An Analysis Method for Sampling in Shift-Invariant Spaces

Abstract

A subspace V of L 2 (R) is called shift-invariant if it is the closed linear span of integershifted 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 for deriving both new results and relatively simple proofs of some previously known results. Among these are both 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 x 0 and an upper bound δ such that any function f ∈ V can be reconstructed from a sequence of sample values (f (x 0 + k + δ k )) k∈Z , either when all δ k = 0 or in the irregular sampling case with an upper bound sup k |δ k | < δ.

Efficient wavelet prefilters with optimal time-shifts

“…This was predicted in [6], but Figure 3 shows that even if x 0 ≈ x max is a good rule of thumb for quick decisions, an optimal x 0 can give much lower approximation errors.…”

“…Both this and analysis of errors caused by irregular sampling is out of the scope of this paper but a planned topic for future papers. For now, we refer to, e.g., [6], [18] for some related error bounds and irregular sampling theorems.…”

“…Remark 1: Under certain conditions on x 0 and ϕ, the initial projection P J in Figure 1, assures that f J can be reproduced from the sample values f J (2 −J (x 0 + k)) k∈Z (see, e.g., [6], [7], [8]). For the Shannon wavelet, this is nothing but wellknown classical sampling theory and P J is simply a lowpass filtering at the Nyquist frequency.…”

“…For convergence of the Neumann series approximation (when N → ∞), it is sufficient to use the stronger condition |1 − m(ξ)| < 1, or equivalently I − Φ < 1, which is known to hold, for example, for Shannon wavelets and for all B-spline wavelets [6]. It is easy to check (for example, in MATLAB) that |1 − m(ξ)| < 1 also for Daubechies wavelets with 1-31 vanishing moments and at least for all Symlet wavelets with up to 36 vanishing moments.…”

Detecting Periodically Expression in Unevenly Spaced Microarray Time Series

“…Signal reconstructing in the shift-invariant space is an active research area and there are many mathematical theories and computer algorithms on the topic [10][11][12][13][14][15][16][17][18][19][20][21][22].…”