Irregular Sampling on Shift Invariant Spaces

We consider irregular sampling in shift invariant spaces V of higher dimensions. The problem that we address is: find ε so that given perturbations (λk) satisfying sup |λk| < ε, we can reconstruct an arbitrary function f of V as a Riesz basis expansions from its irregular sample values f(k + λk). A framework for dealing with this problem is outlined and in which one can explicitly calculate sufficient limits ε for the reconstruction. We show how it works in two concrete situations.

Download Full-text

Regular and Irregular Sampling Theorem for Multiwavelet Subspaces

Mathematical Problems in Engineering ◽

10.1155/2010/757039 ◽

2010 ◽

Vol 2010 ◽

pp. 1-18

Author(s):

Liu Zhanwei ◽

Hu Guoen ◽

Wu Guochang

Keyword(s):

Sampling Theorem ◽

Irregular Sampling ◽

Sufficient Condition ◽

Regular Sampling ◽

Shift Invariant Spaces ◽

Invariant Spaces

We study the sampling theorem for frames in multiwavelet subspaces. Firstly, a sufficient condition under which the regular sampling theorem holds is established. Then, notice that irregular sampling is also useful in practice; we consider the general cases of the irregular sampling and establish a general irregular sampling theorem for multiwavelet subspaces. Finally, using this generalized irregular sampling theorem, we obtain an estimate for the perturbations of regular sampling in shift-invariant spaces.

Download Full-text

Regular and Irregular Sampling Linear Transforms in Series of Shift-Invariant Spaces

Journal of Mathematics and System Science ◽

10.17265/2159-5291/2020.01.005 ◽

2020 ◽

Vol 10 (1) ◽

Author(s):

Adam Zakria ◽

Ahmed Yahia ◽

Ibrahim Elkhalil ◽

Hosam Alfadel

Keyword(s):

Irregular Sampling ◽

Linear Transforms ◽

In Series ◽

Shift Invariant Spaces ◽

Invariant Spaces

Download Full-text

GENERALIZED IRREGULAR SAMPLING IN SHIFT-INVARIANT SPACES

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s021969130700180x ◽

2007 ◽

Vol 05 (03) ◽

pp. 369-387 ◽

Cited By ~ 7

Author(s):

ANTONIO G. GARCÍA ◽

GERARDO PÉREZ-VILLALÓN

Keyword(s):

Sampling Period ◽

Sampling Theory ◽

Irregular Sampling ◽

Natural Consequence ◽

Invariant Space ◽

Starting Point ◽

Regular Sampling ◽

Shift Invariant Spaces ◽

Invariant Spaces ◽

Shift Invariant Space

This article concerns the problem of stable recovering of any function in a shift-invariant space from irregular samples of some filtered versions of the function itself. These samples arise as a perturbation of regular samples. The starting point is the generalized regular sampling theory which allows any function f in a shift-invariant space to be recovered from the samples at {rn}n∈ℤ of s filtered versions [Formula: see text] of f, where the number of channels s is greater or equal than the sampling period r. These regular samples can be expressed as the frame coefficients of a function related to f in L2(0,1) with respect to certain frame for L2(0,1). The irregular samples are also obtained as a perturbation of the aforesaid frame. As a natural consequence, the irregular sampling results arise from the theory of perturbation of frames. The paper concludes by putting the theory to work in some spline examples where Kadec-type results are obtained.

Download Full-text

AN ANALYSIS METHOD FOR SAMPLING IN SHIFT-INVARIANT SPACES

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691305000877 ◽

2005 ◽

Vol 03 (03) ◽

pp. 301-319 ◽

Cited By ~ 12

Author(s):

STEFAN ERICSSON ◽

NIKLAS GRIP

Keyword(s):

Upper Bound ◽

Linear Span ◽

Irregular Sampling ◽

General Nature ◽

Analysis Method ◽

Classical Analysis ◽

Spline Wavelets ◽

Single Function ◽

Shift Invariant Spaces ◽

Invariant Spaces

A subspace V of L2(ℝ) is called shift-invariant if it is the closed linear span of integer-shifted copies of a single function. As a complement to classical analysis techniques for sampling in such spaces, we propose a method which is based on a simple interpolation estimate of a certain coefficient mapping. Then we use this method to derive both new results and relatively simple proofs of some previously known results. Among these are some results of rather general nature and some more specialized results for B-spline wavelets. The main problem under study is to find a shift x0 and an upper bound δ such that any function f ∈ V can be reconstructed from a sequence of sample values (f(x0 + k + δk))k∈ℤ, either when all δk = 0 or in the irregular sampling case with an upper bound sup k|δk| < δ.

Download Full-text