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This paper studies certain aspects of harmonic analysis on nonabelian free groups. We focus on the concept of a positive definite function on the free group and our primary goal is to understand how such functions can be extended from balls of finite radius to the entire group. Previous work showed that such extensions always exist and we study the problem of simultaneous extension of multiple positive definite functions. More specifically, we define a concept of ‘relative energy’ which measures the proximity between a pair of positive definite functions, and show that a pair of positive definite functions on a finite ball can be extended to the entire group without increasing their relative energy. The proof is analytic, involving differentiation of noncommutative Szegö parameters. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to provide necessary and sufficient conditions on a continuous and matrix valued radial kernel on a Euclidean space in order that it be conditionally positive definite of a fixed order. Except for the one dimensional Euclidean space, the strict conditional positive definiteness of the kernel is fully characterized. [ABSTRACT FROM AUTHOR]

The matrix representation of operators in Hilbert spaces is a useful tool in applications. It is important to present the matrix representation by sequences other than orthonormal bases. In this paper, we extend the matrix representation of operators using g-frames and investigate their invertibility and stability. [ABSTRACT FROM AUTHOR]

A.M. Kagan introduced a class of distributions D m , k in R m and proved that if the joint distribution of m linear forms of n independent random variables belongs to the class D m , m - 1 , then the random variables are Gaussian. A.M. Kagan's theorem implies, in particular, the well-known Skitovich–Darmois theorem, where the Gaussian distribution on the real line is characterized by independence of two linear forms of n independent random variables. In the note we describe a wide class of locally compact Abelian groups where A.M. Kagan's theorem is valid. [ABSTRACT FROM AUTHOR]

Positive definite matrix functions on spheres arise naturally in multivariate approximation and spatial statistics. The construction of strictly positive definite models has become one of the most important goals for the analysis and study of vector valued random fields on spheres. This paper derives strictly positive definite models based on hypergeometric functions. In statistical nomenclature, it describes strictly positive definite covariance functions of stationary, isotropic and mean square continuous Gaussian vector random fields on spheres in which the isotropic part of the covariance functions are set through hypergeometric functions. [ABSTRACT FROM AUTHOR]

Compatibility conditions of quantum channels featuring symmetry through covariance are studied. Compatibility here means the possibility of obtaining two or more channels through partial trace out of a broadcasting channel. We see that covariance conditions can be used to simplify compatibility conditions as the broadcasting channel can be assumed to be covariant in a particular way. A particular emphasis is on Weyl covariance and in determining compatibility conditions for Weyl-covariant channels. The concrete examples studied include the case of a non-compact continuous phase space and the case of a finite phase space. [ABSTRACT FROM AUTHOR]

It is well known that for any irrational rotation number α , the noncommutative torus A α must have representations π such that the generated von Neumann algebra π ( A α ) ″ is of type III. Therefore, it could be of interest to exhibit and investigate such kind of representations, together with the associated spectral triples whose twist of the Dirac operator and the corresponding derivation arises from the Tomita modular operator. In the present paper, we show that this program can be carried out, at least when α is a Liouville number satisfying a faster approximation property by rationals. In this case, we exhibit several type I I ∞ and II I λ , λ ∈ [ 0 , 1 ] , factor representations and modular spectral triples. The method developed in the present paper can be generalised to CCR algebras based on a locally compact abelian group equipped with a symplectic form. [ABSTRACT FROM AUTHOR]

We study the strict positive definiteness of matrix-valued covariance functions associated to multivariate random fields defined over d -dimensional spheres of the ( d + 1 ) -dimensional Euclidean space. Characterization of strict positive definiteness is crucial to both estimation and cokriging prediction in classical geostatistical routines. We provide characterization theorems for high dimensional spheres as well as for the Hilbert sphere. We offer a necessary condition for positive definiteness on the circle. Finally, we discuss a parametric example which might turn to be useful for geostatistical applications. [ABSTRACT FROM AUTHOR]

Schoenberg’s theorem for the complex Hilbert sphere proved by Christensen and Ressel in 1982 by Choquet theory is extended to the following result: Let L denote a locally compact group and let D ¯ denote the closed unit disc in the complex plane. Continuous functions f : D ¯ × L → C such that f ( ξ ⋅ η , u − 1 v ) is a positive definite kernel on the product of the unit sphere in ℓ 2 ( C ) and L are characterized as the functions with a uniformly convergent expansion f ( z , u ) = ∑ m , n = 0 ∞ φ m , n ( u ) z m z ¯ n , where φ m , n is a double sequence of continuous positive definite functions on L such that ∑ φ m , n ( e L ) < ∞ ( e L is the neutral element of L ). It is shown how the coefficient functions φ m , n are obtained as limits from expansions for positive definite functions on finite dimensional complex spheres via a Rodrigues formula for disc polynomials. Similar results are obtained for the real Hilbert sphere. [ABSTRACT FROM AUTHOR]

