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Kernels with only a finite number of characteristic values

Published online by Cambridge University Press:  24 October 2008

Dale W. Swann
Dale W. Swann
Affiliation:
Bell Telephone Laboratories, Incorporated, Whippany, New Jersey, 07981
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Extract

Let K(s, t) be a complex-valued L2 kernel on the square ⋜ s, t ⋜ by which we mean

and let {λν}, perhaps empty, be the set of finite characteristic values (f.c.v.) of K(s, t), i.e. complex numbers with which there are associated non-trivial L2 functions øν(s) satisfying

For such kernels, the iterated kernels,

are well-defined (1), as are the higher order traces

Carleman(2) showed that the f.c.v. of K are the zeros of the modified Fredhoim determinant

the latter expression holding only for |λ| sufficiently small (3). The δn in (3) may be calculated, at least in theory, by well-known formulae involving the higher order traces (1). For our later analysis of this case, we define and , respectively, as the minimum and maximum moduli of the zeros of , the nth section of D*(K, λ).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 70 , Issue 2 , September 1971 , pp. 257 - 262
Copyright
Copyright © Cambridge Philosophical Society 1971

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References

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