scholarly journals INTEGRAL REPRESENTATION OF EVEN POSITIVE DEFINITE BOUNDED FUNCTIONS OF AN INFINITE NUMBER OF VARIABLES

2020 ◽  
Vol 8 (2) ◽  
pp. 93-102
Author(s):  
O. Lopotko
Keyword(s):  
Integral Representation ◽  
Infinite Number ◽  
Positive Definite ◽  
Definite Function ◽  
Bounded Functions ◽  
Almost All ◽  
Positive Functions

In this article the integral representation for bounded even positive functions $k(x)$\linebreak $\left(x\in \mathbb{R}^\infty=\mathbb{R}\times\mathbb{R}\times\dots \right)$ is proved. We understand the positive the positive definite in the integral sense with integration respects to measure $d\theta(x)= p(x_1)dx_1\otimes p(x_2)dx_2\otimes \dots$\linebreak $\left(p(x)=\sqrt{\frac{1}{\pi}}e^{-x^2} \right)$. This integral representation has the form \begin{equation}\label{ovl1.0} k(x)=\int\limits_{l_2^+} {\rm Cos}\,\lambda_ix_id\rho(\lambda) \end{equation} Equality stands to reason for almost all $x\in \mathbb{R}^\infty$. $l_2^+$ space consists of those vectors $\lambda\in\mathbb{R}^\infty_+=\mathbb{R}^1_+\times \mathbb{R}^1_+\times\dots\left| \sum\limits_{i=1}^\infty \lambda_i^2 <\infty\right.$. Conversely, every integral of form~\eqref{ovl1.0} is bounded by even positively definite function $k(x)$ $x\in\mathbb{R}^\infty$. As a result, from this theorem we shall get generalization of theorem of R.~A.~Minlos--V.~V.~Sazonov \cite{lov2,lov3} in case of bounded even positively definite functions $k(x)$ $(x\in H)$, which are continuous in $O$ in $j$"=topology.

scholarly journals Complete positivity and distance-avoiding sets

2020 ◽  
Author(s):  
Evan DeCorte ◽  
Fernando Mário de Oliveira Filho ◽  
Frank Vallentin
Keyword(s):  
Convex Optimization ◽  
Euclidean Space ◽  
Optimization Problem ◽  
Maximum Density ◽  
Complete Positivity ◽  
Optimal Solutions ◽  
Convex Optimization Problem ◽  
Completely Positive ◽  
Positive Type ◽  
Positive Functions

Abstract We introduce the cone of completely positive functions, a subset of the cone of positive-type functions, and use it to fully characterize maximum-density distance-avoiding sets as the optimal solutions of a convex optimization problem. As a consequence of this characterization, it is possible to reprove and improve many results concerning distance-avoiding sets on the sphere and in Euclidean space.

scholarly journals ON INTEGRAL ESTIMATES OF NONNEGATIVE POSITIVE DEFINITE FUNCTIONS

2017 ◽  
Vol 96 (1) ◽  
pp. 117-125 ◽  
Author(s):  
ANDREY EFIMOV ◽  
MARCELL GAÁL ◽  
SZILÁRD GY. RÉVÉSZ
Keyword(s):  
Classical Problem ◽  
Positive Definite ◽  
Positive Definite Functions ◽  
Image Position ◽  
The Right ◽  
Positive Functions

Let $\ell >0$ be arbitrary. We introduce the extremal quantities $$\begin{eqnarray}G(\ell ):=\sup _{f}\int _{-\ell }^{\ell }f\,dx\,\bigg/\int _{-1}^{1}f\,dx,\quad C(\ell ):=\sup _{f}\sup _{a\in \mathbb{R}}\int _{a-\ell }^{a+\ell }f\,dx\,\bigg/\int _{-1}^{1}f\,dx,\end{eqnarray}$$ where the supremum is taken over all not identically zero nonnegative positive definite functions. We investigate how large these extremal quantities can be. This problem was originally posed by Yu. Shteinikov and S. Konyagin (for the case $\ell =2$) and is an extension of the classical problem of Wiener. In this note we obtain exact values for the right limits $\overline{\lim }_{\unicode[STIX]{x1D700}\rightarrow 0+}G(k+\unicode[STIX]{x1D700})$ and $\overline{\lim }_{\unicode[STIX]{x1D700}\rightarrow 0+}C(k+\unicode[STIX]{x1D700})$$(k\in \mathbb{N})$ taken over doubly positive functions, and sufficiently close bounds for other values of $\ell$.

scholarly journals The index of product systems of Hilbert modules: Two equivalent definitions

2015 ◽  
Vol 97 (111) ◽  
pp. 49-56
Author(s):  
Biljana Vujosevic
Keyword(s):  
Positive Definite ◽  
Hilbert Modules ◽  
Product System ◽  
Initial Product ◽  
Positive Definite Kernel ◽  
Completely Positive ◽  
C Algebra ◽  
Product Systems ◽  
The One ◽  
Definition Of

We prove that a conditionally completely positive definite kernel, as the generator of completely positive definite (CPD) semigroup associated with a continuous set of units for a product system over a C*-algebra B, allows a construction of a Hilbert B?B module. That construction is used to define the index of the initial product system. It is proved that such definition is equivalent to the one previously given by Keckic and Vujosevic [On the index of product systems of Hilbert modules, Filomat, to appear, ArXiv:1111.1935v1 [math.OA] 8 Nov 2011]. Also, it is pointed out that the new definition of the index corresponds to the one given earlier by Arveson (in the case B = C).

scholarly journals ON ITERATED POWERS OF POSITIVE DEFINITE FUNCTIONS

2015 ◽  
Vol 92 (3) ◽  
pp. 440-443
Author(s):  
MEHRDAD KALANTAR
Keyword(s):  
Compact Group ◽  
Strong Operator Topology ◽  
Positive Definite Function ◽  
Positive Definite ◽  
Locally Compact Group ◽  
Definite Function ◽  
Completely Positive Maps ◽  
Locally Compact ◽  
Completely Positive ◽  
Positive Maps

We prove that if ${\it\rho}$ is an irreducible positive definite function in the Fourier–Stieltjes algebra $B(G)$ of a locally compact group $G$ with $\Vert {\it\rho}\Vert _{B(G)}=1$, then the iterated powers $({\it\rho}^{n})$ as a sequence of unital completely positive maps on the group $C^{\ast }$-algebra converge to zero in the strong operator topology.

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