We study the following question posed by Turan. Suppose K is a convex body in Euclidean space which is symmetric with respect to the origin. Of all positive definite functions supported in K, and with value 1 at the origin, which one has the largest integral? It is probably the case that the extremal function is the indicator of the half-body convolved with itself and properly scaled, but this has been proved only for a small class of domains so far. We add to this class of known "Turan domains" the class of all spectral convex domains. These are all convex domains which have an orthogonal basis of complex exponentials. As a corollary we obtain that all convex domains which tile space by translation are Turan domains. We also give a new proof that the Euclidean ball is a Turan domain.
We study the following question: Given an open set $Ω$, symmetric about 0, and a continuous, integrable, positive definite function $f$, supported in $Ω$ and with $f(0)=1$, how large can $\int f$ be? This problem has been studied so far mostly for convex domains $Ω$ in Euclidean space. In this paper we study the question in arbitrary locally compact abelian groups and for more general domains. Our emphasis is on finite groups as well as Euclidean spaces and $\ZZ^d$. We exhibit upper bounds for $\int f$ assuming geometric properties of $Ω$ of two types: (a) packing properties of $Ω$ and (b) spe…
We study the sharp constant $W_{n}(D)$ in Wiener's inequality for positive definite functions \[ \int_{\mathbb{T}^{n}}|f|^{2}\,dx\le W_{n}(D)|D|^{-1}\int_{D}|f|^{2}\,dx,\quad D\subset \mathbb{T}^{n}. \] N. Wiener proved that $W_{1}([-δ,δ])<\infty$, $δ\in (0,1/2)$. E. Hlawka showed that $W_{n}(D)\le 2^{n}$, where $D$ is an origin-symmetric convex body. We sharpen Hlawka's estimates for $D$ being the ball $B^{n}$ and the cube $I^{n}$. In particular, we prove that $W_{n}(B^{n})\le 2^{(0.401\ldots +o(1))n}$. We also obtain a lower bound of $W_{n}(D)$. Moreover, for a cube $ D=\frac1q I^{n}$ with $…
Let G be a locally compact abelian group (LCA group) and U be an open, 0-symmetric set. Let F:=F(U) be the set of all real valued continuous functions from G to R which are supported in U and are positive definite. The Turan constant T(U) of U is then defined as the supremum of the integral of any f on U, which belongs to F and is normalised to have f(0)=1. Mihalis Kolountzakis and the author has shown that structural properties - like spectrality, tiling or packing with a certain set L - of subsets U in finite or compact groups and on R^d and in Z^d, yield estimates of T(U). However, in these…
For the real, continuous, isotropic and positive definite kernels on a product of spheres, one may consider not only its usual strict positive definiteness but also strict positive definiteness restrict to the points of the product that have distinct components. In this paper, we provide a characterization for strict positive definiteness in these two cases, settling all the cases but those in which one of the spheres is a circle.
We introduce the notion of Dunkl positive definite and strictly positive definite functions on $\mathbb{R}^{d}$. This done by the use of the properties of Dunkl translation. We establish the analogue of Bochner's theorem in Dunkl setting. The case of radial functions is considered. We give a sufficient condition for a function to be Dunkl strictly positive definite on $\mathbb{R}^{d}.$
In this paper we give some conditions for a class of functions related to Bessel functions to be positive definite or strictly positive definite . We present some properties and relationships involving logarithmically completely monotonic functions and strictly positive definite functions. In particular, we are interested with the modified Bessel functions.
This paper is concerned with the construction of positive definite functions on a cartesian product of quasi-metric spaces using generalized Stieltjes and complete Bernstein functions. The results we prove are aligned with a well-established method of T. Gneiting to construct space-time positive definite functions and its many extensions. Necessary and sufficient conditions for the strict positive definiteness of the models are provided when the spaces are metric.
We present, among other things, a necessary and sufficient condition for the strict positive definiteness of an isotropic and positive definite kernel on the cartesian product of a circle and a higher dimensional sphere. The result complements similar results previously obtained for strict positive definiteness on a product of circles [Positivity, to appear, arXiv:1505.01169] and on a product of high dimensional spheres [J. Math. Anal. Appl. 435 (2016), 286-301, arXiv:1505.03695].
