Criteria of Pólya type for radial positive definite functions

By Tilmann Gneiting. 2001

58 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/a8c1c31f28de4f3a9e49c8d5d722b5934537035c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Mathematical Physics

In Numerical Analysis

Citations in the corpus

Gaussian multiplicative chaos and applications: A review (2013, methodology): “By using a Chasles relation in the integral of the right-hand side, proving that this kernel is of σ-positive type thus boils down to considering the possible values of μ > 0 such that the function (1−|x|)+ is of positive type: this is the Kuttner-Golubov problem (see [77]).”
Gaussian multiplicative chaos revisited (2008, background): “We are therefore led to considering the μ > 0 such that (1−|x|)+ is positive definite (the so called Kuttner-Golubov problem: see [10] for an introduction).”
The Wiener algebra of absolutely convergent Fourier integrals: an overview (2012, background): “For the detailed bibliography, see [171] and [51].”
Strictly and non-strictly positive definite functions on spheres (2011, background): “Gneiting (2001) reviews and develops Pólya criteria on Euclidean spaces.”
Spatio-Temporal Covariance and Cross-Covariance Functions of the Great Circle Distance on a Sphere (2016, methodology): “The names given to each of them (first column) are either parenthetical to the spatial margin, according to Gneiting (2013) (second and fourth entries), or to the coefficients bk in the series expansion in Equation (6).”
On a maximum principle and its application to logarithmically critical Boussinesq system (2009, background): “For example in dimension one the celebrated criterion of Pólya [1949] states that if F : [0,+∞[ → R is continuous and convex with F(0)= 1 and limr→+∞ F(r)= 0 then f (x)= F(|x |) is positive definite. This criterion was extended to higher dimensions by numerous authors [Askey 1973; Gneiting 2001; Tri”
Modelling spatio-temporal data: A new variogram and covariance structure proposal (2007, background): “Direct calculation of the Hankel transform is ‘‘often impossible’’ (Gneiting, 2001), so we must recur to a sufficient condition.”
Analogies and correspondences between variograms and covariance functions (2001, background): “Taking k = 0 and l = 1 in Theorem 1.1 of Gneiting (2001), we see that if α ∈ (0, 2) then C(t) = (3 − α)(4 − α) 3(2 − α)2 − |t | α, |t | ≤ 1, α(2 − α)2 48 ( 4 − α 2 − α + 3|t | )( 4 − α 2 − α − |t | )3 , 1 ≤ |t | ≤ 4 − α 2 − α , 0, |t | ≥ 4 − α 2 − α , is a covariance function in R.”