Regularity of Dual Gabor Windows

Abstract

We present a construction of dual windows associated with Gabor frames with compactly supported windows. The size of the support of the dual windows is comparable to that of the given window. Under certain conditions, we prove that there exist dual windows with higher regularity than the canonical dual window. On the other hand, there are cases where no differentiable dual window exists, even in the overcomplete case. As a special case of our results, we show that there exists a common smooth dual window for an interesting class of Gabor frames. In particular, for any value of 𝐾∈ℕ, there is a smooth function ℎ which simultaneously is a dual window for all B-spline generated Gabor frames {𝐸_𝑚𝑏𝑇_𝑛𝐵_𝑁(𝑥/2)}_{𝑚,𝑛∈ℕ} for B-splines 𝐵_𝑁 of order 𝑁=1,…,2𝐾+1 with a fixed and sufficiently small value of 𝑏.

1. Introduction

A frame{𝑓_𝑘} in a separable Hilbert space ℋ leads to expansions of arbitrary elements 𝑓∈ℋ, in a similar fashion as the well-known orthonormal bases. More precisely, there exists at least one so-called dual frame, that is, a frame{ℎ_𝑘}such that 𝑓=𝑓,ℎ_𝑘𝑓_𝑘,∀𝑓∈ℋ. (1) Unless{𝑓_𝑘}is a basis, the dual{ℎ_𝑘}is not unique. This makes it natural to search for duals with special prescribed properties. In this paper, we will consider Gabor frames with translation parameter 𝑎=1, that is, frames for 𝐿²(ℝ) that for a certain fixed parameter 𝑏>0 and a fixed function 𝑔∈𝐿²(ℝ) have the form {𝐸_𝑚𝑏𝑇_𝑛𝑔}_{𝑚,𝑛∈ℤ}∶={𝑒^{2𝜋𝑖𝑚𝑏𝑥}𝑔(𝑥−𝑛)}_{𝑚,𝑛∈ℤ}. The function 𝑔 is called the window function. We will construct dual frames of the form {𝐸_𝑚𝑏𝑇_𝑛ℎ}_{𝑚,𝑛∈ℤ}={𝑒^{2𝜋𝑖𝑚𝑏𝑥}ℎ(𝑥−𝑛)}_{𝑚,𝑛∈ℤ} for a suitable function ℎ∈𝐿²(ℝ) to be called the dual window.

It is known that a frame {𝐸_𝑚𝑏𝑇_𝑛𝑔}_{𝑚,𝑛∈ℤ} will be overcomplete if 𝑏<1, with the redundancy increasing when 𝑏 decreases. Thus, the dual window is not unique. We will investigate whether this freedom can be used to find dual windows with higher regularity than the canonical dual and comparable size of the support. We will present cases where this is possible and other cases where it is not. As a special case of our results, we show that there are certain classes of interesting frames that have the same dual window. For example, for any value of 𝐾∈ℕ there is a smooth function ℎ which simultaneously is a dual window for all B-spline generated Gabor frames {𝐸_𝑚𝑏𝑇_𝑛𝐵_𝑁(𝑥/2)}_{𝑚,𝑛∈ℕ}for B-splines 𝐵_𝑁of order𝑁=1,…,2𝐾+1with a fixed and sufficiently small value of𝑏.

Just to give the reader an impression of the results to come, consider the B-spline 𝐵₂. The function 𝐵₂ is continuous but not differentiable at the points 0,±1. The Gabor system {𝐸_𝑚𝑏𝑇_𝑛𝐵₂}_{𝑚,𝑛∈ℤ} is a frame for all sufficiently small values of 𝑏>0. As 𝑏 tends to zero, the redundancy of the frame {𝐸_𝑚𝑏𝑇_𝑛𝐵₂}_{𝑚,𝑛∈ℤ} increases, meaning that we get more and more freedom in the choice of a dual window ℎ. However, we will show that it is impossible to find a differentiable dual window ℎ supported on supp𝐵₂=[−1,1], regardless of the considered 𝑏>0. On the other hand, by a seemingly innocent scaling we obtain the function 𝑔(𝑥)∶=𝐵₂(𝑥/2). Again, {𝐸_𝑚𝑏𝑇_𝑛𝑔}_{𝑚,𝑛∈ℤ} is a frame for all sufficiently small values of 𝑏>0. But in contrast with the situation for 𝐵₂, we will show that one can find infinitely often differentiable dual windows ℎ that are supported on supp𝑔=[−2,2], for any value of 𝑏∈]0,1/2]. These examples illustrate that the question of differentiability of the dual windows is a nontrivial issue. The examples will be derived as a consequence of more general results; see Example 9.

