Let \ be a nonnegative homogeneous function on R n . General structure of the set of numerical pairs ($, \*), for which the function (1&\ \* (x)) $ + is positive definite on R n is investigated; a criterion for positive definiteness of this function is given in terms of completely monotonic function
On Different Definitions of Positive Definiteness
β Scribed by J. Friedrich; L. Klotz
- Publisher
- John Wiley and Sons
- Year
- 1991
- Tongue
- English
- Weight
- 147 KB
- Volume
- 152
- Category
- Article
- ISSN
- 0025-584X
No coin nor oath required. For personal study only.
π SIMILAR VOLUMES
Let f be a positive definite function on a locally compact abelian group G. In [3] we showed that measurability of 1 on an open neighbourhood of the zero implies measurability of f on G. As a main tool we used a result about the support of f [3, Th. I]. The aim of this note is to simplify the proof
Continuous bizonal positive definite kernels on the spheres in β«ήβ¬ q are shown to be a series of disk polynomials with nonnegative coefficients. These kernels are the complex analogs of the so-called positive definite functions on real spheres intro-Ε½ . duced and characterized by I. J. Schoenberg 19
We give a complete characterization of the strictly positive definite functions on the real line. By Bochner's theorem, this is equivalent to proving that if the separated sequence of real numbers [a n ] describes the points of discontinuity of a distribution function, there exists an almost periodi
Let G be a locally compact commutative group and let g and h be positive definite functions on G, which are not identically zero. We show that continuity of gh implies the existence of a character y of Gd (the discrete version of G) such that yg and y h are continuous. As corollary we get a special