Local reconstruction for sampling in shift-invariant spaces

    Authors

    Q. Y. Sun

    Comments

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    Abbreviated Journal Title

    Adv. Comput. Math.

    Keywords

    Local reconstruction; Sampling; Shift-invariant space; Locally; finitely-generated space; Refinable space; Ripplets; Spline; Wavelets; FINITE RATE; LINEAR INDEPENDENCE; REFINABLE FUNCTIONS; WAVELET; SUBSPACES; GENERAL DILATION; SIGNALS; INNOVATION; THEOREMS; ALGORITHM; SHANNON; Mathematics, Applied

    Abstract

    The local reconstruction from samples is one of most desirable properties for many applications in signal processing, but it has not been given as much attention. In this paper, we will consider the local reconstruction problem for signals in a shift-invariant space. In particular, we consider finding sampling sets X such that signals in a shift-invariant space can be locally reconstructed from their samples on X. For a locally finite-dimensional shift-invariant space V we show that signals in V can be locally reconstructed from its samples on any sampling set with sufficiently large density. For a shift-invariant space V(phi(1),..., phi(N)) generated by finitely many compactly supported functions phi(1),..., phi(N), we characterize all periodic nonuniform sampling sets X such that signals in that shift-invariant space V(phi(1),..., phi(N)) can be locally reconstructed from the samples taken from X. For a refinable shift-invariant space V(phi) generated by a compactly supported refinable function phi, we prove that for almost all (x(0), x(1)) is an element of [0, 1](2), any signal in V(phi) can be locally reconstructed from its samples from {x(0), x(1)} + Z with oversampling rate 2. The proofs of our results on the local sampling and reconstruction in the refinable shift-invariant space V(phi) depend heavily on the linear independent shifts of a refinable function on measurable sets with positive Lebesgue measure and the almost ripplet property for a refinable function, which are new and interesting by themselves.

    Journal Title

    Advances in Computational Mathematics

    Volume

    32

    Issue/Number

    3

    Publication Date

    1-1-2010

    Document Type

    Article

    Language

    English

    First Page

    335

    Last Page

    352

    WOS Identifier

    WOS:000274955200004

    ISSN

    1019-7168

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