Density results for Gabor systems associated with periodic subsets of the real line

Abstract

The well-known density theorem for one-dimensional Gabor systems of the form {e 2πimbx g(x − na)} m,n∈Z , where g ∈ L 2 (R), states that a necessary and sufficient condition for the existence of such a system whose linear span is dense in L 2 (R), or which forms a frame for L 2 (R), is that the density condition a b ≤ 1 is satisfied. The main goal of this paper is to study the analogous problem for Gabor systems for which the window function g vanishes outside a periodic set S ⊂ R which is a Z-shift invariant. We obtain measure-theoretic conditions that are necessary and sufficient for the existence of a window g such that the linear span of the corresponding Gabor system is dense in L 2 (S). Moreover, we show that if this density condition holds, there exists, in fact, a measurable set E ⊂ R with the property that the Gabor system associated with the same parameters a, b and the window g = χ E , forms a tight frame for L 2 (S).

Characterization of Rational Time-Frequency Multi-Window Gabor Frames and Their Duals

“…Zibulski-Zeevi matrix method was also used in the study of subspace singlewindow Gabor frames ( [1], [5] and [6]). In [7] and [16], a different Zak transform and Zak transform matrix from those in [20] and [21] were introduced and used effectively to study Gabor systems on periodic subsets of the real line, while Zibulski-Zeevi matrix method does not work well for such Gabor systems. A variation of the method in [7] and [16] was applied to Gabor systems on discrete periodic sets ( [14] and [15]).…”

“…In [7] and [16], a different Zak transform and Zak transform matrix from those in [20] and [21] were introduced and used effectively to study Gabor systems on periodic subsets of the real line, while Zibulski-Zeevi matrix method does not work well for such Gabor systems. A variation of the method in [7] and [16] was applied to Gabor systems on discrete periodic sets ( [14] and [15]). It was also pointed out in [20] that Zibulski-Zeevi matrix method is not very efficient for Gabor frames G(g, a, b) of the form (1.1) when (1.3) does not hold.…”

“…So we usually say S and S ′ are λZ-congruent in this case. Let us recall the definition of Zak transform in [7]. The Zak transform Z αq :…”

Gabor frame sets for subspaces

“…We deal with the Gabor systems of the form G(χ F , a, b ), where F is a measurable subset of S, and χ F denotes the characteristic function of F. A measurable set F in S is called a Gabor frame set (tight Gabor set) in S if G(χ F , a, b ) is a frame (tight frame) for L 2 (S), and called a Gabor Bessel set in S if G(χ F , a, b ) is a Bessel sequence in L 2 (S). During the last 20 years Gabor systems have been extensively studied in L 2 (R) (see [7][8][9]15] and the references therein), and also in the setting of subspaces [2,10,11,13,18,19]. This paper concerns Gabor analysis in L 2 (S), where S is an aZ-periodic set in R. Such a scenario can be used to model a signal that appears periodically but intermittently.…”

“…Given a, b > 0. For Gabor analysis in an aZ-periodic set S, Gabardo and the second named author of this paper in [13] provided some conditions on S for the existence of a complete Gabor system in L 2 (S) and the existence of tight Gabor sets. This paper is devoted to Gabor frame sets (especially tight Gabor sets) in S. Our results are also new even if S = R.…”