On Gabor frames generated by sign-changing windows and B-splines

Abstract

For a class of compactly supported windows we characterize the frame property for a Gabor system {E mb T na g} m,n∈Z , for translation parameters a belonging to a certain range depending on the support size. We show that the obstructions to the frame property are located on a countable number of "curves." For functions that are positive on the interior of the support these obstructions do not appear, and the considered region in the (a, b) plane is fully contained in the frame set. In particular this confirms a recent conjecture about B-splines by Gröchenig in that particular region. We prove that the full conjecture is true if it can be proved in a certain "hyperbolic strip."

Counterexamples to the B-spline Conjecture for Gabor Frames

“…Kloos and Stöckler [11] and Christensen, Kim, and Kim [4] reported positive results to support of the frame set conjecture for B-splines, adding new parameter values (a, b) ∈ R 2 + to the known parts of F (B n ); these new values are illustrated in Figure 1 for the case of B 2 . Along the same lines, we verify the frame set conjecture for B-spline for a new a b n 2 n 3n 2 1 n 1 2 3 Figure 1: A sketch of the frame set for B n , n = 2, for 0 < ab < 1.…”

“…Kloos and Stöckler [11] and Christensen, Kim, and Kim [4] reported positive results to support of the frame set conjecture for B-splines, adding new parameter values (a, b) ∈ R 2 + to the known parts of F (B n ); these new values are illustrated in Figure 1 The green region is the classical "painless expansions" [6], the yellow region is the result from [4], while the dark green lines follow from [1] or [11]. The blue region is the result in Theorem 3.…”

“…Red, pink and purple indicate (a, b)-values, where G(B 2 , a, b) is not a frame. All other colors indicate the frame property.The green region is the classical "painless expansions"[6], the yellow region is the result from[4], while the dark green lines follow from[1] or[11]. The blue region is the result in Theorem 3.…”

An Invitation to Gabor Analysis

“…Due to the fact that g N ∈ M 1 (R) for N ≥ 2, we know that F(g N ) is an open subset of R 2 + . The current description of points in this set can be found in [2,15,16,36,54,56]. For example, consider the case N = 2 where…”

“…All other colors indicate the frame property. The green region is the classical: "painless expansions" [21], and the yellow region is the result from [15]. The blue and the magenta regions are respectively from [56] and [16].…”