scholarly journals Berezin number inequality for convex function in reproducing Kernel Hilbert Space

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5711-5717 ◽  
Author(s):  
Ulaş Yamancı ◽  
Mehmet Gürdal ◽  
Mubariz Garayev
Keyword(s):  
Hilbert Space ◽  
Convex Function ◽  
Hilbert Spaces ◽  
Convex Functions ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Reproducing Kernel Hilbert Spaces ◽  
Selfadjoint Operators ◽  
Kernel Hilbert Spaces ◽  
Berezin Number

By using Hardy-Hilbert?s inequality, some power inequalities for the Berezin number of a selfadjoint operators in Reproducing Kernel Hilbert Spaces (RKHSs) with applications for convex functions are given.

scholarly journals Cebysev’s type inequalities and power inequalities for the Berezin number of operators

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2307-2316
Author(s):  
Mubariz Garayev ◽  
Ulaş Yamancı
Keyword(s):  
Hilbert Spaces ◽  
Convex Functions ◽  
Reproducing Kernel ◽  
Type Inequality ◽  
Reproducing Kernel Hilbert Spaces ◽  
Selfadjoint Operators ◽  
Kernel Hilbert Spaces ◽  
Berezin Number

We give operator analogues of some classical inequalities, including Cebysev type inequality for synchronous and convex functions of selfadjoint operators in Reproducing Kernel Hilbert Spaces (RKHSs). We obtain some Berezin number inequalities for the product of operators. Also, we prove the Berezin number inequality for the commutator of two operators.

Berezin Number Inequalities of an Invertible Operator and Some Slater Type Inequalities in Reproducing Kernel Hilbert Spaces

2020 ◽  
pp. 55-78
Author(s):  
Ulaş Yamancı ◽  
Mehmet Gürdal
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Invertible Operator ◽  
Reproducing Kernel Hilbert Spaces ◽  
Slater Type ◽  
Riesz Representation ◽  
Kernel Hilbert Spaces ◽  
Berezin Number

A reproducing kernel Hilbert space (shorty, RKHS) H=H(Ω) on some set Ω is a Hilbert space of complex valued functions on Ω such that for every λ∈Ω the linear functional (evaluation functional) f→f(λ) is bounded on H. If H is RKHS on a set Ω, then, by the classical Riesz representation theorem for every λ∈Ω there is a unique element kH,λ∈H such that f(λ)=〈f,kH,λ〉; for all f∈H. The family {kH,λ:λ∈Ω} is called the reproducing kernel of the space H. The Berezin set and the Berezin number of the operator A was respectively given by Karaev in [26] as following Ber(A)={A(λ):λ∈Ω} and ber(A):=|A(λ)|. In this chapter, the authors give the Berezin number inequalities for an invertible operator and some other related results are studied. Also, they obtain some inequalities of the slater type for convex functions of selfadjoint operators in reproducing kernel Hilbert spaces and examine related results.

Data-Driven Optimization: A Reproducing Kernel Hilbert Space Approach

2021 ◽  
Author(s):  
Dimitris Bertsimas ◽  
Nihal Koduri
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Reproducing Kernel Hilbert Spaces ◽  
Data Driven ◽  
Kernel Hilbert Spaces ◽  
Space Approach

Data-Driven Optimization Using Reproducing Kernel Hilbert Spaces

scholarly journals General construction of reproducing kernels on a quaternionic Hilbert space

2017 ◽  
Vol 29 (05) ◽  
pp. 1750017
Author(s):  
K. Thirulogasanthar ◽  
S. Twareque Ali
Keyword(s):  
Hilbert Space ◽  
Positive Operator ◽  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Extension Theorem ◽  
Reproducing Kernels ◽  
Reproducing Kernel Hilbert Spaces ◽  
Positive Operator Valued Measures ◽  
Kernel Hilbert Spaces ◽  
Quaternionic Hilbert Space

A general theory of reproducing kernels and reproducing kernel Hilbert spaces on a right quaternionic Hilbert space is presented. Positive operator-valued measures and their connection to a class of generalized quaternionic coherent states are examined. A Naimark type extension theorem associated with the positive operator-valued measures is proved in a right quaternionic Hilbert space. As illustrative examples, real, complex and quaternionic reproducing kernels and reproducing kernel Hilbert spaces arising from Hermite and Laguerre polynomials are presented. In particular, in the Laguerre case, the Naimark type extension theorem on the associated quaternionic Hilbert space is indicated.

