On the Gabor frame set for compactly supported continuous functions

We identify a class of continuous compactly supported functions for which the known part of the Gabor frame set can be extended. At least for functions with support on an interval of length two, the curve determining the set touches the known obstructions. Easy verifiable sufficient conditions for a function to belong to the class are derived, and it is shown that the B-splines $B_N, N\ge 2,$ as well as certain"continuous and truncated"versions of several classical functions (e.g., the Gaussian and the two-sided exponential function) belong to the class. The sufficient conditions for the frame property guarantees the existence of a dual window with a prescribed size of the support.


Introduction
Frames is a functional analytic tool to obtain representations of the elements in a Hilbert space as a (typically infinite) superposition of building blocks. Frames indeed lead to decompositions that are similar to the ones obtained via orthonormal bases, but with much greater flexibility, due to the fact that the definition is significantly less restrictive. For example, in contrast to the case for a basis, the elements in a frame are not necessarily (linearly) independent, i.e., frames can be redundant.
One of the main manifestations of frame theory is within Gabor analysis, where the aim is to obtain efficient representations of signals in a way that reflects the time-frequency distribution. For any a, b > 0, consider the translation operator T a and the modulation operator E b , both acting on the particular Hilbert space L 2 (R), given by T a f (x) = f (x − a), respectively E b f (x) = e 2πibx f (x). Given g ∈ L 2 (R), the collection of functions {E mb T na g} m,n∈Z is called a (Gabor) frame if there exist constants A, B > 0 such that If at least the upper condition is satisfied, {E mb T na g} m,n∈Z is called a Bessel sequence. It is known that for every frame {E mb T na g} m,n∈Z there exists a dual frame {E mb T na h} m,n∈Z such that each f ∈ L 2 (R) has the decomposition (1.1) The problem of determining g ∈ L 2 (R) and parameters a, b > 0 such that {E mb T na g} m,n∈Z is a frame has attracted a lot of attention over the past 25 years. The frame set for a function g ∈ L 2 (R) is defined as the set F g := (a, b) ∈ R 2 + {E mb T na g} m,n∈Z is a frame for L 2 (R) .
Clearly the "size" of the set F g reflects the flexibility of the function g in regard of obtaining expansions of the type (1.1). In particular it is known that ab ≤ 1 is necessary for {E mb T na g} m,n∈Z to be a frame and that the number (ab) −1 is a measure of the redundance of the frame; the smaller the number is, the more redundant the frame will be. Thus a reasonable function g should lead to a frame {E mb T na g} m,n∈Z for values (ab) −1 that are reasonably close to one. We remark that F g is known to be open if g belongs to the Feichtinger algebra; see [9,1]. Until recently the exact frame set was only known for very few functions: the Gaussian g(x) = e −x 2 [22,26,27], the hyperbolic secant [17], and the functions h(x) = e −|x| , k(x) = e −x χ [0,∞[ (x) [13,16]. In [12] a characterization was obtained for the class of totally positive functions of finite type, and based on [15] the frame set for functions χ [0,c] , c > 0, was characterized in [7].
For the sake of applications of Gabor frames it is essential that the window g is a continuous function with compact support. 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].
To our best knowledge the frame set has not been characterized for any function g ∈ C c (R) \ {0}. We will, among others, consider a class of functions for which we can extend the known set of parameters (a, b) yielding a Gabor frame. The class of functions contains the B-splines B N , N ≥ 2, as well as certain "continuous and compactly supported variants" of the above functions g, h and other classical functions. Furthermore, the results guarantees the existence of dual windows with a support size given in terms of the translation parameter.
In the rest of this introduction we will describe the relevant class of windows and their frame properties. Proofs of the frame properties are in Section 2, and easy verifiable conditions for a function to belong to the class are derived in Section 3.
Let us first collect some of the known results concerning frame properties for continuous compactly supported functions; (i) is classical, and we refer to [2] for a proof. Proposition 1.1 Let N > 0, and assume that g : R → C is a continuous function with supp g ⊆ [− N 2 , N 2 ]. Then the following holds: (i) If {E mb T na g} m,n∈Z is a frame, then ab < 1 and a < N.
