Uncertainty principles and Balian-Low type theorems in principal shift-invariant spaces
Abbreviated Journal Title
Appl. Comput. Harmon. Anal.
Shift-invariant spaces; 1/nZ-invariance; Uncertainty principle; TRANSLATION-INVARIANCE; SUBSPACES; Mathematics, Applied; Physics, Mathematical
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.
Applied and Computational Harmonic Analysis
"Uncertainty principles and Balian-Low type theorems in principal shift-invariant spaces" (2011). Faculty Bibliography 2010s. 1046.