Positive definite functions of compact support are widely used for radial basis function approximation as well as for estimation of spatial processes in geostatistics. Several constructions of such functions for ${\mathbb R}^d$ are based upon recurrence operators. These map functions of such type in a given space dimension onto similar ones in a space of lower or higher dimension. We provide analogs of these dimension hopping operators for positive definite, and strictly positive definite, zonal (radial) functions on the sphere. These operators are then used to provide new families of strictly…
Positive definite functions are very important in both theory and applications of approximation theory, probability and statistics. In particular, identifying strictly positive definite kernels is of great interest as interpolation problems corresponding to these kernels are guaranteed to be poised. A Bochner type result of Schoenberg characterises continuous positive definite zonal functions, $f(\cos \cdot)$, on the sphere $\Sdmone$, as those with nonnegative Gegenbauer coefficients. More recent results characterise strictly positive definite functions on $\Sdmone$ by stronger conditions on t…
We provide walks through dimensions for isotropic positive definite functions defined over complex spheres. We show that the analogues of Montée and Descente operators as proposed by Beatson and zu Castell [J. Approx. Theory 221 (2017), 22-37] on the basis of the original Matheron operator [Les variables régionalisées et leur estimation, Masson, Paris, 1965], allow for similar walks through dimensions. We show that the Montée operators also preserve, up to a constant, strict positive definiteness. For the Descente operators, we show that strict positive definiteness is preserved under some add…
In this note we give a recursive formula for the derivatives of isotropic positive definite functions on the Hilbert sphere. We then use it to prove a conjecture stated by Trübner and Ziegel, which says that for a positive definite function on the Hilbert sphere to be in $C^{2\ell}([0,π])$, it is necessary and sufficient for its $\infty$-Schoenberg sequence to satisfy $\sum\limits_{m=0}^{\infty}a_m m^{\ell}<\infty$.
We investigate inequalities of M.G. Krein, Yu.V. Linnik and E.A. Gorin for positive definite functions. Modifications and generalizations of these inequalities are proved. We also prove that multipoint E.A. Gorin's inequality follows from two--point M.G. Krein's inequality.
In this paper we consider Positive Definite functions on products $Ω_{2q}\timesΩ_{2p}$ of complex spheres, and we obtain a condition, in terms of the coefficients in their disc polynomial expansions, which is necessary and sufficient for the function to be Strictly Positive Definite. The result includes also the more delicate cases in which $p$ and/or $q$ can be $1$ or $\infty$. The condition we obtain states that a suitable set in $\mathbb{Z}^2$, containing the indexes of the strictly positive coefficients in the expansion, must intersect every product of arithmetic progressions.
There exists a positive function $ψ(t)${on}$t\geq0${, with fast decay at infinity, such that for every measurable set}$Ω${in the Euclidean space and}$R>0${, there exist entire functions}$A(x) ${and}$B(x) ${of exponential type}$R${, satisfying\}$A(x)\leq χ_Ω(x)\leq B(x)${and}$| B(x)-A(x)| \leqslantψ(R\operatorname*{dist}(x,\partialΩ)) $. This leads to Erdős Turán estimates for discrepancy of point set distributions in the multi dimensional torus. Analogous results hold for approximations by eigenfunctions of differential operators and discrepancy on compact manifolds.
Convolution is an important tool in the construction of positive definite kernels on a manifold. This contribution provides conditions on an $L^2$-positive definite and zonal kernel on the unit sphere of $\mathbb{C}^q$ in order that the kernel can be recovered as a generalized convolution root of an equally positive definite and zonal kernel.
Positive definite functions are fundamental to many areas of applied mathematics, probability theory, spatial statistics and machine learning, amogst others. Motivated by a problem coming from the maximum likelihood estimation under fixed domain asymptotics, we consider a new operator acting on rescaled weighted differences between two members of the class $Φ_d$ of positive definite radial functions,. In particular, we study the positive definiteness of the operator for the Matérn, Generalized Cauchy and Generalized Wendland families. It turns out that proposed operator allows to govern differ…
Hilbert-Schmidt independence criterion and distance covariance are methods to describe independence of random variables using either the Kronecker product of positive definite kernels or the Kronecker product of conditionally negative definite kernels. In this paper we generalize both methods by providing an independence criteria using a new concept, of positive definite independent kernels. We provide a characterization of the radial kernels that are positive definite independent on all Euclidean spaces and we present several examples.
We present sufficient condition for a family of positive definite kernels on a compact two-point homogeneous space to be strictly positive definite based on their representation as a series of spherical harmonics. The family analyzed is a generalization of the isotropic kernels and the case of a real sphere is analyzed in details.
We show that isotropic positive definite functions on the $d$-dimensional sphere which are $2k$ times differentiable at zero have $2k+[(d-1)/2]$ continuous derivatives on $(0,π)$. 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 Euclidian spaces.
