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Radial Positive Definite Functions Generated by Euclid's Hat
- Source :
- Journal of Multivariate Analysis. 69:88-119
- Publication Year :
- 1999
- Publisher :
- Elsevier BV, 1999.
-
Abstract
- Radial positive definite functions are of importance both as the characteristic functions of spherically symmetric probability distributions, and as the correlation functions of isotropic random fields. The Euclid's hat functionhn(‖x‖),x∈Rn, is the self-convolution of an indicator function supported on the unit ball in Rn. This function is evidently radial and positive definite, and so are its scale mixtures that form the classHn. Our main results characterize the classesHn,n⩾1, andH∞=∩n⩾1Hn. This leads to an analogue of Pólya's criterion for radial functions on Rn,n⩾2: Ifϕ:[0,∞)→R is such thatϕ(0)=1,ϕ(t) is continuous, limt→∞ϕ(t)=0, and(−1)kdkdtk[−ϕ′(t)]is convex fork=[(n−2)/2], the greatest integer less than or equal to (n−2)/2, thenϕ(‖x‖) is a characteristic function in Rn. Along the way, side results on multiply monotone and completely monotone functions occur. We discuss the relations ofHnto classes of radial positive definite functions studied by Askey (Technical Report No. 1262, Math. Res. Center, Univ. of Wisconsin–Madison), Mittal (Pacific J. Math.64(1976), 517–538), and Berman (Pacific J. Math.78(1978), 1–9), and close with hints at applications in geostatistics.
- Subjects :
- Statistics and Probability
Unit sphere
Numerical Analysis
Characteristic function (probability theory)
positive definite
multiply monotone
scale mixture
Center (group theory)
Positive-definite matrix
Function (mathematics)
completely monotone
radial
Askey's theorem
Combinatorics
characteristic function
Correlation function (statistical mechanics)
Monotone polygon
Indicator function
Pólya's criterion
geostatistics
isotropic
correlation function
Statistics, Probability and Uncertainty
Euclid's hat
Mathematics
Subjects
Details
- ISSN :
- 0047259X
- Volume :
- 69
- Database :
- OpenAIRE
- Journal :
- Journal of Multivariate Analysis
- Accession number :
- edsair.doi.dedup.....d050df593bd68e47d078ee978618c86a