(ii) [21] Assume that 0 < a < N, (iii) [6] Assume that N 2 ≤ a < N and We will now introduce the window class that will be used in the current paper; it is a subset of the set of functions g considered in Proposition 1.1 (iii). The definition is inspired by certain explicit estimates for B-splines, given by Trebels and Steidl in [28]; this point will be clear in Proposition 3.1. First, fix N > 0 and 0 < a < N . Consider the first order difference ∆ a f and the second order difference ∆ 2 a f, given by We define the window class as the set of functions where (A1) f is symmetric around the origin; Note that by the symmetry condition (A1) a function f ∈ V N,a is completely determined by its behavior for ; however, if desired, the symmetry condition allows to formulate the condition ∆ 2 because the argument x − 2a of the last term in the second order difference is less than − N 2 . The definition of V N,a is technical, but we will derive easy verifiable conditions for a function g to belong to this set in Proposition 3.1, and also provide several natural examples of such functions. Our main result extends the range of b > 0 yielding a frame, compared with Proposition 1.1 (ii): Theorem 1.2 For N > 0, let 0 < a < N and 2 N +a < b ≤ 4 N +3a . Assume that g ∈ V N,a . Then the Gabor system {E mb T na g} m,n∈Z is a frame for L 2 (R), and there is a unique dual window h ∈ L 2 (R) such that supp h ⊆ [− 3a 2 , 3a 2 ]. Membership of a function g in a set V N,a for some a ∈]0, N [ only gives information about the frame properties of {E mb T na g} m,n∈Z for this specific value of the translation parameter a. In order to get an impression of the frame properties of {E mb T na g} m,n∈Z in a region in the (a, b)-plane, we need to consider a function g that belongs to V N,a for an interval of a-values, preferably for all a ∈]0, N [. Fortunately several natural functions have this property. The following list collects some of the results we will obtain in Section 3. Considering any N ∈ N \ {1}, • The B-spline B N of order N belongs to 0<a<N V N,a ; In particular, Proposition 1.1 and Theorem 1.2 imply that for N ∈ N\{1} the functions B N , f N , and h N generate frames whenever 0 < a < N and 0 < b ≤ 4 N +3a ; and g N generates a frame whenever 3N 7 ≤ a < N and 0 < b ≤ 4 N +3a . Note that the limit curve b = 4 N +3a in Theorem 1.2 touches the known obstructions for Gabor frames. In fact, for N = 2 we obtain that b → 2 whenever a → 0. Since it is known that the B-spline B 2 does not generate a frame for b = 2 [8,11] we can not go beyond this. 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. These considerations indicate that the frame region in Theorem 1.2 in a quite accurate way describes the maximally possible frame set below b = 2 that is valid for all the functions in V N,a , at least for N = 2.
2 Frame properties for functions g ∈ V N,a The purpose of this section is to prove Theorem 1.2. Since the functions g ∈ V N,a are bounded and have compact support, they generate Bessel sequences {E mb T na g} m,n∈Z for all a, b > 0. By the duality conditions [25,14], two bounded functions g, h with compact support generate dual frames in particular, a function g ∈ V N,a and a bounded real hold for ℓ = 0, ±1. Given g ∈ V N,a we will therefore consider the 3 × 3 matrix-valued function G on [− a 2 , a 2 ] defined by In terms of the G(x) the condition (2.1) simply means that ; this will ultimately give us a bounded and compactly supported function h satisfying (2.1) and hereby prove Theorem 1.2. The invertibility of G(x) will be derived as a consequence of a series of lemmas, where we first consider x ∈ [− a 2 , 0]. Note that the proof of the first result does not use the property (A3): . Then the following hold: Proof. For (a), let x ∈ [− a 2 , 0]. Using b ≤ 4 N +3a and a < N , It follows that By (A2) we know that g is strictly decreasing on [0, N 2 ]. If x + 1 b − a ≤ 0, then we have by (2.3), (2.4) and the symmetry of g that . Hence (a) holds. Similarly, (b) and (c) hold.
We now show that if g ∈ V N,a and a ≥ N/3, the condition (A3) automatically holds on a larger interval. ].
We first note that (A1) and (A2) imply that for ).
Together with (2.5) this shows that as desired.
Let G ij (x) denote the ij-th minor of G(x), the determinant of the submatrix obtained by removing the i-th row and the j-th column from G(x).
Assume that g ∈ V N,a and let x ∈ [− a 2 , 0]. Then the following hold: (a) G 21 (x) ≥ 0, and equality holds iff g(x Proof. Since g ≥ 0, (a) and (b) follow from Lemma 2.1 (a) & (b). For (c), we note that ∆ a g(x) = g(x)−g(x−a) = g(−x)−g(−x+a) = −∆ a g(−x+a) by the symmetry of g. Now a direct calculation shows that Hence it suffices to show that , this means precisely that We note that for 2 Together with Lemma 2.2 this implies that (2.6) holds, as desired.
After this preparation we can now show that G(x) is indeed invertible for x ∈ [− a 2 , a 2 ] under the assumptions in Theorem 1.2.
Using Lemma 2.1 (c) and Lemma 2.3 (a) & (b), , 0] the proof is completed; thus the rest of the proof will focus on the case where A N (x 0 ) = 0 for some x 0 ∈ [− a 2 , 0]. In this case Lemma 2.1 (c) shows that either The case ( Inserting this information into the entries of the matrix G(x 0 ) and applying Lemma 2.1 yields that as desired. This completes the proof that G(x) > 0 for x ∈ [− a 2 , 0]. Since g is symmetric around the origin, we have Thus G(x) is also invertible for x ∈]0, a 2 ].
We are now ready to prove Theorem 1.2.
Proof of Theorem 1.2: By Corollary 2.4 and continuity of g, which is a bounded function. On R \ [− 3a 2 , 3a 2 ], put h(x) = 0. It follows immediately by definition of h that then g and h are dual windows.

