Title

Uncertainty Principles And Balian-Low Type Theorems In Principal Shift-Invariant Spaces

Keywords

1nZ-invariance; Shift-invariant spaces; Uncertainty principle

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 φ, but φ satisfies ∫ℝ|φ(x)|2|x|1+∈dx = ∞ for any ∞>0 and its Fourier transform φ̂ cannot decay as fast as (1+|ξ|)-r for any r>12. 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. © 2010 Elsevier Inc. All rights reserved.

Publication Date

5-1-2011

Publication Title

Applied and Computational Harmonic Analysis

Volume

30

Issue

3

Number of Pages

337-347

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1016/j.acha.2010.09.003

Socpus ID

79953075685 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/79953075685

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