scholarly journals Generalized sampling: From shift-invariant to U-invariant spaces

2015 ◽  
Vol 13 (03) ◽  
pp. 303-329 ◽  
Author(s):  
H. R. Fernández-Morales ◽  
A. G. García ◽  
M. A. Hernández-Medina ◽  
M. J. Muñoz-Bouzo
Keyword(s):  
Invariant Subspaces ◽  
Invertible Operator ◽  
Continuous Group ◽  
Sampling Frame ◽  
Generalize Convolution ◽  
Time Jitter ◽  
Generalized Sampling ◽  
Frame Expansions ◽  
Invariant Spaces ◽  
Convolution Systems

The aim of this article is to derive a sampling theory in U-invariant subspaces of a separable Hilbert space ℋ where U denotes a unitary operator defined on ℋ. To this end, we use some special dual frames for L2(0, 1), and the fact that any U-invariant subspace with stable generator is the image of L2(0, 1) by means of a bounded invertible operator. The used mathematical technique mimics some previous sampling work for shift-invariant subspaces of L2(ℝ). Thus, sampling frame expansions in U-invariant spaces are obtained. In order to generalize convolution systems and deal with the time-jitter error in this new setting we consider a continuous group of unitary operators which includes the operator U.

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.

On Regular Generalized Sampling in T-Invariant Subspaces of a Hilbert Space: An Overview

2019 ◽  
Vol 41 (6) ◽  
pp. 685-709
Author(s):  
Antonio G. García ◽  
María J. Muñoz-Bouzo ◽  
Gerardo Pérez-Villalón
Keyword(s):  
Hilbert Space ◽  
Invariant Subspaces ◽  
Generalized Sampling

scholarly journals On Generalizations of Sampling Theorem and Stability Theorem in Shift-Invariant Subspaces of Lebesgue and Wiener Amalgam Spaces with Mixed-Norms

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 331
Author(s):  
Junjian Zhao ◽  
Marko Kostić ◽  
Wei-Shih Du
Keyword(s):  
Invariant Subspaces ◽  
Sampling Theorem ◽  
Stability Theorem ◽  
Wiener Amalgam Spaces ◽  
Mixed Norms ◽  
Sampling Theorems ◽  
Generalized Sampling ◽  
Stability Theorems ◽  
Amalgam Spaces ◽  
New Inequalities

In this paper, we establish generalized sampling theorems, generalized stability theorems and new inequalities in the setting of shift-invariant subspaces of Lebesgue and Wiener amalgam spaces with mixed-norms. A convergence theorem of general iteration algorithms for sampling in some shift-invariant subspaces of Lp→(Rd) are also given.

SHIFT INVARIANT SPACES FOR LOCAL FIELDS

2011 ◽  
Vol 09 (03) ◽  
pp. 417-426 ◽  
Author(s):  
A. AHMADI ◽  
A. ASKARI HEMMAT ◽  
R. RAISI TOUSI
Keyword(s):  
Sufficient Conditions ◽  
Invariant Subspaces ◽  
Open Subgroup ◽  
Local Fields ◽  
Locally Compact Abelian Group ◽  
Compact Open Subgroup ◽  
Parseval Frame ◽  
Shift Invariant Spaces ◽  
Necessary And Sufficient ◽  
Invariant Spaces

This paper is an investigation of shift invariant subspaces of L2(G), where G is a locally compact abelian group, or in general a local field, with a compact open subgroup. In this paper we state necessary and sufficient conditions for shifts of an element of L2(G) to be an orthonormal system or a Parseval frame. Also we show that each shift invariant subspace of L2(G) is a direct sum of principle shift invariant subspaces of L2(G) generated by Parseval frame generators.

scholarly journals Operadores uniformes estables y subespacios invariantes

2019 ◽  
Vol 4 (1) ◽  
pp. 75
Author(s):  
Edixo Rosales
Keyword(s):  
Key Words ◽  
Banach Spaces ◽  
Invariant Subspaces ◽  
Palabras Clave ◽  
Invariant Spaces ◽  
Uniformly Stable

  Este trabajo estudia operadores uniformemente estables sobre espacios de Banach en general, con el propósito de caracterizar algunos que tengan subespacios invariantes no triviales.   Palabras clave: Operadores Uniformemente estables, subespacios invariantes.   Abstract:   This paper studies uniformly stable operators on Banach spaces in general, with the purpose of characterizing some that have non-trivial invariant subspaces.   Key words: Uniformly stable operators, sub invariant spaces

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