Counterexamples to the B-spline conjecture for Gabor frames

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1 Counterexamples to the B-spline conjecture for Gaor frames Jako Lemvig, Kamilla Haahr Nielsen arxiv: v3 [math.fa] 19 Aug 015 April 6, 018 Astract: The frame set conjecture for B-splines B n, n, states that the frame set is the maximal set that avoids the known ostructions. We show that any hyperola of the form a = r, where r is a rational numer smaller than one and a and denote the sampling and modulation rates, respectively, has infinitely many pieces, located around =,3,..., not elonging 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 elonging to the frame set for B-splines B n, n. 1 Introduction One of the fundamental prolems in Gaor analysis is to determine for which sampling and modulation rates, the corresponding time-frequency shifts of a given generator constitute a frame. The so-called frame set of a generator g L (R) is the parameter values (a,) R + for which the associated Gaor system G(g,a,) := { e πim g( ak) } k,m Z is a frame for L (R). We denote the frame set y F(g) and refer to [,7] for an introduction to frames and Gaor analysis. In this note we are interested in the frame set of Gaor systems generated y B-splines B n. The B-splines are given inductively as B 1 = χ [ 1/,1/], and B n+1 = B n B 1, for n N. Dai and Sun [5] recently gave a complete characterization of the frame set for the first B- spline B 1. For the higher order B-splines, the picture is a lot less complete. Since, for n, B n elongs to the Feichtinger algera M 1 (R), the set F(B n ) is open in R andf(b n ) { (a,) R + : a < 1 }, see e.g., [8]. Christensen put focus on the prolem of characterizing F(B n ) in [3], while Gröchenig, in [8], went one step further conjecturing that the frame 010 Mathematics Suject Classification. Primary 4C15. Secondary: 4A60 Key words and phrases. B-spline, frame, frame set, Gaor system, Ziulski-Zeevi matrix Technical University of Denmark, Department of Applied Mathematics and Computer Science, Matematiktorvet 303B, 800 Kgs. Lyngy, Denmark, jakle@dtu.dk Technical University of Denmark, Department of Applied Mathematics and Computer Science, Matematiktorvet 303B, 800 Kgs. Lyngy, Denmark, kamillahn@villagok.dk 1 of 10

2 set for B-splines of order n is F(B n ) = { (a,) R + : a < 1, a < n,,3,... }. On the other hand, the set F(B 1 ) has a very complicated structure. This phenomenon is partly explained y the fact that B 1 is the only B-spline that does not elong to M 1 (R). Indeed, F(B 1 ) is not open and (a,) = (1,1) F(B 1 ). The main messages of this note are that the B-spline conjecture is false and that the frame set for B-splines of all orders must have a very complicated structure, sharing several similarities with F(B 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,) R + to the known parts of F(B n ); these new values are illustrated in Figure 1 for the case of B. Along the same lines, we verify the frame set conjecture for B-spline for a new n n n 3n a Figure 1: A sketch of the frame set for B n, n =, for 0 < a < 1. Red, pink and purple indicate (a,)-values, where G(B,a,) 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 lue region is the result in Theorem 3. The pink hyperola pieces a = p q are the counterexamples from Theorem 7 (only illustrated for q 6). The purple hyperolic curve through ( 1 3, 5 ) is the counterexamples in Theorem 8. region, marked with lue in Figure 1 for the case B. The idea of the proof is a painless construction of an alternate dual frame; the details are presented in Section. date/time:6-apr-018/5:43 of 10