The results will be based on a construction of dual windows, as presented in Section 2. In Section 3, we consider a particular case where it is possible to obtain smooth dual windows, regardless of the regularity of the given window𝑔. This is much more than one can hope for in the general case. A general approach to the question of differentiability of the dual windows is given in Section 5. Since the necessary conditions are quite involved and not very intuitive, we first, in Section 4, state a version for the case of windows𝑔 that are supported in [−1,1]. For this case, we can provide concrete examples demonstrating that the desired conclusions might fail if any of the constraints is removed.

Note that a complementary approach to duality for Gabor frames that also deals with the issue of regularity is considered by Laugesen [1] and Kim [2]. For more information about the theory for Gabor analysis and its applications, see [3–5].

2. Construction of Dual Frames

In the literature, various characterizations of the pairs of dual Gabor frames are available. For general frames, Li gave a characterization in [6], which in the special case of Gabor frames also lead to a class of dual Gabor frames; later, in [7], it was shown that these duals actually characterize all duals. In order to start our analysis, we need the duality conditions for Gabor frames by Ron and Shen [8] and Janssen [9], respectively.

Lemma 1. Two Bessel sequences{𝐸_𝑚𝑏𝑇_𝑛𝑎𝑔}_{𝑚,𝑛∈ℤ}and {𝐸_𝑚𝑏𝑇_𝑛𝑎ℎ}_{𝑚,𝑛∈ℤ} form dual frames for𝐿²(ℝ)if and only if _𝑘∈ℤ𝑔𝑛𝑥+𝑏+𝑘𝑎ℎ(𝑥+𝑘𝑎)=𝑏𝛿_𝑛,0,𝑎.𝑒.𝑥∈[0,𝑎]. (2)

We will use the following to apply Lemma 1.

Lemma 2. Let𝐺be a real-valued bounded function, and assume that for some constant𝐴>0, 𝐺(𝑥−1)+𝐺(𝑥)≥𝐴,𝑥∈[0,1]. (3) Then, there exists a real-valued bounded function 𝐻 with suppH⊆supp𝐺∩[−1,1] such that 𝐺(𝑥−1)𝐻(𝑥−1)+𝐺(𝑥)𝐻(𝑥)=1,𝑥∈[0,1]. (4)

Proof. Consider𝑥∈[0,1].We will define𝐻(𝑥)and𝐻(𝑥−1)simultaneously. In case|𝐺(𝑥−1)|≥𝐴/2,put𝐻(𝑥)=0and𝐻(𝑥−1)=1/𝐺(𝑥−1).On the other hand, if |𝐺(𝑥−1)|<𝐴/2, we know that |𝐺(𝑥)|≥𝐴/2. In this case, put 𝐻(𝑥−1)=0 and 𝐻(𝑥)=1/𝐺(𝑥). Clearly, we can take 𝐻=0 outside [−1,1].

We will now present a general result about the existence of frames with a dual window of a special form. Note that, in contrast with most results from the literature, we do not need to assume that the integer translates of the window function form a partition of unity.

Associated to a function 𝑔 with support on an interval [−(2𝐾+1),2𝐾+1],𝐾∈ℕ, we will in the rest of the paper use the function 𝐺(𝑥)∶=_𝑘∈ℤ𝑔(𝑥+2𝑘),𝑥∈ℝ. (5) Due to the periodicity of𝐺, we are mainly interested in𝑥∈[−1,1]. Note that by the compact support of𝑔, 𝐺(𝑥)=^𝐾_𝑘=−𝐾𝑔(𝑥+2𝑘),𝑥∈[−1,1]. (6)

Theorem 3. Let 𝐾∈ℕ∪{0}, and let 𝑏∈]0,1/(4𝐾+2)]. Let𝑔be a real-valued bounded function withsupp𝑔⊆[−(2𝐾+1),2𝐾+1], for which _𝑛∈ℤ𝑔(𝑥+𝑛)≥𝐴,𝑥∈[0,1], (7) for a constant𝐴>0. Then, the following hold. (i)The function𝐺in (5) satisfies the conditions in Lemma 2. (ii)Take𝐻as in Lemma 2, and let ℎ(𝑥)=𝑏^𝐾_𝑘=−𝐾𝑇_2𝑘𝐻(𝑥). (8)
Then,{𝐸_𝑚𝑏𝑇_𝑛𝑔}_{𝑚,𝑛∈ℤ}and{𝐸_𝑚𝑏𝑇_𝑛ℎ}_{𝑚,𝑛∈ℤ}form dual frames for𝐿²(ℝ),andℎ is supported in[−(2𝐾+1),2𝐾+1].

