Geodesic
Counterexample to a Conjecture on Positive Definite Functions
Canadian journal of mathematics, Tome 24 (1972) no. 5, pp. 926-929

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Stewart, James. Counterexample to a Conjecture on Positive Definite Functions. Canadian journal of mathematics, Tome 24 (1972) no. 5, pp. 926-929. doi: 10.4153/CJM-1972-092-2
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[1] 1. Cooper, J. L. B., Positive definite functions of a real variable, Proc. London Math. Soc. (3) 10 (1960), 53–66. Google Scholar

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