3 In Section 3 we give the proof that the frame set conjecture is false. Indeed, using a Ziulski-Zeevi representation and properties of the Zak transform of B-splines, we show in Theorem 7 that any hyperola of the form a = p < 1 with relative prime p and q q has infinitely many pieces not in the frame set F(B n ). The pieces are hyperolic curves located around =,3,..., with the range of each of these hyperolic curve determined y 1, where R(x) denotes the round function to the nearest integer of x R; nq a few of the hyperolic curves are marked with pink in Figure 1 for the case B. Note that these pink curves color a region that leaves a white region (etween = m and = m + 1 for m =,3,...) looking like a ow tie, similar to Janssen s tie for B 1 [10], ut thicker at the knot. We end Section 3 y presenting a counterexample for B that is not covered y Theorem 7; in that susection we also formulate a new conjecture on the frame set of B. Painless expansions for alternate duals For n > 0, define the parameter set: { Σ n = (a,) R + : 0 < a < n,a < 1,n+a }. For n N,n, we prove that G(B n,a,) is a frame for (a,) Σ n y constructing a function h L (R) such that G(B n,a,) and G(h,a,) are dual frames. For this, we need the following well-known characterization result of dual Gaor frames; we refer the reader to [] for a proof. Theorem 1. Let a, > 0, and let g,h L (R). Suppose G(g,a,) and G(h,a,) are Bessel sequences. Then they are dual frames if and only if, for all m Z, [ g(x m/ ka)h(x ka) = δ m,0 for a.e. x a, a ]. (.1) k Z The following result provides a painless expansion of L -functions for a class of compactly supported generators using alternate duals. The announced frame property of B-splines for (a,) in the region Σ n will follow as a special case. Theorem. Let n > 0, (a,) Σ n, and g L (R). Suppose that suppg [ n, n ] and c := inf x [ a,a ] g(x) > 0. Define h L (R) y h(x) = { g(x) x [ a, a ], 0 otherwise. Then G(g,a,) and G(h,a,) are dual frames. In particular, G(g,a,) is a frame with lower ound A g satisfying A g c. date/time:6-apr-018/5:43 3 of 10

4 Proof. It follows y, e.g., [7, Proposition 6..], that G(g,a,) and G(h,a,) are Bessel sequences. To show that these systems are dual frames, we will show that the equations (.1) hold for a.e. x [ a, ] [ a. For a.e. x a, ] [ a, since supph = a, a ], we only have one nonzero term in the sum in (.1). Thus the equations we have to verify are g ( x m ) [ h(x) = δm,0, a.e. x a, a ] for m Z. For m = 0, we have y definition of h that g(x)h(x) = g(x) [ g(x) =, for a.e. x a, a ]. For m Z\{0}, it follows from our assumption m + a 1 that the support sets ( suppg m ) [ = m + m, m + m ] [ and supph = a, a ] are disjoint. Hence, for m 0, the equations (.1) are trivially satisfied. The canonical dual of G(h,a,) has optimal lower frame ound c /. Since G(g,a,) is an alternate dual of G(h,a,), the ound A g c / follows. Remark 1. In the classical painless non-orthogonal expansion for compactly supported generators [6], the Walnut representation of the frame operator S g,g only has one nonzero term due to the assumptions on the parameter values (a,) and on the support of g. Indeed, in this case, S g,g ecomes a multiplication operator. In Theorem 1 it is the Walnut representation of the mixed frame operator S g,h that, due the support of g and h, is guaranteed to have only one nonzero term. Corollary 3. Let n N,n, and let (a,) Σ n. Then G(B n,a,) is a frame for L (R). 3 The counterexamples 3.1 Preliminaries We first need to set up notation and recall the Ziulski-Zeevi representation. The Zak transform of a function f L (R) is defined as (Z λ f)(x,ν) = λ k Zf(λ(x k))e πikν, a.e. x,ν R, (3.1) with convergence in L loc (R). The Zak transform Z λ is a unitary map of L (R) onto L ([0,1) ), and it has the following quasi-periodicity: Z λ f(x+1,ν) = e πiν Z λ f(x,ν), Z λ f(x,ν +1) = Z λ f(x,ν) for a.e. x,ν R. We consider rationally oversampled Gaor systems, i.e., G(g,a,) with a Q, a = p q gcd(p,q) = 1. date/time:6-apr-018/5:43 4 of 10

