Counterexamples to the B-spline Conjecture for Gabor Frames

Abstract

The frame set conjecture for B-splines B n , n ≥ 2, states that the frame set is the maximal set that avoids the known obstructions. We show that any hyperbola of the form ab = r, where r is a rational number smaller than one and a and b denote the sampling and modulation rates, respectively, has infinitely many pieces, located around b = 2, 3, . . . , not belonging to the frame set of the nth order B-spline. This, in turn, disproves the frame set conjecture for B-splines. On the other hand, we uncover a new region belonging to the frame set for B-splines B n , n ≥ 2.2010 Mathematics Subject Classification. Primary 42C15. Secondary: 42A60

On the Gabor frame set for compactly supported continuous functions

“…The region C is a part of the region determined by Proposition 1.1 (ii), corresponding to the new findings in the paper [21].…”

“…We also know that at least for some functions g ∈ 0<a<N V N,a parts of the region determined by the inequalities b < 2, a < 2, ab < 1 do not belong to the frame set. Considering for example the B-spline B 2 , [21] shows that the point (a, b) = ( 2 7 , 7 4 ) does not belong to the frame set. For a = 2 7 Theorem 1.2 guarantees the frame property for b < 7 5 , which is close to the obstruction.…”

“…Most of the related literature deals with special types of functions like truncated trigonometric functions or various types of splines, see [8,20,19,18]. Various classes of functions have also been considered, e.g., functions yielding a partition of unity [11,5], functions with short support or a finite number of sign-changes [3,4,6], or functions that are bounded away from zero on a specified part of the support [21]. The case of B-spline generated Gabor systems has attracted special attention, see, e.g., [24,19,6,21,10].…”

Generalized shift-invariant systems and approximately dual frames

“…In particular the result implies that for any choice of translation parameter a > 0 and modulation parameter b > 0 such that ab < 1, the Gabor system generated by the scaled B-splines generate frames whenever the order of the B-spline is sufficiently high. This result is rather surprising in view of the many known obstructions to the frame property for B-splines, see [8,12,11,17,15]. We also note that the arguments used in the proof are of a general nature, which allows for a similar formulation for general frames in Hilbert spaces [5].…”

Gabor Frames: Characterizations and Coarse Structure

“…Clearly, inequality (22) The Zak transform has been used frequently to derive theoretical properties of Gabor frames. The Zeevi-Zibulski matrices in particular are very useful for computational issues, and several important counter-examples have been discovered first through numerical tests before being proved rigorously [32,33]. On the other hand, it seems to be very difficult to apply directly and decide rigorously whether a concrete Gabor system is a frame or not.…”