scholarly journals Sampling Associated with a Unitary Representation of a Semi-Direct Product of Groups: A Filter Bank Approach

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 529
Author(s):  
Antonio G. García ◽  
Miguel Angel Hernández-Medina ◽  
Gerardo Pérez-Villalón
Keyword(s):  
Unitary Representation ◽  
Direct Product ◽  
Filter Bank ◽  
Separable Hilbert Space ◽  
Frame Analysis ◽  
Mathematical Problem ◽  
Dual Frames ◽  
Crystallographic Groups ◽  
Suitable Expression ◽  
Product Of Groups

An abstract sampling theory associated with a unitary representation of a countable discrete non abelian group G, which is a semi-direct product of groups, on a separable Hilbert space is studied. A suitable expression of the data samples, the use of a filter bank formalism and the corresponding frame analysis allow for fixing the mathematical problem to be solved: the search of appropriate dual frames for ℓ 2 ( G ) . An example involving crystallographic groups illustrates the obtained results by using either average or pointwise samples.

UNDECIDABILITY OF THE THEORIES OF CLASSES OF STRUCTURES

2014 ◽  
Vol 79 (4) ◽  
pp. 1001-1019 ◽  
Author(s):  
ASHER M. KACH ◽  
ANTONIO MONTALBÁN
Keyword(s):  
Direct Product ◽  
Disjoint Union ◽  
Second Order ◽  
Linear Orders ◽  
Second Order Arithmetic ◽  
Order Arithmetic ◽  
Natural Classes ◽  
Product Of Groups

AbstractMany classes of structures have natural functions and relations on them: concatenation of linear orders, direct product of groups, disjoint union of equivalence structures, and so on. Here, we study the (un)decidability of the theory of several natural classes of structures with appropriate functions and relations. For some of these classes of structures, the resulting theory is decidable; for some of these classes of structures, the resulting theory is bi-interpretable with second-order arithmetic.

On Nilpotent Products of Cyclic Groups

1960 ◽  
Vol 12 ◽  
pp. 447-462 ◽  
Author(s):  
Ruth Rebekka Struik
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Finite Abelian Group ◽  
Nilpotent Groups ◽  
Multiplication Table ◽  
Direct Products ◽  
Free Products ◽  
Cyclic Groups ◽  
Special Cases ◽  
Product Of Groups

In this paper G = F/Fn is studied for F a free product of a finite number of cyclic groups, and Fn the normal subgroup generated by commutators of weight n. The case of n = 4 is completely treated (F/F2 is well known; F/F3 is completely treated in (2)); special cases of n > 4 are studied; a partial conjecture is offered in regard to the unsolved cases. For n = 4 a multiplication table and other properties are given.The problem arose from Golovin's work on nilpotent products ((1), (2), (3)) which are of interest because they are generalizations of the free and direct product of groups: all nilpotent groups are factor groups of nilpotent products in the same sense that all groups are factor groups of free products, and all Abelian groups are factor groups of direct products. In particular (as is well known) every finite Abelian group is a direct product of cyclic groups. Hence it becomes of interest to investigate nilpotent products of finite cyclic groups.

scholarly journals Certain Groups of Orthonormal Step Functions

1957 ◽  
Vol 9 ◽  
pp. 413-425 ◽  
Author(s):  
J. J. Price
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Compact Abelian Group ◽  
Walsh Functions ◽  
Cyclic Groups ◽  
Step Functions ◽  
Product Of Groups

It was first pointed out by Fine (2), that the Walsh functions are essentially the characters of a certain compact abelian group, namely the countable direct product of groups of order two. Later Chrestenson (1) considered characters of the direct product of cyclic groups of order α (α = 2, 3, …). In general, his results show that the analytic properties of these generalized Walsh functions are basically the same as those of the ordinary Walsh functions.

scholarly journals On Some Sampling-Related Frames inU-Invariant Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
H. R. Fernández-Morales ◽  
A. G. García ◽  
M. A. Hernández-Medina ◽  
M. J. Muñoz-Bouzo
Keyword(s):  
Unitary Operator ◽  
Separable Hilbert Space ◽  
General Setting ◽  
Continuous Group ◽  
Unitary Operators ◽  
Dual Frames ◽  
Generalized Sampling ◽  
Sampling Problem ◽  
Perturbation Problems ◽  
Invariant Spaces

This paper is concerned with the characterization as frames of some sequences inU-invariant spaces of a separable Hilbert spaceℋwhereUdenotes an unitary operator defined onℋ; besides, the dual frames having the same form are also found. This general setting includes, in particular, shift-invariant or modulation-invariant subspaces inL2ℝ, where these frames are intimately related to the generalized sampling problem. We also deal with some related perturbation problems. In doing so, we need the unitary operatorUto belong to a continuous group of unitary operators.

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