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Mercer's Theorem and Fredholm resolvents

Published online by Cambridge University Press:  17 April 2009

C.S. Withers
C.S. Withers
Affiliation:
Applied Mathematics Division, Department of Scientific and Industrial Research, Wellington, New Zealand.
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Abstract

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Multivariate versions of Mercer's Theorem and the usual expansions of the resolvent and Fredholm determinant are shown to hold for an n × n symmetric kernel N(x, y) with arbitrary domain in Rp under weakened continuity conditions. Further, the resolvent and determinant of N(x, y) − a(x)b(y) are given in terms of those of N(x, y).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 11 , Issue 3 , December 1974 , pp. 373 - 380
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Carleman, T., “Zur Theorie der linearen Integralgleichungen”, Math. Z. 9 (1921), 196–217.CrossRefGoogle Scholar
[2]Deutsch, Ralph, Estimation theory (Prentice-Hall, Englewood Cliffs, New Jersey, 1965).Google Scholar
[3]Hobson, E.W., “On the representation of the symmetrical nucleus of a linear integral equation”, Proc. London Math. Soc. (2) 14 (1915), 5–30.CrossRefGoogle Scholar
[4]Michlin, S., “On the convergence of Fredholm series”, C.R. (Doklady) Acad. Sci. URSS (NS) 42 (1944), 373–376.Google Scholar
[5]Pogorzelski, W., Integral equations and their applications. Vol. 1 (International Series of Monographs in Pure and Applied Mathematics, 88. Pergamon, Oxford, New York, Frankfurt; PWN-Polish Scientific Publishers, Warsaw, 1966).Google Scholar
[6]Riesz, Frigyes and Sz.-Nagy, Béla, Functional analysis (Frederick Ungar, New York, 1955).Google Scholar
[7]Smithies, F., Integral equations (Cambridge Tracts in Mathematics and Mathematical Physics, 49. Cambridge University Press, Cambridge, 1958).Google Scholar
[8]Withers, C.S., “The characteristic function of the L 2 -norm of a Gaussian process”, submitted.Google Scholar
[9]Withers, C.S., “On the asymptotic power of statistics which are L 2 -norms”, submitted.Google Scholar