Norm dependent positive definite functions and measures on vector spaces

About the density of spectral measure of the two-dimensional SaS random vector

“…Using now the characterization (1) we have that exp − (|a| p + |b| p ) 1/p is a characteristic function of some 1-stable random vector Y = (Y 1 , Y 2 ). This fact was very well known for p ∈ (0, 2], for p > 2 this is a part of the main result in [6].…”

“…The proof of this theorem given in [6] implies that for the norm c smooth enough the measure µ is absolutely continuous with respect to the Lebesgue measure and it has the density function given by:…”

A Short Proof of Schoenberg's Conjecture on Positive Definite Functions

“…In particular, for very q > 2 the class φ ∞ (q) contains no functions besides f ≡ 1. For some partial results on the classes φ n (q), 0 < q < 2, see Richards [26,27] and Misiewicz [21]. Misiewicz [22] proved that for n ≥ 3 a function f (max(|x 1 |, .…”

“…In the case n = 2 we have to prove only that B 2 (q) ∩ (1, ∞) = ∅. In fact, it is well-known that exp(− x β ) is a positive definite function for every two-dimensional norm and every β ∈ (0, 1], see Ferguson [9], Hertz [12], Lindenstrauss and Tzafriri [20], Dor [7], Misiewicz and Ryll-Nardziewski [23], Yost [31], Koldobsky [14] for different proofs.…”

Second Derivative Test for Intersection Bodies

“…For a long time, the connection with stable random vectors and positive definite functions had been the main source of results on isometric embedding in L p (see [1,11,15,17,18,23,24]). However, it turns out to be quite difficult to check whether exp(− x p ) is positive definite for certain norms.…”