We determine a necessary and sufficient condition for the strict positive definiteness of a continuous and positive definite kernel on the torus. [ABSTRACT FROM AUTHOR]

We extended the definition of regular dilation to graph products of N , which is an important class of quasi-lattice ordered semigroups. Two important results in dilation theory are unified under our result: namely, Brehmer's regular dilation on N k and Frazho–Bunce–Popescu's dilation of row contractions. We further show that a representation of a graph product has an isometric Nica-covariant dilation if and only if it is ⁎-regular. A special case of our result was considered by Popescu, and we studied the connection with Popescu's work. [ABSTRACT FROM AUTHOR]

The authors provide a characterization of the continuous and isotropic multivariate covariance functions associated to a Gaussian random field with index set varying over a compact two-point homogeneous space. Sufficient conditions for the strict positive definiteness based on this characterization are presented. Under the assumption that the space is not a sphere, a necessary and sufficient condition is given for the continuous and isotropic multivariate covariance function to be strictly positive definite. Under the same assumption, an alternative necessary and sufficient condition is also provided for the strict positive definiteness of a continuous and isotropic bivariate covariance function based on the main diagonal entries in the matrix representation for the covariance function. [ABSTRACT FROM AUTHOR]

In this note, we refine the notion of conditionally negative definite kernels to the notion of conditionally strictly negative definite kernels and study its properties. We show that the class of these kernels carries some surprising rigidity, in particular, the word metric function on Coxeter groups is conditionally strictly negative definite if and only if the group is a free product of a number of copies of’s and that the class of conditionally strictly negative definite kernels on a finite set is a one-parameter perturbation of the class of strictly positive definite kernels on this set. We also discuss several examples. [ABSTRACT FROM AUTHOR]

Edit distances provide us with an established method to capture structural features of data, and a distance between data objects represents their dissimilarity. In contrast, kernels form a category of similarity functions, and a positive definite kernel enables us to leverage abundant techniques of multivariate analysis. This paper aims to fill the gap between distances and kernels. In the literature, we have several formulas that convert a negative definite distance function into a positive definite kernel. Edit distance functions, however, are not necessarily negative definite, and our first contribution is to introduce an alternative method to derive positive definite kernels from edit distance functions that are not necessarily negative definite. The method is equipped with an easy-to-check and strong sufficient condition for positive definiteness, and the condition turns out to be tightly related with the triangle inequality. In fact, to our knowledge, all of the edit distance functions in the literature that support the triangle inequality meet the condition for positive definiteness. Secondly, we apply this method to four well-known edit distance functions for trees to introduce four novel kernels and show that three of them are positive definite. Thirdly, we develop a theory of subtree matching to study these kernels. Our kernels count matchings between subtrees of the input trees with weights determined according to individual matchings. Although the number of such matchings is an exponential function of the size of the input trees (the number of vertices), our theory enables us to develop dynamic-programming-based algorithms, whose asymptotic computational complexities fall between a quadratic function and a cubic function of the size. [ABSTRACT FROM AUTHOR]

In this paper a Jessen's type inequality for normalized positive $C_{0}$-semigroups is obtained. An adjoint of Jessen's type inequality has also been derived for the corresponding adjoint semigroup, which does not give the analogous results but the behavior is still interesting. Moreover, it is followed by some results regarding positive definiteness and exponential convexity of complex structures involving operators from a semigroup. [ABSTRACT FROM AUTHOR]

It was one of great successes of Kirillov's orbit method to see that the unitary dual of an exponential Lie group is in bijective correspondence with the orbit space associated with the linear dual of the Lie algebra of the group in question. To show that this correspondence is an homeomorphism turned out to be unexpectedly difficult. Only in 1994 H. Leptin and J. Ludwig gave a proof using the notion of variable groups. In this article their proof in the case of completely solvable Lie group is reorganized, some 'philosophy' and some new arguments are added. The purpose is to contribute to a better understanding of this proof. [ABSTRACT FROM AUTHOR]

We study differentiability of functions in the reproducing kernel Hilbert space (RKHS) associated with a smooth Mercer-like kernel on the sphere. We show that differentiability up to a certain order of the kernel yields both, differentiability up to the same order of the elements in the series representation of the kernel and a series representation for the corresponding derivatives of the kernel. These facts are used to embed the RKHS into spaces of differentiable functions and to deduce reproducing properties for the derivatives of functions in the RKHS. We discuss compactness and boundedness of the embedding and some applications to Gaussian-like kernels. [ABSTRACT FROM AUTHOR]

We consider approximation methods defined by translates of a positive definite function on a compact group. A characterization of the native space generated by a positive definite function on a compact group is presented. Starting from Bochner's theorem, we construct examples of well-localized positive definite central functions on the rotation group [image omitted](3). Finally, the stability of the interpolation problem and the error analysis for the given examples are studied in detail. [ABSTRACT FROM AUTHOR]