A Factorization Theorem for Multiplier Algebras of Reproducing Kernel Hilbert Spaces

2013 ◽  
Vol 56 (2) ◽  
pp. 400-406
Author(s):  
Bebe Prunaru
Keyword(s):  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Finite Measure ◽  
Reproducing Kernel Hilbert Spaces ◽  
Property A ◽  
Multiplier Algebra ◽  
Finite Measure Space ◽  
Kernel Hilbert Spaces ◽  
Multiplier Algebras

Abstract.Let (X;B; μ) be a σ-finite measure space and let H ⊂ L2(X; μ) be a separable reproducing kernel Hilbert space on X. We show that the multiplier algebra of H has property (A1(1)).

scholarly journals Reproducing Kernel Hilbert Space Associated with a Unitary Representation of a Groupoid

2021 ◽  
Vol 15 (5) ◽  
Author(s):  
Monika Drewnik ◽  
Tomasz Miller ◽  
Zbigniew Pasternak-Winiarski
Keyword(s):  
Unitary Representation ◽  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Reproducing Kernels ◽  
Positive Definite ◽  
Reproducing Kernel Hilbert Spaces ◽  
Locally Compact Group ◽  
Positive Definite Kernel ◽  
Kernel Hilbert Spaces

AbstractThe aim of the paper is to create a link between the theory of reproducing kernel Hilbert spaces (RKHS) and the notion of a unitary representation of a group or of a groupoid. More specifically, it is demonstrated on one hand how to construct a positive definite kernel and an RKHS for a given unitary representation of a group(oid), and on the other hand how to retrieve the unitary representation of a group or a groupoid from a positive definite kernel defined on that group(oid) with the help of the Moore–Aronszajn theorem. The kernel constructed from the group(oid) representation is inspired by the kernel defined in terms of the convolution of functions on a locally compact group. Several illustrative examples of reproducing kernels related with unitary representations of groupoids are discussed in detail. The paper is concluded with the brief overview of the possible applications of the proposed constructions.

ESTIMATING THE APPROXIMATION ERROR IN LEARNING THEORY

2003 ◽  
Vol 01 (01) ◽  
pp. 17-41 ◽  
Author(s):  
STEVE SMALE ◽  
DING-XUAN ZHOU
Keyword(s):  
Hilbert Space ◽  
Learning Theory ◽  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Interpolation Space ◽  
Reproducing Kernel Hilbert Space ◽  
Approximation Error ◽  
Approximation Problem ◽  
Linear Operators ◽  
Kernel Hilbert Spaces

Let B be a Banach space and (ℋ,‖·‖ℋ) be a dense, imbedded subspace. For a ∈ B, its distance to the ball of ℋ with radius R (denoted as I(a, R)) tends to zero when R tends to infinity. We are interested in the rate of this convergence. This approximation problem arose from the study of learning theory, where B is the L2 space and ℋ is a reproducing kernel Hilbert space. The class of elements having I(a, R) = O(R-r) with r > 0 is an interpolation space of the couple (B, ℋ). The rate of convergence can often be realized by linear operators. In particular, this is the case when ℋ is the range of a compact, symmetric, and strictly positive definite linear operator on a separable Hilbert space B. For the kernel approximation studied in Learning Theory, the rate depends on the regularity of the kernel function. This yields error estimates for the approximation by reproducing kernel Hilbert spaces. When the kernel is smooth, the convergence is slow and a logarithmic convergence rate is presented for analytic kernels in this paper. The purpose of our results is to provide some theoretical estimates, including the constants, for the approximation error required for the learning theory.

scholarly journals New characterizations of reproducing kernel Hilbert spaces and applications to metric geometry

2021 ◽  
Vol 41 (3) ◽  
pp. 283-300
Author(s):  
Daniel Alpay ◽  
Palle E.T. Jorgensen
Keyword(s):  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Positive Definite ◽  
Reproducing Kernel Hilbert Spaces ◽  
Bilinear Forms ◽  
Metric Geometry ◽  
Positive Definite Kernels ◽  
Positive Definite Kernel ◽  
Kernel Hilbert Spaces

We give two new global and algorithmic constructions of the reproducing kernel Hilbert space associated to a positive definite kernel. We further present a general positive definite kernel setting using bilinear forms, and we provide new examples. Our results cover the case of measurable positive definite kernels, and we give applications to both stochastic analysis and metric geometry and provide a number of examples.

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