Proof. To prove (i), we note that by (7) and the definition of𝐺, for𝑥∈[0,1], 𝐺(𝑥−1)+𝐺(𝑥)=_𝑘∈ℤ𝑔(𝑥−1+2𝑘)+_𝑘∈ℤ𝑔(≥𝑥+2𝑘)_𝑘∈ℤ𝑔(𝑥+𝑘)≥𝐴. (9) Thus,𝐺satisfies the condition (3) in Lemma 2. Also, it is clear that𝐺is bounded, real-valued, and 2-periodic. Therefore, we can choose a function𝐻which is supported in [−1,1]and such that 𝐺(𝑥−1)𝐻(𝑥−1)+𝐺(𝑥)𝐻(𝑥)=1,𝑥∈[0,1]. (10) Defineℎas in (8). In order to prove (ii), we will apply Lemma 1. By assumption, the function 𝑔 has compact support and is bounded; by the construction, the function ℎ shares these properties. It follows that{𝐸_𝑚𝑏𝑇_𝑛𝑔}_{𝑚,𝑛∈ℤ}and{𝐸_𝑚𝑏𝑇_𝑛ℎ}_{𝑚,𝑛∈ℤ} are Bessel sequences. In order to verify that these sequences form dual frames, we need to check that for𝑥∈[0,1], _𝑘∈ℤ𝑔𝑛𝑥+𝑏+𝑘ℎ(𝑥+𝑘)=𝑏𝛿_𝑛,0,a.e.𝑥∈[0,1]. (11) By assumption and construction,𝑔andℎhave support in[−(2𝐾+1),2𝐾+1]; thus, (11) is satisfied for𝑛≠0whenever1/𝑏≥4𝐾+2, that is, if𝑏∈]0,1/(4𝐾+2)]. For𝑛=0, and using the compact support of𝑔, we need to check that ^2𝐾_{𝑘=−2𝐾−1}𝑔(𝑥+𝑘)ℎ(𝑥+𝑘)=𝑏,𝑥∈[0,1]. (12) For each𝑘∈{−𝐾,−𝐾+1,…,𝐾}, if𝑥∈[0,1], then 𝑥−1+2𝑘∈[2𝑘−1,2𝑘],𝑥+2𝑘∈[2𝑘,2𝑘+1]; (13) Thus, we have ℎ(𝑥−1+2𝑘)=𝑏𝐻(𝑥−1),ℎ(𝑥+2𝑘)=𝑏𝐻(𝑥). (14) This together with (6) and (10) implies, for𝑥∈[0,1], that ^2𝐾_{𝑘=−2𝐾−1}𝑔()ℎ()=𝑥+𝑘𝑥+𝑘_{𝐾𝑘=−𝐾}𝑔()ℎ()+𝑥−1+2𝑘𝑥−1+2𝑘_{𝐾𝑘=−𝐾}𝑔(𝑥+2𝑘)ℎ(𝑥+2𝑘)=𝑏^𝐾_𝑘=−𝐾𝑔(𝑥−1+2𝑘)𝐻(𝑥−1)+𝑏^𝐾_𝑘=−𝐾𝑔()𝐻(𝑥)𝐺()𝐻()+𝐺(𝑥)𝐻(𝑥𝑥+2𝑘=𝑏𝑥−1𝑥−1)=𝑏. (15) Hence, (12) holds. Therefore,{𝐸_𝑚𝑏𝑇_𝑛𝑔}_{𝑚,𝑛∈ℤ}and {𝐸_𝑚𝑏𝑇_𝑛ℎ}_{𝑚,𝑛∈ℤ}form dual frames for𝐿²(ℝ).

3. Smooth Dual Windows for a Class of Gabor Frames

Before we start the general analysis of the dual windows in Theorem 3, we will consider a particular case, where we can construct smooth compactly supported dual windows, regardless of the regularity of the window itself.

Theorem 4. Let 𝐾∈ℕ∪{0}, and let 𝑏∈]0,1/(4𝐾+2)]. Let 𝑔 be a real-valued bounded function with supp𝑔⊆[−(2𝐾+1),2𝐾+1], for which _𝑛∈ℤ𝑔(𝑥+2𝑛)=1,𝑥∈[−1,1]. (16) Let𝑓∶ℝ→ℝbe any bounded function for which 𝑓(𝑥)=0,𝑥≤0,𝑓(𝑥)=1,𝑥≥1. (17) Define the function𝐻by 𝐻(𝑥)=12𝑓(2(1𝑥+1)),−1≤𝑥<−2,11−2𝑓()1−2𝑥,−21≤𝑥<0,1−2𝑓()12𝑥,0≤𝑥<2,12𝑓(2(,11−𝑥))2≤𝑥<1,0,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒, (18) and let ℎ(𝑥)∶=𝑏^𝐾_𝑘=−𝐾𝑇_2𝑘𝐻(𝑥). (19) Then, the following holds. (i)ℎis a symmetric function withsuppℎ⊆[−(2𝐾+1),2𝐾+1].(ii){𝐸_𝑚𝑏𝑇_𝑛𝑔}_{𝑚,𝑛∈ℤ}and{𝐸_𝑚𝑏𝑇_𝑛ℎ}_{𝑚,𝑛∈ℤ}are dual frames for𝐿²(ℝ).(iii)If𝑓is chosen to be smooth, then the function𝐻in (18) is smooth, and consequently the dual windowℎin (19) is smooth as well.

