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The Schoenberg problem on positive definite functions. (Russian) Zbl 0741.60010

For every \(n\geq 3\), \(q>2\), \(\beta>0\), the function \(\exp(-(| x_ 1|^ q+\dots+| x_ n|^ q)^{\beta/q})\) is not positive definite. This result gives an answer to I. J. Schoenberg’s question posed in [Trans. Am. Math. Soc. 44, 522-536 (1938; Zbl 0019.41502)].

MSC:

60E10 Characteristic functions; other transforms
42A82 Positive definite functions in one variable harmonic analysis

Citations:

Zbl 0019.41502
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