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A short proof of Schoenberg’s conjecture on positive definite functions. (English) Zbl 1020.42005

In 1938, I. J. Schoenberg asked for which positive numbers \(p\) is the function \(\exp(-\|x\|^p)\) positive definite, where the norm is taken from one of the spaces \(\ell^n_q\), \(q> 2\). The solution of the problem was completed in 1991 [A. L. Koldobskij, St. Petersbg. Math. J. 3, No. 3, 563-570 (1992); translation from Algebra Anal. 3, No. 3, 78-85 (1991; Zbl 0741.60010)], by showing that for every \(p\in (0,2]\) the function \(\exp(-\|x\|^p)\) is not positive definite for the \(\ell^n_q\) norms with \(q> 2\) and \(n\geq 3\). We prove a similar result for a more general class of norms, which contains some Orlicz spaces and \(q\)-sums, and, in particular, present a simple proof of the answer to the original Schoenberg’s question. Some consequences concerning isometric embeddings in \(L_p\) spaces for \(0< p\leq 2\) are discussed as well.

MSC:

42A82 Positive definite functions in one variable harmonic analysis
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Citations:

Zbl 0741.60010
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