Proof. By the assumption (16), we have 𝐺(𝑥)=1,𝑥∈[−1,1]. (20) Take𝐻as in (18). For𝑥∈]0,1/2[, 𝐻()+𝐻(𝑥)=𝑓()𝑥−12𝑥2+𝑓()1−2𝑥2=1; (21) for 𝑥∈]1/2,1[, 𝐻()+𝐻(𝑥)=𝑓((𝑥−11−−2𝑥−1))2+𝑓(2(1−𝑥))2=1. (22) That is,𝐻(𝑥−1)+𝐻(𝑥)=1,𝑥∈[0,1].Combining this with (20) yields 𝐺()𝐻()+𝐺(𝑥)𝐻(𝑥)=𝐻()+𝐻(𝑥)[].𝑥−1𝑥−1𝑥−1=1,𝑥∈0,1 (23) By Theorem 3, ℎ is a dual window of𝑔.By construction,ℎis a symmetric function with suppℎ⊆[−(2𝐾+1),2𝐾+1]. The result in (iii) follows by direct investigation of the derivatives at−1,−1/2,0,1/2,1.

An example of a smooth function𝑓satisfying condition (17) is (see [10, page 36]) 𝑓(𝑥)=−𝑥expexp(1−𝑥)−1⁻¹,0<𝑥<1,0,𝑥≤0,1,𝑥≥1. (24)

As noted in the introduction, the possibility of constructing a smooth dual window is a significant improvement compared to the use of the canonical dual window, which might not even be continuous. We return to this point in Example 9.

Another interesting feature of the construction in Theorem 4 is that the dual windowℎin (19) is independent of the window𝑔that generates the frame{𝐸_𝑚𝑏𝑇_𝑛𝑔}_{𝑚,𝑛∈ℤ}! In other words, we can construct a windowℎthat generates a dual frame for all the frames {𝐸_𝑚𝑏𝑇_𝑛𝑔}_{𝑚,𝑛∈ℤ} satisfying the conditions in Theorem 4 for fixed values of𝐾and𝑏.

Corollary 5. Let 𝐾∈ℕ, and let 𝑏∈]0,1/(4𝐾+2)]. Consider a bounded real-valued function 𝜙 that is supported on [−𝐾,𝐾] and satisfies the partition of unity condition, _𝑛∈ℤ𝜙(𝑥−𝑛)=1,𝑥∈ℝ. (25) Then, the function𝑔(𝑥)∶=𝜙(𝑥/2)generates a Gabor frame{𝐸_𝑚𝑏𝑇_𝑛𝑔}_{𝑚,𝑛∈ℤ}.Choosing𝐻 as in (18), the functionℎ in (19) is a dual window of𝑔.

Proof. By the choice of the function𝜙,supp𝑔⊆[−2𝐾,2𝐾]⊆[−2𝐾−1,2𝐾+1], and _𝑛∈ℤ𝑔(𝑥−2𝑛)=_𝑛∈ℤ𝜙𝑥2−𝑛=1. (26) Choose𝐻as in (18). By Theorem 4, the functionℎdefined byℎ(𝑥)=𝑏∑^𝐾_𝑘=−𝐾𝑇_2𝑘𝐻(𝑥)is a dual window of𝑔.

It is known that the partition of unity condition (25) is satisfied for a large class of functions, for example, any scaling function for a multiresolution analysis. As a concrete example, recall that the centered B-splines 𝐵_𝑁, 𝑁∈ℕ are given inductively by𝐵₁=𝜒_[−1/2,1/2],𝐵_𝑁+1=𝐵_𝑁∗𝐵₁.Any B-spline satisfies the partition of unity condition, and the B-spline𝐵_𝑁 has support on the interval[−𝑁/2,𝑁/2]. Thus, for each fixed value of𝐾∈ℕ, the function ℎ in (19) is a dual window for each of the B-splines𝐵_𝑁,𝑁=1,…,2𝐾 and a fixed choice of 𝑏≤1/(4𝐾+2).

4. Regularity of the Dual Windows if supp𝑔⊆[−1,1]

Based on Theorem 3, we now aim at a general analysis of the relationship between the regularity of a window𝑔and the associated dual windowsℎwith comparable support size. We will exhibit cases where the smoothness can be increased and other cases where these dual windows cannot have higher smoothness than the window itself. The general version of our result, to be stated in Theorem 10, is quite complicated, and the role of the conditions is not intuitively clear. Therefore, we will first present, in Theorem 6, the corresponding version for windows 𝑔 that are supported in [−1,1]. An advantage of this approach is that for each of the requirements in Theorem 6 we can provide an example showing that the desired conclusion might fail if the condition is removed.