Remark 2.5
We note that the above approach is tailored to the region of parameters (a, b) in Theorem 1.2. For example, it does not apply to the region considered in Proposition 1.1 (ii). In fact, if 0 < b ≤ 2 N +a , then the first row of G( , such that the duality conditions (2.1) hold. In order to obtain a contradiction, let us assume that such a dual window indeed exists. Let x 0 = 0. Then and consequently By the continuity of g, there exist continuous functions ǫ ij (x), 1 ≤ i, j ≤ 3 such that Then (2.2) implies that . By elementary row operations, this leads to  Ignoring a possible set of measure zero and using that h is a bounded function, this implies that This is a contradiction.
On the other hand, the condition g ∈ V N,a is not a necessary condition for {E mb T na g} m,n∈Z to be a frame in the considered region.
, one can prove that {E mb T na g 1 } m,n∈Z is a frame by following the steps in the proof of Theorem 1.2.

The set V N,a
In this section we give easy verifiable sufficient conditions for a function g to belong to V N,a . Recall that a continuous function f : is piecewise continuously differentiable if there exist finitely many (2) the one-sided limits lim x→x + i−1 f ′ (x) and lim x→x − i f ′ (x) exist for every i ∈ {1, · · · , n}.
Note that if g is a continuous and piecewise continuously differentiable function, the fundamental theorem of calculus yields that In order to avoid a tedious presentation, we will forego to mention the points where a piecewise continuously differentiable function is not differentiable, e.g., in conditions (c) and (d) in the following Proposition 3.1. The result is inspired by explicit calculations for B-splines, due to Trebels and Steidl; see Lemma 1 in [28].
Proposition 3.1 Let N > 0 and assume that a continuous and piecewise continuously differentiable function g : R → R with supp g = [− N 2 , N 2 ] satisfies the following conditions: (a) g is symmetric around the origin;

Proof.
Note that the conditions (a) and (b) are exactly the same as (A1) and (A2). Thus we will prove (A3). In the entire argument we will assume that g is differentiable; an elementary consideration then extends the result to the case of piecewise differentiable functions. Let us first consider x ≤ − N 4 . Then by the mean value theorem We now consider the terms (3.2) and (3.3) separately. For (3.2), by the mean value theorem, x + a > 0; thus, we conclude that the term in (3.2) indeed is nonnegative.
For (3.3) we split into two cases. If x − a ≥ −x − N 2 , exactly the same argument as for (3.2) works. If x − a < −x − N 2 we perform the same argument after a rearrangement of the terms. Indeed, As before this implies that (3.3) is nonnegative.
We now consider x = − N 4 + 3a 4 ; according to (1.2) we must prove that Thus, this can again be expressed in terms of a difference g ′ and is hence positive. Finally, we assume that g ′ (t)dt = ( * * ).
Proposition 3.1 immediately leads to the following simple criterion for a function to belong to ∩ 0<a<N V N,a . Corollary 3.2 Let N > 0, and assume that a continuous and piecewise continuously differentiable function g : R → R with supp g = [− N 2 , N 2 ] satisfies the following conditions: (a) g is symmetric around the origin; We will now describe several functions belonging to V N,a , either for all a ∈]0, N [ or a subinterval hereof.
In [28,Lemma 1] it is proved that for N ∈ N \ {1}, In the following examples we consider continuous and compactly supported "variants" of the two-sided exponential function, the Gaussian, and other classical functions.
Then k N ∈ 0<a<N V N,a by Corollary 3.2.
In the following examples the simple sufficient conditions in Proposition 3.1 and Corollary 3.2 are not satisfied. We will use the definition directly to show that the considered functions belong to V N,a for certain ranges of the parameter a.
Example 3.7 Let N > 0, and consider We will show that It is clear that (A1) and (A2) hold, so for the considered values for a we now check (A3). In fact the argument below will prove more, namely that it is now enough to prove that . We see that . (3.7) In order to prove (a), we now assume that 3N 7 ≤ a < N . Since x 2 ≤ ( N 2 ) 2 and (x − a) 2 − x 2 ≥ 0, we have Note that the quadratic function h is symmetric around x = 2a. Since − N 4 + 3a 4 < 2a and 3N 7 ≤ a < N , we have Therefore (3.5) holds, i.e., the proof of (a) is completed.