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Counterexample to a Conjecture on Positive Definite Functions
Published online by Cambridge University Press: 20 November 2018
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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.
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- Copyright © Canadian Mathematical Society 1972