Radial Positive Definite Functions Research Articles

Continuation of Radial Positive Definite Functions and Their Characterization

This paper delves into the extension and characterization of radial positive definite functions into non-integer dimensions. We provide a thorough investigation by employing the Riemann–Liouville fractional integral and fractional Caputo derivatives, enabling a comprehensive understanding of these functions. Additionally, we introduce a secondary characterization based on the Bernstein characterization of completely monotone functions. The practical significance of our study is showcased through an examination of the positivity of the fundamental solution of the space-fractional Bessel diffusion equation, highlighting the real-world applicability of the developed concepts. Through this work, we contribute to the broader understanding of radial positive definite functions and their utility in diverse mathematical and applied contexts.

Some generalizations of the problem of positive definiteness of a piecewise linear function

In this paper, we consider the following problem concerning functions that are positive definite on a real linear space E (the set Φ(E)).Letf,g∈Φ(E)be such thatf(0)=g(0)andf≢g. We are looking for the setPf,g:={λ∈C:fλ∈Φ(E)}, wherefλ(x):=λf(x)+(1−λ)g(x),λ∈C. A particular case of this problem is g(x)≡f(ax), with a>0, a≠1, and we want to find the set Pf,a:=Pf,g. When f(x)=(1−|x|)+ we come up with the positive definiteness of a piecewise liner function. It is well-known that the set Pf,g is always an interval of the real axis. In case E=Rn, the expressions for endpoints of that interval were found. We also provide examples of functions f∈Φ(Rn), for which the set Pf,a was found, including those functions that depend on Euclidean norm (radial positive definite functions).

Radial positive definite functions and Schoenberg matrices with negative eigenvalues

The main object under consideration is a class Φ n ∖ Φ n + 1 \Phi _n\backslash \Phi _{n+1} of radial positive definite functions on R n \mathbb {R}^n which do not admit radial positive definite continuation on R n + 1 \mathbb {R}^{n+1} . We find certain necessary and sufficient conditions on the Schoenberg representation measure ν n \nu _n of f ∈ Φ n f\in \Phi _n for f ∈ Φ n + k f\in \Phi _{n+k} , k ∈ N k\in \mathbb {N} . We show that the class Φ n ∖ Φ n + k \Phi _n\backslash \Phi _{n+k} is rich enough by giving a number of examples. In particular, we give a direct proof of Ω n ∈ Φ n ∖ Φ n + 1 \Omega _n\in \Phi _n\backslash \Phi _{n+1} , which avoids Schoenberg’s theorem; Ω n \Omega _n is the Schoenberg kernel. We show that Ω n ( a ⋅ ) Ω n ( b ⋅ ) ∈ Φ n ∖ Φ n + 1 \Omega _n(a\cdot )\Omega _n(b\cdot )\in \Phi _n\backslash \Phi _{n+1} for a ≠ b a\not =b . Moreover, for the square of this function we prove the surprisingly much stronger result Ω n 2 ( a ⋅ ) ∈ Φ 2 n − 1 ∖ Φ 2 n \Omega _n^2(a\cdot )\in \Phi _{2n-1}\backslash \Phi _{2n} . We also show that any f ∈ Φ n ∖ Φ n + 1 f\in \Phi _n\backslash \Phi _{n+1} , n ≥ 2 n\ge 2 , has infinitely many negative squares. The latter means that for an arbitrary positive integer N N there is a finite Schoenberg matrix S X ( f ) := ‖ f ( | x i − x j | n + 1 ) ‖ i , j = 1 m \mathcal {S}_X(f) := \|f(|x_i-x_j|_{n+1})\|_{i,j=1}^{m} , X := { x j } j = 1 m ⊂ R n + 1 X := \{x_j\}_{j=1}^m \subset \mathbb {R}^{n+1} , which has at least N N negative eigenvalues.

