  24 October 2008

J. R. Higgins
J. R. Higgins
Affiliation:
The Cambridgeshire College of Arts and Technology
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Some orthogonal sets of Bessel functions of real order v are identified using the equation Δ2u = utt of the vibrating plate. Our main concern is with the L2 completeness of such sets, and we prove that the well known ‘clamped edge’ type is complete for v > -1, thus completing a result of E. Dahlberg. We also study a very closely related set and show that it needs an extra (non-Bessel) function for completeness.

Our method for proving the completeness is based on one given by H. Hochstadt in connection with Dini functions. We have found it necessary to reorganize Hoch-stadt's method and correct some errors contained in it.

Certain isolated values of v require special attention and we treat these by subjecting the Dalzell completeness criterion to a continuity argument.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 91 , Issue 3 , May 1982 , pp. 503 - 515
Copyright
Copyright © Cambridge Philosophical Society 1982

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References

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