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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

REFERENCES

(1)Courant, R. and Hilbert, D.Methods of mathematical physics, vol. 1 (Inter-science, New York, 1953).Google Scholar
(2)Dahlberg, E.On the completeness of eigenfunctions of some higher order operators. J. Math. Anal. Appl. 44 (1973), 227–237.CrossRefGoogle Scholar
(3)Dalzell, D. P.On the completeness of Dini's series. J. London Math. Soc. 20 (1945), 213–218.CrossRefGoogle Scholar
(4)Higgins, J. R.Completeness and basis properties of sets of special functions (Cambridge University Press, 1977).CrossRefGoogle Scholar
(5)Hochstadt, H.The mean convergence of Fourier-Bessel series. Siam Rev. 9 (1967), 211–218.CrossRefGoogle Scholar
(6)Hochstadt, H.The functions of mathematical physics (Wiley-Interscience, New York, 1971).Google Scholar
(7)Magnus, W., Oberhettinger, F. and Soni, R. P.Formulas and theorems for the special functions of mathematical physics (Springer, Berlin, 1966).CrossRefGoogle Scholar
(8)Mclachlan, N. W.Bessel functions for engineers, 2nd ed. (Clarendon Press, Oxford, 1961).Google Scholar
(9)Rayleigh, J. W. S. (3rd Baron). The theory of sound, vol. 1, 2nd ed. (Macmillan, London, 1894).Google Scholar
(10)Riesz, F. and Sz.-Nagy, B.Functional analysis, 2nd ed., trans. Boron, L. F. (Blackie, London and Glasgow, 1956).Google Scholar
(11)Watson, G. N.A treatise on the theory of Bessel functions, 2nd ed. (Cambridge University Press, 1944).Google Scholar
(12)Whittaker, E. T. and Watson, G. N.A course of modern analysis, 4th ed. (Cambridge University Press, 1927).Google Scholar