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Counterexample to a Conjecture on Positive Definite Functions

Published online by Cambridge University Press:  20 November 2018

James Stewart
James Stewart*
Affiliation:
McMaster University, Hamilton, Ontario
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Cooper [1] called a complex-valued function f on the real line Rpositive definite for F, where F is a set of complex-valued functions on R, if the integral

exists as a Lebesgue integral and is nonnegative for every ϕ in F. Let us denote by P(F) the set of functions positive definite for F, by the set of functions in LP(R) with compact support, and by the set of functions which are locally in LP(R), i.e., for every compact subset K of R. Cooper showed that for any p ≧ 2 and that each function in is essentially bounded and hence equal almost everywhere to an ordinary continuous positive definite function in the sense of Bochner.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 5 , October 1972 , pp. 926 - 929
Copyright
Copyright © Canadian Mathematical Society 1972

References

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