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.

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.

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.

ANALYTIC TRIDIAGONAL REPRODUCING KERNELS

2001 ◽  
Vol 64 (3) ◽  
pp. 722-738 ◽  
Author(s):  
GREGORY T. ADAMS ◽  
PAUL J. MCGUIRE
Keyword(s):  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Bounded Operator ◽  
Reproducing Kernels ◽  
Reproducing Kernel Hilbert Spaces ◽  
Unilateral Shift ◽  
Orthonormal Bases ◽  
Kernel Hilbert Spaces ◽  
Primary Focus

The paper characterizes the reproducing kernel Hilbert spaces with orthonormal bases of the form {(an,0+an,1z+…+an,JzJ)zn, n [ges ] 0}. The primary focus is on the tridiagonal case where J = 1, and on how it compares with the diagonal case where J = 0. The question of when multiplication by z is a bounded operator is investigated, and aspects of this operator are discussed. In the diagonal case, Mz is a weighted unilateral shift. It is shown that in the tridiagonal case, this need not be so, and an example is given in which the commutant of Mz on a tridiagonal space is strikingly different from that on any diagonal space.

INDEFINITE KERNEL NETWORK WITH DEPENDENT SAMPLING

2013 ◽  
Vol 11 (05) ◽  
pp. 1350020 ◽  
Author(s):  
HONGWEI SUN ◽  
QIANG WU
Keyword(s):  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Function Class ◽  
Positive Definiteness ◽  
Reproducing Kernel Hilbert Spaces ◽  
Strong Mixing ◽  
Mixing Condition ◽  
Indefinite Kernel ◽  
Learning Rates ◽  
Kernel Hilbert Spaces

We study the asymptotical properties of indefinite kernel network with coefficient regularization and dependent sampling. The framework under investigation is different from classical kernel learning. Positive definiteness is not required by the kernel function and the samples are allowed to be weakly dependent with the dependence measured by a strong mixing condition. By a new kernel decomposition technique introduced in [27], two reproducing kernel Hilbert spaces and their associated kernel integral operators are used to characterize the properties and learnability of the hypothesis function class. Capacity independent error bounds and learning rates are deduced.

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