Strictly positive definite functions on compact abelian groups

On a class of non-Hermitian matrices with positive definite Schur complements

Given a positive definite matrix $A\in \mathbb{C}^{n\times n}$ and a Hermitian matrix $D\in \mathbb{C}^{m\times m}$, we characterize under which conditions there exists a strictly contractive matrix $K\in \mathbb{C}^{n\times m}$ such that the non-Hermitian block-matrix \[ \left[ \begin{array}{cc} A & -AK \\ K^*A & D \end{array} \right] \] has a positive definite Schur complement with respect to its submatrix~$A$. Additionally, we show that~$K$ can be chosen such that diagonalizability of the block-matrix is guaranteed and we compute its spectrum. Moreover, we show a connection to the recently …

固定维数下的矩阵正性保持子。II：正定性与Schur函数比的严格单调性

Matrix positivity preservers in fixed dimension. II: positive definiteness and strict monotonicity of Schur function ratios

本文研究固定维数矩阵上保持半正定性的实多项式扰动问题，证明当扰动系数严格大于临界负下界时变换矩阵在一般性条件下正定，并通过加强Schur单调性结果给出秩的刻画。

Positive definite (p.d.) functions vs p.d. distributions

We give explicit transforms for Hilbert spaces associated with positive definite functions on $\mathbb{R}$, and positive definite tempered distributions, incl., generalizations to non-abelian locally compact groups. Applications to the theory of extensions of p.d. functions/distributions are included. We obtain explicit representation formulas for positive definite tempered distributions in the sense of L. Schwartz, and we give applications to Dirac combs and to diffraction. As further applications, we give parallels between Bochner's theorem (for continuous p.d. functions) on the one hand, an…

Integral comparisons of nonnegative positive definite functions on LCA groups

In this paper we investigate the following questions. Let $μ, ν$ be two regular Borel measures of finite total variation. When do we have a constant $C$ satisfying $$\int f dν\le C \int f dμ$$ whenever $f$ is a continuous nonnegative positive definite function? How the admissible constants $C$ can be characterized, and what is their optimal value? We first discuss the problem in locally compact abelian groups. Then we make further specializations when the Borel measures $μ, ν$ are both either purely atomic or absolutely continuous with respect to a reference Haar measure. In addition, we prove…

正定算子的Wasserstein重心

On the Wasserstein barycenter of positive definite operators

该研究将正定矩阵的Bures-Wasserstein均值推广到希尔伯特空间正定算子情形，证明其存在唯一性，分析其生成的ODE半群的指数收缩性，还证明相关极限定理并建立算子不等式。

Characteristic kernels on Hilbert spaces, Banach spaces, and on sets of measures

We present new classes of positive definite kernels on non-standard spaces that are integrally strictly positive definite or characteristic. In particular, we discuss radial kernels on separable Hilbert spaces, and introduce broad classes of kernels on Banach spaces and on metric spaces of strong negative type. The general results are used to give explicit classes of kernels on separable $L^p$ spaces and on sets of measures.

Wiener's theorem for positive definite functions on hypergroups

The following theorem on the circle group $\mathbb{T}$ is due to Norbert Wiener: If $f\in L^{1}\left( \mathbb{T}\right) $ has non-negative Fourier coefficients and is square integrable on a neighbourhood of the identity, then $f\in L^{2}\left( \mathbb{T}\right) $. This result has been extended to even exponents including $p=\infty$, but shown to fail for all other $p\in\left( 1,\infty\right] .$ All of this was extended further (appropriately formulated) well beyond locally compact abelian groups. In this paper we prove Wiener's theorem for even exponents for a large class of commutative hyperg…

Equivalence of modes of convergence on reproducing kernel Hilbert spaces

Let $(X,μ)$ be a strictly-positive Borel measure space. We show that the modes of convergence in a reproducing kernel Hilbert (RKHS) space, pointwise, weak and strong are all equivalents. From this we describe some important consequences such as an association with positive operators and positive definite kernels and a compact embedding condition for a RKHS in $L^{2}(X,μ)$.

AF-embedding of the crossed products of AH-algebras by finitely generated abelian groups

Let $X$ be a compact metric space and let $Λ$ be a $\Z^k$ ($k\ge 1$) action on $X.$ We give a solution to a version of Voiculescu's problem of AF-embedding: The crossed product $C(X)\rtimes_Λ\Z^k$ can be embedded into a unital simple AF-algebra if and only if $X$ admits a strictly positive $Λ$-invariant Borel probability measure. Let $C$ be a unital AH-algebra, let $G$ be a finitely generated abelian group and let $Λ: G\to Aut(C)$ be a monomorphism. We show that $C\rtimes_Λ G$ can be embedded into a unital simple AF-algebra if and only if $C$ admits a faithful $Λ$-invariant tracial state.

A generalized Fourier inversion theorem

In this work we define operator-valued Fourier transforms for suitable integrable elements with respect to the Plancherel weight of a (not necessarily Abelian) locally compact group. Our main result is a generalized version of the Fourier inversion Theorem for strictly-unconditionally integrable Fourier transforms. Our results generalize and improve those previously obtained by Ruy Exel in the case of Abelian groups.

黎曼对称空间上核的普适性

Universality of kernels on Riemannian symmetric spaces

该研究刻画了黎曼对称空间上连续正定不变核的普适性，对紧与非紧对称空间分别给出等价条件，并提供实例说明普适核的构造准则。

局部紧阿贝尔群上Delsarte极值问题的对偶性

Duality for Delsarte's extremal problem on locally compact Abelian groups

本文在局部紧阿贝尔群上推广了Delsarte极值问题，通过泛函分析方法证明了强对偶定理，统一并扩展了先前在有限群和ℝ^d等情形下的结果。

严格正性与$D$-优化排序

Strict Positivity and $D$-Majorization

该研究受量子热力学启发，探究严格正性的判定与性质，并将其用于推广$D$-优化排序，给出二维情形下该排序的迹范数刻画及序性质。

Strictly proper kernel scores and characteristic kernels on compact spaces

Strictly proper kernel scores are well-known tool in probabilistic forecasting, while characteristic kernels have been extensively investigated in the machine learning literature. We first show that both notions coincide, so that insights from one part of the literature can be used in the other. We then show that the metric induced by a characteristic kernel cannot reliably distinguish between distributions that are far apart in the total variation norm as soon as the underlying space of measures is infinite dimensional. In addition, we provide a characterization of characteristic kernels in t…