scholarly journals Regular and Irregular Sampling Theorem for Multiwavelet Subspaces

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.

GENERALIZED IRREGULAR SAMPLING IN SHIFT-INVARIANT SPACES

2007 ◽  
Vol 05 (03) ◽  
pp. 369-387 ◽  
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.

SAMPLING AND OVERSAMPLING IN SHIFT-INVARIANT AND MULTIRESOLUTION SPACES I: VALIDATION OF SAMPLING SCHEMES

2005 ◽  
Vol 03 (02) ◽  
pp. 257-281 ◽  
Author(s):  
J. A. HOGAN ◽  
J. D. LAKEY
Keyword(s):  
Summation Formula ◽  
Sampling Theorem ◽  
Poisson Summation Formula ◽  
Zak Transform ◽  
Self Similarity ◽  
Critical Rate ◽  
Sampling Schemes ◽  
Bandlimited Signals ◽  
Shift Invariant Spaces ◽  
Invariant Spaces

We ask what conditions can be placed on generators φ of principal shift invariant spaces to ensure the validity of analogues of the classical sampling theorem for bandlimited signals. Critical rate sampling schemes lead to expansion formulas in terms of samples, while oversampling schemes can lead to expansions in which function values depend only on nearby samples. The basic techniques for validating such schemes are built on the Zak transform and the Poisson summation formula. Validation conditions are phrased in terms of orthogonality, smoothness, and self-similarity, as well as bandlimitedness or compact support of the generator. Effective sampling rates which depend on the length of support of the generator or its Fourier transform are derived.

IRREGULAR SAMPLING IN SHIFT INVARIANT SPACES OF HIGHER DIMENSIONS

2008 ◽  
Vol 06 (01) ◽  
pp. 121-136 ◽  
Author(s):  
STEFAN ERICSSON
Keyword(s):  
Arbitrary Function ◽  
Riesz Basis ◽  
Higher Dimensions ◽  
Irregular Sampling ◽  
Shift Invariant Spaces ◽  
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.

A sampling theorem for the twisted shift-invariant space

2017 ◽  
Vol 8 (4) ◽  
Author(s):  
Radha Ramakrishnan ◽  
Saswata Adhikari
Keyword(s):  
Sampling Theorem ◽  
Invariant Space ◽  
Shift Invariant Spaces ◽  
Invariant Spaces ◽  
Shift Invariant Space

AbstractRecently, a characterization of frames in twisted shift-invariant spaces in

scholarly journals Two-channel sampling in wavelet subspaces

2015 ◽  
Vol 23 (1) ◽  
pp. 115-126
Author(s):  
J.M. Kim ◽  
K.H. Kwon
Keyword(s):  
Sampling Theory ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Expansion Formula ◽  
Multi Resolution Analysis ◽  
Resolution Analysis ◽  
Shift Invariant Spaces ◽  
Wavelet Subspaces ◽  
Necessary And Sufficient ◽  
Invariant Spaces

Abstract We develop two-channel sampling theory in the wavelet subspace V1 from the multi resolution analysis {Vj}j∈𝕫. Extending earlier results by G. G. Walter [11], W. Chen and S. Itoh [2] and Y. M. Hong et al [5] on the sampling theory in the wavelet or shift invariant spaces, we find a necessary and sufficient condition for two-channel sampling expansion formula to hold in V1.

SAMPLING EXPANSION IN SHIFT INVARIANT SPACES

2008 ◽  
Vol 06 (02) ◽  
pp. 223-248 ◽  
Author(s):  
J. M. KIM ◽  
K. H. KWON
Keyword(s):  
Truncation Error ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Expansion Formula ◽  
Sampling Series ◽  
Shift Invariant Spaces ◽  
Necessary And Sufficient ◽  
Invariant Spaces

For any ϕ(t) in L2(ℝ), let V(ϕ) be the closed shift invariant subspace of L2(ℝ) spanned by integer translates {ϕ(t - n) : n ∈ ℤ} of ϕ(t). Assuming that ϕ(t) is a frame or a Riesz generator of V(ϕ), we first find conditions under which V(ϕ) becomes a reproducing kernel Hilbert space. We then find necessary and sufficient conditions under which an irregular or a regular shifted sampling expansion formula holds on V(ϕ) and obtain truncation error estimates of the sampling series. We also find a sufficient condition for a function in L2(ℝ) that belongs to a sampling subspace of L2(ℝ). Several illustrating examples are also provided.

scholarly journals AN ANALYSIS METHOD FOR SAMPLING IN SHIFT-INVARIANT SPACES

2005 ◽  
Vol 03 (03) ◽  
pp. 301-319 ◽  
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.

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