A sharp Balian–Low uncertainty principle for shift-invariant spaces

Abstract

A sharp version of the Balian-Low theorem is proven for the generators of finitely generated shift-invariant spaces. If generators {f k } K k=1 ⊂ L 2 (R d ) are translated along a lattice to form a frame or Riesz basis for a shift-invariant space V , and if V has extra invariance by a suitable finer lattice, then one of the generators f k must sat-Similar results are proven for frames of translates that are not Riesz bases without the assumption of extra lattice invariance. The best previously existing results in the literature give a notably weaker conclusion using the Sobolev space H d/2+ǫ (R d ); our results provide an absolutely sharp improvement with H 1/2 (R d ). Our results are sharp in the sense that H 1/2 (R d ) cannot be replaced by H s (R d ) for any s < 1/2. 2010 Mathematics Subject Classification. Primary 42C15.

Uncertainty Principles for Fourier Multipliers

“…This theorem is sharp and the non-triviality of the extra invariance is necessary, [26]. Our next results extend the Balian-Low theorem for shift-invariant spaces from the setting of Riesz bases to the settings of exact systems and exact pC q q-systems.…”

“…To prove the particular case of W P rW s,2 pT d qs KˆK we may, without loss of generality, assume that 0 ă s ă 1, since Theorem 2.4 allows us to replace W s,r with W s,r where s ă 1 and s ´d{r " s ´d{r. It is straightforward to check that the assumptions on s and r and the restriction on q are all invariant under this change. Since we assume 0 ă s ă 1, Lemma 4.3 of [26] implies that the eigenvalue functions tλ k u K k"1 of W each satisfy λ k P W s,r pT d q. (In fact, this lemma is given for the space W s,2 pT d q, but the calculations remain unchanged when replacing the seminorm for W s,2 pT d q by that of W s,r pT d q.)…”

“…For more information on extra invariance in shift-invariant spaces, see [2,4,5]. Analogues of the Balian-Low theorem for shift-invariant spaces were studied in [3,26,41]. Specifically, in [26] D. Hardin, A. M. Powell, and the author prove that if V pF q admits nontrivial extra invariance, and T pF q is a Riesz basis for V pF q, then for t ě 1 there exists at least one f k P F which satisfies ż…”

Sharper uncertainty principles associated with Lp‐norm

“…The future work will be focused on the following aspects: (i) develop our results in the quaternionic algebra framework; (ii) study our results' extension to the time‐frequency domain characterized by the fractional FT or the linear canonical transform; (iii) extend our results to the high‐dimensional case …”

N-dimensional Heisenberg’s uncertainty principle for fractional Fourier transform

“…Uncertainty principle plays an important role in harmonic analysis, quantum mechanics, and time-frequency analysis [1,2,3]. The classical N-dimensional Heisenberg's uncertainty principle is given by the inequality [4,5] R N x − a 2 |f (x)| 2 dx…”