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Functions with a Finite Number of Negative Squares

Published online by Cambridge University Press:  20 November 2018

James Stewart*
Affiliation:
McMaster University, Hamilton, Ontario
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Let f be a complex-valued function defined on the real line R with the property that for every x∊R. If k is a nonnegative integer,f is said to have k negative squares, or to be indefinite of order k, if the Hermitian form

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Banach, S., Théorie des opérations linéaires, Monografje Matematyczne, Warsaw, 1932.Google Scholar
2. Cohen, P. J., Factorization in group algebras, Duke Math. J. 26 (1959), 199-205.Google Scholar
3. Cooper, J. L. B., Positive definite functions of a real variable, Proc. London Math. Soc. (3) 10 (1960), 53-66.Google Scholar
4. Gorbachuk, V.I., On integral representations of Hermitian-indefinite kernels (the case of several variables), (Russian) Ukrain. Mat. Ž. 16 (1964), 232-236.Google Scholar
5. Hewitt, E. and Ross, K. A., Abstract harmonic analysis, Vol. I, Academic Press, New York, 1963.Google Scholar
6. Iohvidov, I. S. and Krein, M. G., Spectral theory of operators in spaces with an indefinite metric I, II (Russian) Trudy Moskov. Mat. Obšč. 5 (1956), 367-432; 8 (1959), 413-496. English translation: Amer. Math. Soc. Transi. (2) 13 (1960), 105-175; (2) 34 (1963), 283-373.Google Scholar
7. Krein, M. G., The integral representation of a continuous Hermitian-indefinite function with a finite number of negative squares (Russian) Dokl. Akad. Nauk SSSR 125 (1959), 31-34.Google Scholar
8. Krein, M. G., Screw lines in infinite-dimensional Lobachevski space and the Lorentz transformation (Russian) Uspehi Mat. Nauk 3 (1948), 158-160.Google Scholar
9. Naimark, M. A., Self-adjoint extensions of the second kind of a symmetric operator (Russian) Izv. Akad. Nauk SSSR, Ser. Mat. 4 (1940), 53-89. (English summary, 90-104.)Google Scholar
10. Pontryagin, L. S., Hermitian operators in spaces with indefinite metric, (Russian) Izv.Akad. Nauk SSSR, Ser. Mat. 8 (1944), 243-280.Google Scholar
11. Shah Tao-Shing, On Conditionally Positive-Definite Generalized Functions, Sci. Sinica 11 (1962), 1147-1168.Google Scholar
12. Stewart, J., Unbounded positive definite functions, Canad. J. Math. 21 (1969), 1309-1318.Google Scholar
13. Titchmarsh, E. C., Theory of Fourier integrals, Oxford Univ. Press, London, 1937.Google Scholar