scholarly journals Translation-invariant function algebras on abelian groups

1960 ◽  
Vol 79 ◽  
pp. 345-370 ◽  
Author(s):  
K. De Leeuw ◽  
H. Mirkil
Keyword(s):  
Abelian Groups ◽  
Invariant Function ◽  
Translation Invariant ◽  
Function Algebras

scholarly journals Translation Invariant Spaces and Asymptotic Properties of Variational Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-36 ◽  
Author(s):  
Adina Luminiţa Sasu ◽  
Bogdan Sasu
Keyword(s):  
Control Systems ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Invariant Function ◽  
Function Spaces ◽  
Skew Product ◽  
Translation Invariant ◽  
Variational Equations ◽  
Skew Product Flows ◽  
New Perspective

We present a new perspective concerning the study of the asymptotic behavior of variational equations by employing function spaces techniques. We give a complete description of the dichotomous behaviors of the most general case of skew-product flows, without any assumption concerning the flow, the cocycle or the splitting of the state space, our study being based only on the solvability of some associated control systems between certain function spaces. The main results do not only point out new necessary and sufficient conditions for the existence of uniform and exponential dichotomy of skew-product flows, but also provide a clear chart of the connections between the classes of translation invariant function spaces that play the role of the input or output classes with respect to certain control systems. Finally, we emphasize the significance of each underlying hypothesis by illustrative examples and present several interesting applications.

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.

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