Title

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

Authors

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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