Given a function𝑔that is supported in [−1,1], let [](𝑥),[].𝑍∶=𝑥∈−1,1∶𝑔=0𝐸∶=𝑥∈−1,1∶𝑔isnotdifferentiableat𝑥 (27)

Theorem 6. Let𝑔be a real-valued bounded function with supp𝑔⊆[−1,1]. Assume that for some constant𝐴>0, 𝑔(𝑥−1)²+𝑔(𝑥)²≥𝐴,𝑥∈[0,1]. (28) Then, the following assertions hold. (1) If 𝑔 is not differentiable at 0, then there does not exist a differentiable dual window ℎ with suppℎ⊆[−1,1] for any 𝑏>0; (2) Assume that 𝑔 is differentiable at 0. Assume further that the set 𝑍 is a finite union of intervals and points not containing 0, that the set 𝐸 is finite, and that  (a)𝐸∩(𝐸+1)=∅;  (b)𝑍∩(𝐸±1)=∅.  Then, for any 𝑏∈]0,1/2], there exists a differentiable dual window ℎ withsuppℎ⊆supp𝑔.

Note that (28) is a necessary condition for {𝐸_𝑚𝑏𝑇_𝑛𝑔}_{𝑚,𝑛∈ℤ} being a frame; thus, it is not a restriction in this context. To support the conditions in Theorem 6, we will now provide a series of examples, where just one of these conditions breaks down and the conclusion in Theorem 6 fails. We first give an example where condition (a) is not satisfied.

Example 7. Consider the function 𝑔(𝑥)1∶=2𝑥+2,𝑥∈−1,−2;−11,𝑥∈2,12;1−2𝑥+2,𝑥∈2;[].,10,𝑥∉−1,1 (29) Then, 𝐾=0,𝑍={±1}, and 𝐸={±1/2,±1}, so 𝐸∩(𝐸+1)={1/2},𝑍∩(𝐸±1)=∅. Hence,𝑔satisfies condition (b) but not (a) in Theorem 6. Now we will show that𝑔does not have a differentiable dual of any form. Suppose that there exists such a dualℎ. By the duality condition, we obtain 𝑔(𝑥−1)ℎ(𝑥−1)+𝑔(𝑥)ℎ(𝑥)=𝑏,𝑥∈[0,1]. (30) Letting𝐷₋𝑔 and𝐷₊𝑔 denote the left, respectively, and right derivatives of𝑔, we note that 𝑔−121=𝑔2=1,𝐷₋𝑔−12𝐷=2,₊𝑔−12=0,𝐷₋𝑔12=0,𝐷₊𝑔12=−2. (31) Taking the left and right derivatives of (30) at 𝑥=1/2, this implies that −1𝐷ℎ21+𝐷ℎ2−1=−2ℎ2,−1𝐷ℎ21+𝐷ℎ21=2ℎ2. (32) But by conditions (30) and (31), we haveℎ(−1/2)+ℎ(1/2)=𝑏. This is a contradiction. Thus, a differentiable dual window does not exist.

Note that the conclusion in Example 7 is even stronger than what we asked for: no dual window at all can be differentiable, regardless of its form and support size. In the next example condition, (b) in Theorem 6 is not satisfied, and the conclusion breaks down.

Example 8. Consider 𝑔(𝑥)5∶=23𝑥+2−3,𝑥∈51,−5;−11,𝑥∈5,15;−523𝑥+21,𝑥∈5,35;−30,𝑥∉5,35. (33) Easy considerations as in Example 7 show that 𝑔 satisfies condition (a) but not (b) in Theorem 6 and that 𝑔 does not have a differentiable dual of any form. We leave the details to the interested reader.

Let us now provide the details for the example mentioned in Section 1.

Example 9. Consider the B-spline𝐵₂, which is continuous but not differentiable at the points 0,±1. It is an easy consequence of the results in the literature (see, e.g., [11]) that the Gabor system {𝐸_𝑚𝑏𝑇_𝑛𝐵₂}_{𝑚,𝑛∈ℤ} is a frame for all sufficiently small values of 𝑏>0. As 𝑏 tends to zero, the redundancy of the frame {𝐸_𝑚𝑏𝑇_𝑛𝐵₂}_{𝑚,𝑛∈ℤ} increases, meaning that we get more and more freedom in the choice of ℎ. However, since 𝐵₂ is nondifferentiable at 𝑥=0, Theorem 6 implies that none of the dual windows supported on [−1,1] are differentiable, for any 𝑏∈]0,1/2].
On the other hand, consider the scaled B-spline 𝑔(𝑥)∶=𝐵₂(𝑥/2). Again, {𝐸_𝑚𝑏𝑇_𝑛𝑔}_{𝑚,𝑛∈ℤ} is a frame for all sufficiently small values of 𝑏>0. The requirements in Theorem 4 are satisfied for 𝑏∈]0,1/2], implying that infinitely often differentiable dual windows exists.

