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A SHORT PROOF OF SCHOENBERG'S CONJECTURE ON POSITIVE DEFINITE FUNCTIONS

Published online by Cambridge University Press:  01 November 1999

ALEXANDER KOLDOBSKY  and
YOSSI LONKE
ALEXANDER KOLDOBSKY
Affiliation:
Division of Mathematics and Statistics, University of Texas at San Antonio, San Antonio, TX 78249, USA Current address Department of Mathematics, University of Missouri at Columbia, Columbia, MO 65211, Cleveland, USA.
YOSSI LONKE
Affiliation:
Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel Current address Department of Mathematics, Case Western Reserve University, 10900 Euclid Avenue, OH 44106-7058, USA.
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Abstract

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 [lscr ]np, q>2. The solution of the problem was completed in 1991, by showing that for every p∈(0, 2], the function exp(−∥x∥p) is not positive definite for the [lscr ]nq norms with q>2 and n[ges ]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 Schoenberg's original question. Some consequences concerning isometric embeddings in Lp spaces for 0<p[les ]2 are also discussed.

Type
NOTES AND PAPERS
Information
Bulletin of the London Mathematical Society , Volume 31 , Issue 6 , November 1999 , pp. 693 - 699
Copyright
© The London Mathematical Society 1999

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