scholarly journals VECTOR VALUED REPRODUCING KERNEL HILBERT SPACES AND UNIVERSALITY

2010 ◽  
Vol 08 (01) ◽  
pp. 19-61 ◽  
Author(s):  
C. CARMELI ◽  
E. DE VITO ◽  
A. TOIGO ◽  
V. UMANITÀ
Keyword(s):  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Abelian Groups ◽  
Reproducing Kernel Hilbert Spaces ◽  
Feature Maps ◽  
Translation Invariant ◽  
Kernel Hilbert Spaces ◽  
Vector Valued ◽  
Invariant Kernels

This paper is devoted to the study of vector valued reproducing kernel Hilbert spaces. We focus on two aspects: vector valued feature maps and universal kernels. In particular, we characterize the structure of translation invariant kernels on abelian groups and we relate it to the universality problem.

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.

VECTOR VALUED REPRODUCING KERNEL HILBERT SPACES OF INTEGRABLE FUNCTIONS AND MERCER THEOREM

2006 ◽  
Vol 04 (04) ◽  
pp. 377-408 ◽  
Author(s):  
CLAUDIO CARMELI ◽  
ERNESTO DE VITO ◽  
ALESSANDRO TOIGO
Keyword(s):  
Integral Operator ◽  
Hilbert Spaces ◽  
Spectral Decomposition ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Spaces ◽  
Integrable Functions ◽  
Kernel Hilbert Spaces ◽  
Vector Valued

We characterize the reproducing kernel Hilbert spaces whose elements are p-integrable functions in terms of the boundedness of the integral operator whose kernel is the reproducing kernel. Moreover, for p = 2, we show that the spectral decomposition of this integral operator gives a complete description of the reproducing kernel, extending the Mercer theorem.

Sign in / Sign up

Export Citation Format

Share Document