Representations of hypergeometric functions for arbitrary parameter values and their use

Integral representations of hypergeometric functions proved to be a very useful tool for studying their properties. The purpose of this paper is twofold. First, we extend the known representations to arbitrary values of the parameters and show that the extended representations can be interpreted as examples of regularizations of integrals containing Meijer’s G function. Second, we give new applications of both, known and extended representations. These include: inverse factorial series expansion for the Gauss type function, new information about zeros of the Bessel and Kummer type functions, connection with radial positive definite functions and generalizations of Luke’s inequalities for the Kummer and Gauss type functions.

Derivatives of isotropic positive definite functions on spheres

We show that isotropic positive definite functions on the d d -dimensional sphere which are 2 k 2k times differentiable at zero have 2 k + [ ( d − 1 ) / 2 ] 2k+[(d-1)/2] continuous derivatives on ( 0 , π ) (0,\pi ) . This result is analogous to the result for radial positive definite functions on Euclidean spaces. We prove optimality of the result for all odd dimensions. The proof relies on montée, descente and turning bands operators on spheres which parallel the corresponding operators originating in the work of Matheron for radial positive definite functions on Euclidean spaces.

An analog of Rudin’s theorem for continuous radial positive definite functions of several variables

Let Open image in new window be the class of radial real-valued functions of m variables with support in the unit ball \(\mathbb{B}\) of the space ℝ m that are continuous on the whole space ℝ m and have a nonnegative Fourier transform. For m ≥ 3, it is proved that a function f from the class Open image in new window can be presented as the sum \(\sum {f_k \tilde *f_k } \) of at most countably many self-convolutions of real-valued functions f k with support in the ball of radius 1/2. This result generalizes the theorem proved by Rudin under the assumptions that the function f is infinitely differentiable and the functions f k are complex-valued.

Schröbinger operators with point interactions and radial positive definite functions

Schröbinger operators with point interactions and radial positive definite functions

Spectral theory of Schrödinger operators with infinitely many point interactions and radial positive definite functions

A number of results on radial positive definite functions on Rn related to Schoenbergʼs integral representation theorem are obtained. They are applied to the study of spectral properties of self-adjoint realizations of two- and three-dimensional Schrödinger operators with countably many point interactions. In particular, we find conditions on the configuration of point interactions such that any self-adjoint realization has purely absolutely continuous non-negative spectrum. We also apply some results on Schrödinger operators to obtain new results on completely monotone functions.

Radial positive definite functions and spectral theory of the Schrödinger operators with point interactions

AbstractWe complete the classical Schoenberg representation theorem for radial positive definite functions. We apply this result to study spectral properties of self‐adjoint realizations of two‐ and three‐dimensional Schrödinger operators with point interactions on a finite set. In particular, we prove that any realization has purely absolute continuous nonnegative spectrum.

The Dagum family of isotropic correlation functions

A function $ρ:[0, ∞)→(0, 1]$ is a completely monotonic function if and only if $ρ(‖\mathbf{x}‖^2)$ is positive definite on $ℝ^d$ for all $d$ and thus it represents the correlation function of a weakly stationary and isotropic Gaussian random field. Radial positive definite functions are also of importance as they represent characteristic functions of spherically symmetric probability distributions. In this paper, we analyze the function $$\rho(\beta ,\gamma)(x)=1-\biggl(\frac{x^{\beta}}{1+x^{\beta}}\biggr)^{\gamma},\qquad x\ge 0, \beta,\gamma>0,$$ called the Dagum function, and show those ranges for which this function is completely monotonic, that is, positive definite, on any $d$-dimensional Euclidean space. Important relations arise with other families of completely monotonic and logarithmically completely monotonic functions.

Radial basis functions and corresponding zonal series expansions on the sphere

Since radial positive definite functions on R d remain positive definite when restricted to the sphere, it is natural to ask for properties of the zonal series expansion of such functions which relate to properties of the Fourier–Bessel transform of the radial function. We show that the decay of the Gegenbauer coefficients is determined by the behavior of the Fourier–Bessel transform at the origin.

