Uncertainty principles and Balian-Low type theorems in principal shift-invariant spaces

    Authors

    A. Aldroubi; Q. Y. Sun;H. C. Wang

    Comments

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    Abbreviated Journal Title

    Appl. Comput. Harmon. Anal.

    Keywords

    Shift-invariant spaces; 1/nZ-invariance; Uncertainty principle; TRANSLATION-INVARIANCE; SUBSPACES; Mathematics, Applied; Physics, Mathematical

    Abstract

    In this paper, we consider the time-frequency localization of the generator of a principal shift-invariant space on the real line which has additional shift-invariance. We prove that if a principal shift-invariant space on the real line is translation-invariant then any of its orthonormal (or Riesz) generators is non-integrable. However, for any n > = 2, there exist principal shift-invariant spaces on the real line that are also 1/nZ-invariant with an integrable orthonormal (or a Riesz) generator phi, but phi satisfies integral(R)vertical bar phi(x)vertical bar(2)vertical bar x vertical bar(1+epsilon) dx = infinity for any epsilon > 0 and its Fourier transform (phi) over cap cannot decay as fast as (1 + vertical bar xi vertical bar)(-r) for any r > 1/2. Examples are constructed to demonstrate that the above decay properties for the orthonormal generator in the time domain and in the frequency domain are optimal. (C) 2010 Elsevier Inc. All rights reserved.

    Journal Title

    Applied and Computational Harmonic Analysis

    Volume

    30

    Issue/Number

    3

    Publication Date

    1-1-2011

    Document Type

    Article

    Language

    English

    First Page

    337

    Last Page

    347

    WOS Identifier

    WOS:000289030200005

    ISSN

    1063-5203

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