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.

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.

scholarly journals On the Isomorphism Problem for Multiplier Algebras of Nevanlinna-Pick Spaces

2017 ◽  
Vol 69 (1) ◽  
pp. 54-106 ◽  
Author(s):  
Michael Hartz
Keyword(s):  
Hilbert Spaces ◽  
Special Class ◽  
Reproducing Kernel ◽  
Reproducing Kernels ◽  
Isomorphism Problem ◽  
Reproducing Kernel Hilbert Spaces ◽  
Universal Space ◽  
Kernel Hilbert Spaces ◽  
Arveson Space ◽  
Multiplier Algebras

AbstractWe continue the investigation of the isomorphism problem for multiplier algebras of reproducing kernel Hilbert spaces with the complete Nevanlinna-Pick property. In contrast to previous work in this area, we do not study these spaces by identifying them with the restrictions of a universal space, namely theDrury-Arveson space. Instead, we work directly with theHilbert spaces and their reproducing kernels. In particular, we show that two multiplier algebras of Nevanlinna-Pick spaces on the same set are equal if and only if the Hilbert spaces are equal. Most of the article is devoted to the study of a special class of complete Nevanlinna-Pick spaces on homogeneous varieties. We provide a complete answer to the question of when two multiplier algebras of spaces of this type are algebraically or isometrically isomorphic.This generalizes results of Davidson, Ramsey,Shalit, and the author.

scholarly journals ON THE INCLUSION RELATION OF REPRODUCING KERNEL HILBERT SPACES

2013 ◽  
Vol 11 (02) ◽  
pp. 1350014 ◽  
Author(s):  
HAIZHANG ZHANG ◽  
LIANG ZHAO
Keyword(s):  
Applied Sciences ◽  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Reproducing Kernels ◽  
Reproducing Kernel Hilbert Spaces ◽  
Inclusion Relation ◽  
Feature Maps ◽  
Translation Invariant ◽  
Inclusion Relations ◽  
Kernel Hilbert Spaces

To help understand various reproducing kernels used in applied sciences, we investigate the inclusion relation of two reproducing kernel Hilbert spaces. Characterizations in terms of feature maps of the corresponding reproducing kernels are established. A full table of inclusion relations among widely-used translation invariant kernels is given. Concrete examples for Hilbert–Schmidt kernels are presented as well. We also discuss the preservation of such a relation under various operations of reproducing kernels. Finally, we briefly discuss the special inclusion with a norm equivalence.

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 spaces on manifolds: Sobolev and diffusion spaces

2020 ◽  
pp. 1-34
Author(s):  
Ernesto De Vito ◽  
Nicole Mücke ◽  
Lorenzo Rosasco
Keyword(s):  
Sobolev Spaces ◽  
Riemannian Manifolds ◽  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Differential Operators ◽  
Reproducing Kernels ◽  
Reproducing Kernel Hilbert Spaces ◽  
Kernel Hilbert Spaces ◽  
Euclidean Setting ◽  
And Diffusion

We study reproducing kernel Hilbert spaces (RKHS) on a Riemannian manifold. In particular, we discuss under which condition Sobolev spaces are RKHS and characterize their reproducing kernels. Further, we introduce and discuss a class of smoother RKHS that we call diffusion spaces. We illustrate the general results with a number of detailed examples. While connections between Sobolev spaces, differential operators and RKHS are well known in the Euclidean setting, here we present a self-contained study of analogous connections for Riemannian manifolds. By collecting a number of results in unified a way, we think our study can be useful for researchers interested in the topic.

Refined Generalization Bounds of Gradient Learning over Reproducing Kernel Hilbert Spaces

2015 ◽  
Vol 27 (6) ◽  
pp. 1294-1320 ◽  
Author(s):  
Shao-Gao Lv
Keyword(s):  
Hilbert Spaces ◽  
Upper Bound ◽  
Dimensional Reduction ◽  
Reproducing Kernel ◽  
Positive Definite ◽  
Reproducing Kernel Hilbert Spaces ◽  
Positive Definite Kernels ◽  
Generalization Bounds ◽  
Kernel Hilbert Spaces ◽  
Gradient Learning

Gradient learning (GL), initially proposed by Mukherjee and Zhou ( 2006 ) has been proved to be a powerful tool for conducting variable selection and dimensional reduction simultaneously. This approach presents a nonparametric version of a gradient estimator with positive definite kernels without estimating the true function itself, so that the proposed version has wide applicability and allows for complex effects between predictors. In terms of theory, however, existing generalization bounds for GL depend on capacity-independent techniques, and the capacity of kernel classes cannot be characterized completely. Thus, this letter considers GL estimators that minimize the empirical convex risk. We prove generalization bounds for such estimators with rates that are faster than previous results. Moreover, we provide a novel upper bound for Rademacher chaos complexity of order two, which also plays an important role in general pairwise-type estimations, including ranking and score problems.

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