arXiv 0801.09412008-01-07math.CAmath.FAmath.PR

On the extremal rays of the cone of positive, positive definite functions

Philippe Jaming, Maté Matolcsi, Szilard Gy. Révesz

发表信息：: Journal of fourier analysis and applications 15 (2009) 561-582
DOI：: 10.1007/s00041-008-9057-6

英文摘要：

The aim of this paper is to investigate the cone of non-negative, radial, positive-definite functions in the set of continuous functions on $\R^d$. Elements of this cone admit a Choquet integral representation in terms of the extremals. The main feature of this article is to characterize some large classes of such extremals. In particular, we show that there many other extremals than the gaussians, thus disproving a conjecture of G. Choquet and that no reasonable conjecture can be made on the full set of extremals. The last feature of this article is to show that many characterizations of positive definite functions available in the literature are actually particular cases of the Choquet integral representations we obtain.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

关于 LCA 群上正定函数的 Delsarte 型极值问题

On extremal problems of Delsarte type for positive definite functions on LCA groups

本文建立局部紧 Abel 群上 Delsarte 与 Turán 极值问题的统一框架，提出变体并分析表述等价性，证明一般拓扑条件下极值函数的存在性。

2024-11-30math.CA

Turán's extremal problem for positive definite functions on groups

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…

2003-12-10math.CAmath.NT

Geometries on the cone of positive-definite matrices derived from the power potential and their relation to the power means

We study a Riemannian metric on the cone of symmetric positive-definite matrices obtained from the Hessian of the power potential function $(1-\det(X)^β)/β$. We give explicit expressions for the geodesics and distance function, under suitable conditions. In the scalar case, the geodesic between two positive numbers coincides with a weighted power mean, while for matrices of size at least two it yields a notion of weighted power mean different from the ones given in the literature. As $β$ tends to zero, the power potential converges to the logarithmic potential, that yields a well-known metric …

2021-12-11math.CAmath.DG

齐性空间上的Delsarte型极值问题与卷积根

Delsarte-type extremal problems and convolution roots on homogeneous spaces

本文研究局部紧群$G$与紧子群$K$构成的齐性空间$G/K$上的Delsarte型极值问题，通过对应关系将其转化为群$G$上的问题并证明极值函数存在，还在紧Gelfand对情形下得到正定核的卷积根存在性。

2025-11-17math.CA

关于Delsarte极值问题极值函数的存在性

On the Existence of an Extremal Function for the Delsarte Extremal Problem

本文研究局部紧阿贝尔群上Delsarte极值问题，证明当约束集Ω为闭集时极值函数存在，推广了先前已知的存在性结果。

2024-11-25math.CAmath.FA

On integral estimates of non-negative positive definite functions

Let $\ell>0$ be arbitrary. We introduce the extremal quantities $$ G(\ell):=\frac{\sup_{f} \int_{-\ell}^{\ell} f\,dx}{\int_{-1}^1 f\,dx},\quad C(\ell):=\frac{\sup_{f} \sup_{a\in {\mathbb R}} \int_{a-\ell}^{a+\ell} f\,dx}{\int_{-1}^1 f\,dx}, $$ where the supremum is taken over all not identically zero non-negative positive definite functions. We are interested in the question: how large can the above extremal quantities be? This problem was originally posed by Yu. Shteinikov and S. Konyagin for the case $\ell=2$. In this note we obtain exact values for the right limits $G(k+0)$ and $C(k+0)$ at …

2016-12-01math.CA

更多相关论文收起更多论文

On pointwise estimates of positive definite functions with given support

The following problem originated from a question due to Paul Turan. Suppose $Ω$ is a convex body in Euclidean space $\RR^d$ or in $\TT^d$, which is symmetric about the origin. Over all positive definite functions supported in $Ω$, and with normalized value 1 at the origin, what is the largest possible value of their integral? From this Arestov, Berdysheva and Berens arrived to pose the analogous pointwise extremal problem for intervals in $\RR$. That is, under the same conditions and normalizations, and for any particular point $z\inΩ$, the supremum of possible function values at $z$ is to be …

2003-02-17math.CA

Semi-definite representations for sets of cubics on the 2-sphere

The compact set of homogeneous quadratic polynomials in $n$ real variables with modulus bounded by 1 on the unit sphere $S^{n-1}$ is trivially semi-definite representable. The compact set of homogeneous ternary quartics with modulus bounded by 1 on the unit sphere $S^2$ is also semi-definite representable. This suggests that the compact set of homogeneous ternary cubics with modulus bounded by 1 on $S^2$ is semi-definite representable. We deduce an explicit semi-definite representation of this norm ball. More generally, we provide a semi-definite description of the cone of inhomogeneous ternar…

2021-03-24math.CAmath.OC

紧Gelfand对上正定函数的杜拉兹极值问题的对偶性

Duality for Delsarte's extremal problem on compact Gelfand pairs

University of Cape Town（开普敦大学）；University of Wuppertal（伍珀塔尔大学）；HUN-REN Rényi Institute of Mathematics（匈牙利研究与教育网络雷尼数学研究所）

本文研究紧Gelfand对上正定函数的杜拉兹型问题作为无限维线性规划问题，探讨了其对偶性及强对偶性定理。

2026-07-07math.CA

On a problem of Turan about positive definite functions

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 bas…

2002-04-08math.CA

On the existence of an extremal function in the delsarte extremal problem

This paper is concerned with a Delsarte type extremal problem. Denote by $\mathcal{P}(G)$ the set of positive definite continuous functions on a locally compact abelian group $G$. We consider the function class, which was originally introduced by Gorbachev, \begin{multline*} \mathcal{G}(W, Q)_G = \left\{ f \in \mathcal{P}(G) \cap L^1(G) ~ : \right. ~ \left. f(0) = 1, ~ \text{supp}f_+ \subseteq W,~ \text{supp}\hat{f} \subseteq Q \right\} \end{multline*} where $W\subseteq G$ is closed and of finite Haar measure and $Q\subseteq \hat{G}$ is compact. We also consider the related Delsarte type probl…

2019-12-04math.CA

Laguerre Functions on Symmetric Cones and recursion relations in the Real Case

In this article we derive differential recursion relations for the Laguerre functions on the cone C of positive definite real matrices. The highest weight representations of the group Sp(n,R) play a fundamental role. Each such representation acts on a Hilbert space of holomorphic functions on the tube domain C+ i Sym(n,R). We then use the Laplace transform to carry the Lie algebra action over to L^2(C ,dm_t). The differential recursion relations result by restricting to a distinguished three dimensional subalgebra, which is isomorphic to sl(2,R).

2004-12-09math.CAmath.RT

圆周上Bernstein算子对正定性的保持

Preservation of Positive-Definiteness by Bernstein Operators on the Circle

University of Wisconsin–Madison（威斯康星大学麦迪逊分校）；George Mason University（乔治梅森大学）；University of Technology Sydney（悉尼科技大学）

该研究证明了圆周上的n次Bernstein算子可保持正定性，通过显式系数公式和两区间正性论证完成证明，并指出朴素仿射Bernstein算子在二维球面S²上无法保持正定性。

2026-08-06math.CA