Premium
A Short Proof of Schoenberg's Conjecture on Positive Definite Functions
Author(s) -
Koldobsky Alexander,
Lonke Yossi
Publication year - 1999
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609399006037
Subject(s) - mathematics , positive definite matrix , conjecture , combinatorics , simple (philosophy) , norm (philosophy) , mathematics subject classification , function (biology) , pure mathematics , eigenvalues and eigenvectors , law , philosophy , physics , epistemology , quantum mechanics , evolutionary biology , political science , biology
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 spacesl p n , 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 thel q nnorms with q > 2 and n ⩾ 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 L p spaces for 0 < p ⩽ 2 are also discussed. 1991 Mathematics Subject Classification 42A82, 46B04.