5. Regularity of the Dual Windows in the General Case

We will now present the general version of Theorem 6. The main difference between the results is that the conditions in the general version are stated in terms of the function 𝐺 in (5) rather than 𝑔 itself. The proof is in the appendix. Given a compactly supported function 𝑔∶ℝ→ℂ, define 𝐺 as in (5), and let []∶𝐺(𝑥),[]∶𝑍∶=𝑥∈−1,1=0𝐸∶=𝑥∈−1,1𝐺𝜒_[−1,1]isnotdifferentiableat𝑥. (34)

Theorem 10. Let 𝐾∈ℕ∪{0}, and let 𝑏∈]0,1/(4𝐾+2)]. Let 𝑔 be a real-valued bounded function with supp𝑔⊆[−(2𝐾+1),2𝐾+1]. Define 𝐺 as in (5). Then, the following assertions hold.
(1) If 𝐺 is not differentiable at 0, then there does not exist a differentiable dual windowℎdefined as in (8).
Assume that, for some constant𝐴>0, 𝐺(𝑥−1)+𝐺(𝑥)≥𝐴,𝑥∈[0,1] (35) and that the set𝑍is a finite union of intervals and points not containing0.
(2) Assume that 𝐺 is differentiable at 0, that the set of points 𝐸 is finite, and that (a)𝐸∩(𝐸−1)=∅; (b)𝐺(𝑥)≠0,𝑥∈(𝐸−1)∪(𝐸+1).
Then,{𝐸_𝑚𝑏𝑇_𝑛𝑔}_{𝑚,𝑛∈ℤ} is a frame for 𝐿²(ℝ), and there exists a differentiable dual windowℎ of the form (8).

Note that the conditions in Theorem 10 (2) are void if 𝐺 is differentiable, that is, the standing assumptions alone imply the existence of a differentiable dual window.

For nonnegative functions, the conditions in Theorem 10 can be formulated in an easier way, where we again refer directly to properties of the function 𝑔 rather than 𝐺.

Corollary 11. Let 𝐾∈ℕ∪{0}, and let 𝑏∈]0,1/(4𝐾+2)]. Let 𝑔 be a nonnegative bounded function withsupp𝑔⊆[−(2𝐾+1),2𝐾+1]. Assume that for some constant𝐴>0, ^2𝐾_{𝑘=−2𝐾−1}𝑔(𝑥+𝑘)≥𝐴,𝑥∈[0,1]. (36)
Assume that the set 𝑍 of zeros of 𝑔 on [−(2𝐾+1),2𝐾+1] is a finite union of intervals and points, that 𝑔 is differentiable except on a finite set 𝐸 of points, and that the sets 𝐸 and 𝑍 satisfy the following conditions: (a)0∉(⋂^𝐾_𝑘=−𝐾(𝑍−2𝑘))∪(⋃^𝐾_𝑘=−𝐾(𝐸−2𝑘)); (b)(⋃^𝐾_𝑘=−𝐾(𝐸−2𝑘))∩(⋃^𝐾_𝑘=−𝐾(𝐸−2𝑘+1))=∅; (c)(⋂^𝐾_𝑘=−𝐾(𝑍−2𝑘))∩(⋃^𝐾_𝑘=−𝐾(𝐸−2𝑘±1))=∅.
Then, {𝐸_𝑚𝑏𝑇_𝑛𝑔}_{𝑚,𝑛∈ℤ}, is a frame for 𝐿²(ℝ), and there exists a differentiable dual window ℎ supported on [−(2𝐾+1),2𝐾+1].

Proof. We check condition (2) in Theorem 10. Let 𝐺(𝑥)∶=_𝑘∈ℤ𝑔(𝑥+2𝑘). (37) By an argument similar to the one at the beginning of the proof of Theorem 3, (36) together with (6) implies that 𝐺(𝑥−1)+𝐺(𝑥)≥𝐴,𝑥∈[0,1]. (38) Since𝑔is nonnegative, the zeros of𝐺restricted to[−1,1]are 𝑍∶=^𝐾_𝑘=−𝐾(𝑍−2𝑘)∩[−1,1]. (39) A direct calculation shows that if𝑔is differentiable except on the set𝐸, then𝐺is differentiable outside the set⋃^𝐾_𝑘=−𝐾(𝐸−2𝑘). Let 𝐸∶=^𝐾_𝑘=−𝐾(𝐸−2𝑘)∩[−1,1]. (40) Then, the sets𝐸and𝑍satisfy the conditions (a) and (b) of (2) in Theorem 10. Hence, there exists a differentiable dual windowℎ.

Example 12. Let 𝑔(𝑥)5∶=2𝑥+2𝜒_{[−4/5,−2/5]}(𝑥)+𝜒_[−2/5,2/5](𝑥)+−52𝑥+2𝜒_[2/5,4/5](𝑥). (41) A direct calculation shows that ±2𝐸=54,±51,𝐸+1=5,35,75,95,−9𝐸−1=57,−53,−51,−5,4𝑍=−1,−5∪45,1. (42) Thus, the conditions (a)–(c) in Corollary 11 are satisfied. Hence, for any 𝑏∈]0,1/2], there exists a differentiable dual windowℎ.

