Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRelations between Schoenberg Coefficients on Real and Complex Spheres of Different Dimensions
62
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Positive definite functions on spheres have received an increasing interest in many branches of mathematics and statistics. In particular, the Schoenberg sequences in the spectral representation of positive definite functions have been studied by several mathematicians in the last years. This paper provides a set of relations between Schoenberg sequences defined over real as well as complex spheres of different dimensions. We illustrate our findings describing an application to strict positive definiteness.
rate research
Read More
We consider the class $Psi_d$ of continuous functions $psi colon [0,pi] to mathbb{R}$, with $psi(0)=1$ such that the associated isotropic kernel $C(xi,eta)= psi(theta(xi,eta))$ ---with $xi,eta in mathbb{S}^d$ and $theta$ the geodesic distance--- is positive definite on the product of two $d$-dimensional spheres $mathbb{S}^d$. We face Problems 1 and 3 proposed in the essay Gneiting (2013b). We have considered an extension that encompasses the solution of Problem 1 solved in Fiedler (2013), regarding the expression of the $d$-Schoenberg coefficients of members of $Psi_d$ as combinations of $1$-Schoenberg coefficients. We also give expressions for the computation of Schoenberg coefficients of the exponential and Askey families for all even dimensions through recurrence formula. Problem 3 regards the curvature at the origin of members of $Psi_d$ of local support. We have improved the current bounds for determining this curvature, which is of applied interest at least for $d=2$.
In the present paper we study the minimization of energy integrals on the sphere with a focus on an interesting clustering phenomenon: for certain types of potentials, optimal measures are discrete or are supported on small sets. In particular, we prove that the support of any minimizer of the $p$-frame energy has empty interior whenever $p$ is not an even integer. A similar effect is also demonstrated for energies with analytic potentials which are not positive definite. In addition, we establish the existence of discrete minimizers for a large class of energies, which includes energies with polynomial potentials.
Comparing probability distributions is an indispensable and ubiquitous task in machine learning and statistics. The most common way to compare a pair of Borel probability measures is to compute a metric between them, and by far the most widely used notions of metric are the Wasserstein metric and the total variation metric. The next most common way is to compute a divergence between them, and in this case almost every known divergences such as those of Kullback--Leibler, Jensen--Shannon, Renyi, and many more, are special cases of the $f$-divergence. Nevertheless these metrics and divergences may only be computed, in fact, are only defined, when the pair of probability measures are on spaces of the same dimension. How would one quantify, say, a KL-divergence between the uniform distribution on the interval $[-1,1]$ and a Gaussian distribution on $mathbb{R}^3$? We will show that, in a completely natural manner, various common notions of metrics and divergences give rise to a distance between Borel probability measures defined on spaces of different dimensions, e.g., one on $mathbb{R}^m$ and another on $mathbb{R}^n$ where $m, n$ are distinct, so as to give a meaningful answer to the previous question.
In this paper, we establish the sharp conditions for the inclusion relations between Besov spaces $B_{p,q}$ and Wiener amalgam spaces $W_{p,q}^s$. We also obtain the optimal inclusion relations between local hardy spaces $h^p$ and Wiener amalgam spaces $W_{p,q}^s$, which completely improve and extend the main results obtained by Cunanana, Kobayashib and Sugimotoa in [J. Funct. Anal. 268 (2015), 239-254]. In addition, we establish some mild characterizations of inclusion relations between Triebel-Lizorkin and Wiener amalgam spaces, which relates some modern inequalities to classical inequalities.
We study distance spheres: the set of points lying at constant distance from a fixed arbitrary subset $K$ of $[0,1]^d$. We show that, away from the regions where $K$ is too dense and a set of small volume, we can decompose $[0,1]^d$ into a finite number of sets on which the distance spheres can be straightened into subsets of parallel $(d-1)$-dimensional planes by a bi-Lipschitz map. Importantly, the number of sets and the bi-Lipschitz constants are independent of the set $K$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