Convolution roots of radial positive definite functions with compact support

A classical theorem of Boas, Kac, and Krein states that a characteristic function φ with φ(x) = 0 for |x| > T admits a representation of the form φ(x) = ∫u(y)u(y + x) dy, x ∈ R, where the convolution root u ∈ L 2 (R) is complex-valued with u(x) = 0 for |x| ≥ τ/2. The result can be expressed equivalently as a factorization theorem for entire functions of finite exponential type. This paper examines the Boas-Kac representation under additional constraints: If φ is real-valued and even, can the convolution root u be chosen as a real-valued and/or even function? A complete answer in terms of the zeros of the Fourier transform of φ is obtained. Furthermore, the analogous problem for radially symmetric functions defined on R d is solved. Perhaps surprisingly, there are compactly supported, radial positive definite functions that do not admit a convolution root with half-support. However, under the additional assumption of nonnegativity, radially symmetric convolution roots with half-support exist. Further results in this paper include a characterization of extreme points, pointwise and integral bounds (Turan's problem), and a unified solution to a minimization problem for compactly supported positive definite functions. Specifically, if f is a probability density on R d whose characteristic function φ vanishes outside the unit ball, then ∫|x| 2 f(x) dx = -Δφ(0) ≥ 4j 2 (d-2)/2 where j v denotes the first positive zero of the Bessel function J v , and the estimate is sharp. Applications to spatial moving average processes, geostatistical simulation, crystallography, optics, and phase retrieval are noted. In particular, a real-valued half-support convolution root of the spherical correlation function in R 2 does not exist.

Recurrence relations for radial positive definite functions

In the theory of radial basis functions as well as in the theory of spherically symmetric characteristic functions recurrence relations are used to construct d-dimensional functions starting with lower-dimensional ones. We show that the operators used so far are special cases of one step recurrence relations for ℓ 2-radial positive definite functions. We further give the analogue for ℓ 1-radial functions and thereby define a turning bands operator for 1-symmetric characteristic functions.

On the Derivatives of Radial Positive Definite Functions

A radial positive definite function ϕ(‖·‖) on Rd which is 2k times differentiable at zero has 2k+[(d−1)/2] continuous derivatives away from zero, and the estimate is sharp. Furthermore, ϕ(iv)(0)≥3d|ϕ″(0)|2/(d+2) if the derivatives exist; this proves a conjecture of Paul Switzer and has implications on the smoothness of isotropic random fields and their local averages. The techniques used have evolved in geostatistics, and side results refer to Matheron's turning bands operator and the montée.

Radial Positive Definite Functions Generated by Euclid's Hat

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.

On integral representation of the continuous O(n)-positive definite functions

The aim of this paper is to study integral representation for continuous O(n)-positive definite functions. Using the Krein method of directing functionals we prove the theorem about integral representation, give the criteria of uniqueness of this representation and description of all representations in the case of ununiqueness. On the basis of these results we solve analogous problems for continuous radial positive definite functions.

On Functions Positive Definite Relative to the Orthogonal Group and the Representation of Functions as Hankel-Stieltjes Transforms

To every continuous function $f$ on an interval $0 \leq x < a(0 < a \leq \infty )$ and every positive number $\nu$ associate the kernel \[ f(x,y) = \int _0^\pi f\left ( (x^2 + y^2 - 2xy \cos \theta )^{1/2}\right ) (\sin \theta )^{2\nu - 1},d\theta ,\quad 0 < x, y < a/2. \] Let $\Omega (z) = \Gamma (\nu + 1/2) (2/z)^{\nu - 1/2} J_{\nu -1/2} J_{\nu - 1/2}(z)$, where $J_{\nu - 1/2}(z)$ is the Bessel function of index $\nu - 1/2$. It is shown that $f$ has an integral representation $f(x) = \int _{-\infty }^\infty \Omega (x\sqrt \lambda )d\gamma (\lambda )$, where $\gamma$ is a finite, positive Radon measure on $R$, if and only if the kernel $f(x,y)$ is positive definite. If $\nu = (N - 1)/2$, where $N$ is an integer $\geq 2$, this condition is equivalent to ${f_N}(x) = f(|x|),\;x \in {R^N},\;|x| < \alpha$, is positive definite relative to the orthogonal group $O(N)$. The results of this investigation extend the preceding one of the author on functions positive definite relative to the orthogonal group. In particular they yield the result of Rudin on the extensions of radial positive definite functions.