Appendix

Proof of Theorem 10

The full proof of Theorem 10 is notationally complicated. We will therefore formulate the proof for a function 𝑔 for which (i)𝐾=0in (6), that is,𝐺(𝑥)=𝑔(𝑥), 𝑥∈[−1,1]; (ii)𝐺 is differentiable except at one point, that is, 𝐸={𝑐}; (iii)the zeroset of 𝐺 within ]−1,1[ consists of just one interval[𝑎,𝑏] (containing the degenerate case of just one point 𝑎=𝑏 as a special case); that is, 𝑍={−1}∪[𝑎,𝑏]∪{1}.

We leave the obvious modifications to the general case to the reader.

We use the following abbreviation: 𝜓(𝑎⁻)∶=lim_{𝑥→𝑎−}𝜓(𝑥),𝜓𝑎⁺∶=lim_𝑥→𝑎+𝜓(𝑥). (A.1)

Proof of Theorem 10. (1) We will prove the contrapositive result, so suppose that a dual windowℎ defined as in (8) with𝐾=0, that is, ℎ(𝑥)=𝑏𝐻(𝑥) (A.2) with supp𝐻⊆[−1,1], is differentiable on ℝ. Then,𝐻 is also differentiable onℝ. Then, we have 𝐻(−1)=𝐻(1)=𝐷𝐻(−1)=𝐷𝐻(1)=0. (A.3) The duality condition can for 𝑛=0 be written as 𝑔(𝑥−1)ℎ(𝑥−1)+𝑔(𝑥)ℎ(𝑥)=𝑏,𝑥∈[0,1], (A.4) or, in terms of𝐺and𝐻, 𝐺(𝑥−1)𝐻(𝑥−1)+𝐺(𝑥)𝐻(𝑥)=1,𝑥∈[0,1]. (A.5) It follows from (A.3) that 𝐺(0)𝐻(0)=1. (A.6) Putting this into (A.5), we have 𝐺()𝐻()+𝐺(𝑥𝐻(𝑥)−𝐻(0+𝐻(0)𝐺(𝑥)−𝐺(0)][,𝑥−1𝑥−1))=0,𝑥∈0,1 (A.7) or, by dividing with 𝑥, 𝐺()𝐻()𝑥−1𝑥−1𝑥+𝐺(𝑥)𝐻(𝑥)−𝐻(0)𝑥+𝐻(0𝐺(𝑥)−𝐺(0))𝑥=0,𝑥∈]0,1[. (A.8) This implies that 𝐺(𝑥)−𝐺(0)𝑥𝐺()𝐻()+𝐺(𝑥)𝐻(𝑥)−𝐻(0)=−𝑥−1𝑥−1/𝑥/𝑥𝐻(0),𝑥∈]0,1[. (A.9) Using (A.3) it now follows that 𝐷₊𝐺(0)𝐺0=−⁺𝐷₊𝐻(0)+𝐺(−1)⁺𝐷₊𝐻(−1)𝐻(0)𝐺0=−⁺𝐷𝐻(0)𝐻(0). (A.10) Similarly, 𝐷₋𝐺(0)𝐺(0=−⁻)𝐷𝐻(0)𝐻(0). (A.11) The conditions (A.3) and (A.4) and the continuity of 𝐻 show that 𝐺 is continuous at 𝑥=0. Hence, 𝐷₊𝐺(0)=𝐷₋𝐺(0), which implies that 𝐺 is differentiable at 0.
(2) We construct a real-valued differentiable function𝐻withsupp𝐻⊆[−1,1]so that 𝐺(𝑥−1)𝐻(𝑥−1)+𝐺(𝑥)𝐻(𝑥)=1,𝑥∈[0,1]. (A.12)
Let 𝑍∶={−1}∪[𝑎,𝑏]∪{1},𝐸∶={𝑐}. (A.13) There are various scenarios concerning the location of the points 𝑎,𝑏,and 𝑐. They are treated in a similar way, and we will assume that −1<𝑎≤𝑏<0<𝑐<1. By the condition (2) in Theorem 10, one of the following cases occurs: (1)𝑏<𝑐−1<0; (2)−1<𝑐−1<𝑎. The cases are similar, so we only consider the case (1), that is, −1<𝑎≤𝑏<𝑐−1<0<𝑐<1. (A.14) Let {}∪[]∪{𝑐}∪{1},[]∪∩[]=}∪[∪}∪{0.𝑊∶=𝑍∪𝐸=−1𝑎,𝑏𝑉∶=𝑊∩−1,0𝑊−1−1,0({−1𝑎,𝑏])({𝑐−1}) (A.15)
