Extremal Problems for Positive-Definite Bandlimited Functions. I. Eventually Positive Functions with Zero Integral

1983 ◽  
Vol 14 (2) ◽  
pp. 249-252 ◽  
Author(s):  
B. F. Logan
Keyword(s):  
Extremal Problems ◽  
Positive Definite ◽  
Bandlimited Functions ◽  
Zero Integral ◽  
Positive Functions

scholarly journals APPLICATIONS OF AN INTERSECTION FORMULA TO DUAL CONES

2017 ◽  
Vol 97 (1) ◽  
pp. 94-101
Author(s):  
DÁNIEL VIROSZTEK
Keyword(s):  
Banach Spaces ◽  
Duality Theorem ◽  
Extremal Problems ◽  
Positive Definite ◽  
Positive Definite Functions ◽  
Trigonometric Polynomials ◽  
Dual Cones ◽  
Intersection Formula

We give a succinct proof of a duality theorem obtained by Révész [‘Some trigonometric extremal problems and duality’, J. Aust. Math. Soc. Ser. A 50 (1991), 384–390] which concerns extremal quantities related to trigonometric polynomials. The key tool of our new proof is an intersection formula on dual cones in real Banach spaces. We show another application of this intersection formula which is related to integral estimates of nonnegative positive-definite functions.

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 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 On Pointwise Estimates of Positive Definite Functions With Given Support

2006 ◽  
Vol 58 (2) ◽  
pp. 401-418 ◽  
Author(s):  
Mihail N. Kolountzakis ◽  
Szilárd Gy. Révész
Keyword(s):  
Extremal Problems ◽  
Positive Definite ◽  
Positive Definite Functions ◽  
Trigonometric Polynomials ◽  
Symmetric Convex Body ◽  
Symmetric Convex ◽  
Convex Domains ◽  
Extremal Values ◽  
Limiting Case ◽  
The Relationship

AbstractThe following problem has been suggested by Paul Turán. Let Ω be a symmetric convex body in the Euclidean space ℝd or in the torus . Then, what is the largest possible value of the integral of positive definite functions that are supported in Ω and normalized with the value 1 at the origin? From this, Arestov, Berdysheva and Berens arrived at the analogous pointwise extremal problem for intervals in ℝ. That is, under the same conditions and normalizations, the supremum of possible function values at z is to be found for any given point z ∈ Ω. However, it turns out that the problem for the real line has already been solved by Boas and Kac, who gave several proofs and also mentioned possible extensions to ℝd and to non-convex domains as well.Here we present another approach to the problem, giving the solution in ℝd and for several cases in . Actually, we elaborate on the fact that the problemis essentially one-dimensional and investigate non-convex open domains as well. We show that the extremal problems are equivalent to some more familiar ones concerning trigonometric polynomials, and thus find the extremal values for a few cases. An analysis of the relationship between the problem for ℝd and that for is given, showing that the former case is just the limiting case of the latter. Thus the hierarchy of difficulty is established, so that extremal problems for trigonometric polynomials gain renewed recognition.

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