We define𝐻(𝑥)on [−1,1] as follows: first, we defineℎ(𝑥)on𝑉∪(𝑉+1) by 𝐻(𝑥)[]={}∪[];1∶=0,𝑥∈𝑊∩−1,0−1𝑎,𝑏𝐺(𝑥)∩[]={}∪{0};1,𝑥∈𝑊−1−1,0𝑐−1𝐺(𝑥)∩[]={0}∪[];[]={𝑐}∪{1},,𝑥∈𝑊+10,1𝑎+1,𝑏+10,𝑥∈𝑊∩0,1 (A.16) which is well defined by (A.14). Then, 𝐺(𝑥−1)𝐻(𝑥−1)+𝐺(𝑥)𝐻(𝑥)=1,𝑥∈𝑉+1. (A.17)
We now extend 𝐻(𝑥)on [−1,1]⧵(𝑉∪(𝑉+1))=([−1,0]⧵𝑉)∪([0,1]⧵(𝑉+1))∶ choose 𝐻(𝑥)on[−1,0]⧵𝑉so that𝐻 is differentiable on[−1,0]⧵𝑉 and 𝐷₊𝐻(−1)=𝐷₋𝐻(𝑎)=𝐷₊𝐻(𝑏)𝐷𝐻()1=0,𝑐−1=𝐷𝐺(),𝐷𝑐−1₋𝐻(0)1=𝐷𝐺(0). (A.18)(A.19) Then, (A.16) and (A.18)-(A.19) imply that𝐻is differentiable on]−1,0[and that 𝐷₊𝐻(−1)=0,𝐷₋𝐻(0)1=𝐷𝐺(0). (A.20) We then define𝐻(𝑥)on[0,1]⧵(𝑉+1)by 𝐻(𝑥)=1−𝐺(𝑥−1)𝐻(𝑥−1)𝐺(𝑥), (A.21) which is well defined since 𝑊∩[0,1]={𝑐,1}⊆𝑉+1. (A.22) This implies that 𝐺()𝐻()+𝐺(𝑥)𝐻(𝑥)[]⧵.𝑥−1𝑥−1=1,𝑥∈0,1𝑉+1 (A.23) Recall that 𝐺(𝑥−1),𝐺(𝑥), and 𝐻(𝑥−1) are differentiable on [0,1]⧵(𝑉+1). By (A.21), the same is the case for 𝐻(𝑥). Since supp𝐻⊆[−1,1], it remains to show that 𝐻 is differentiable on 𝑉+1={0}∪[𝑎+1,𝑏+1]∪{𝑐}∪{1} and that 𝐷₋𝐻(1)=0. Note that 𝐺 is differentiable on {0}∪[𝑎+1,𝑏+1]. Thus, by (A.16) and (A.20), it suffices to show that 𝐷₊𝐻(0)1=𝐷𝐺(0),𝐷₋𝐻()1𝑎+1=𝐷𝐺),𝐷(𝑎+1₊𝐻()1𝑏+1=𝐷𝐺(),𝐷𝑏+1₋𝐻(𝑐)=𝐷₊𝐻(𝑐),𝐷₋𝐻(1)=0. (A.24)(A.25)(A.26) We first show (A.24). From (A.16), we get 𝐻(−1)=0,𝐺(0)𝐻(0)=1. (A.27) Putting this into (A.23), we have, for 𝑥∈[0,1]⧵(𝑉+1), 𝐺(𝐻()−𝐻(+𝐺(𝑥)𝐻(𝑥)−𝐻(0)+𝐻(0)𝐺(𝑥)−𝐺(0)𝑥−1)𝑥−1−1)=0, (A.28) or by dividing with𝑥, 𝐺()𝐻()−𝐻()𝑥−1𝑥−1−1𝑥+𝐺(𝑥)𝐻(𝑥)−𝐻(0)𝑥+𝐻(0𝐺(𝑥)−𝐺(0))𝑥=0. (A.29) This implies that 𝐻(𝑥)−𝐻(0)𝑥𝐺()𝐻()−𝐻()+𝐻(0𝐺(𝑥)−𝐺(0×𝐺(𝑥)=−𝑥−1𝑥−1−1/𝑥))/𝑥₋₁,𝑥∈[0,1]⧵𝑉+1. (A.30) Using (A.18), it follows that 𝐷₊𝐻(0)𝐺()=−−1⁺𝐷₊𝐻(−1)+𝐻(0)𝐷₊𝐺(0)𝐺(0⁺)𝐻(0)𝐷=−⁺𝐺(0)𝐺(0⁺). (A.31) Since 𝐺(𝑥) is differentiable at 0∉𝑊, we have 𝐺(0⁺)=𝐺(0),𝐷₊𝐺(0)=𝐷𝐺(0). This together with (A.27) implies that 𝐷₊𝐻(0)𝐻(0)𝐷𝐺(0)=−𝐺(0)𝐷𝐺(0)=−𝐺²(0)1=𝐷𝐺(0). (A.32) This proves that (A.24) holds. The results in (A.25) and (A.26) can be shown in a similar way.

Acknowledgments

The authors would like to thank the reviewers for their useful comments and suggestions. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2010-0007614).