5 For g L (R), we define column vectors φ g l (x,ν) Cp for l {0,1,...,q 1} y ( φ g l (x,ν) = p 1 (Z1 g)(x l p q,ν + k ) p 1 p ) k=0 and column vectors ψ g l (x,ν) Cp for l {0,1,...,q 1} y a.e. x,ν R, ) p 1 ψ g l ( (x,ν) = 1 g(x+aqn+al+k/)e πiaqnν n Z k=0 a.e. x,ν R. The p q matrix defined y Φ g (x,ν) = [φ g l (x,ν)]q 1 l=0 is the so-called Ziulski-Zeevi matrix, while the p q matrix Ψ g (x,ν) = [ψ g l (x,ν)]q 1 l=0 is closely related to the Ziulski-Zeevi matrix and appears in [9,1]. Theorem 4 elow says that the Gaor system G(g,a,) is a frame with (optimal) ounds A and B for L (R) if and only if A is the infimum over a.e. (x,ν) R of the smallest singular value of Φ g (x,ν) (or Ψ g (x,ν)) and B is the supremum over a.e. (x,ν) R of the largest singular value of Φ g (x,ν) (or Ψ g (x,ν)). Theorem 4. Let A,B 0, and let g L (R). Suppose G(g,a,) is a rationally oversampled Gaor system. Then the following assertions are equivalent: (i) G(g,a,) is a Gaor frame for L (R) with ounds A and B, (ii) {φ g l (x,ν)}q l=0 is a frame for Cp with uniform ounds A and B for a.e. (x,ν) [0,1). (iii) {ψ g l (x,ν)}q l=0 is a frame for Cp with uniform ounds A and B for a.e. (x,ν) [0,a) [0, 1 aq ). 3. A partition of unity property and zeros of the Zak transform We let N n denote the nth order cardinal B-spline, i.e., N n (x) = B n (x n ) for each n N. Since the frame set F(g) is invariant under translation of the generator g, it follows that F(N n ) = F(B n ). The results in this section are formulated for N n ; the results can, of course, also e formulated for B n. Integer translations of the B-splines form a partition of unity. The following result shows that the partition of unity property for B-splines is somewhat stale under perturations of the translation lattice; indeed, the translations of B-splines along c 1 Z yield a partly partition of unity (up to a constant) whenever c is sufficiently close to an integer. For x R let R(x) denote the round function to the nearest integer, i.e., R(x) = x+ 1, and let F(x) = x R(x) ( 1, 1 ] denote the (signed) fractional part of x. Lemma 5. Let n N and c > 0. Assume that F(c) 1 n. (i) If F(c) 0, then k ZN n ((x+k)/c) = const for x m Z [m+nf(c),m+1] (3.) date/time:6-apr-018/5:43 5 of 10

6 (ii) If F(c) 0, then k ZN n ((x+k)/c) = const for x m Z [m,m+1+nf(c)] (3.3) Proof. (i): Assume that F(c) 0. The proof goes y induction. Forn = 1 it is immediate that (3.) holds. Suppose that (3.) holds for n N. Hence, there exist a 1-periodic function g L (R) and a constant d R such that N n ((x+k)/c) = dχ A (x)+g(x)χ A c(x), for x R, k Z where A := m Z [m+nf(c),m+1]. We convolve the left and right hand side of this equation with N 1 ( ). The left hand side then ecomes c k Z N n+1((x+k)/c). The right hand side ecomes (N 1 ( c ) (dχ A +g χ A c))(x) = x c+x (dχ A (t)+g(t)χ A c(t)) dt, For x B := m Z [m+(n+1)f(c),m+1], we have x A and c + x A. Thus N 1 ( ) (cχ c A +g χ A c) is constant on B. This completes the proof of the inductive step, and (i) is proved. The proof of (ii) is similar. Remark. The Zak transform of B-splines Z λ N n is defined pointwise and is ounded on R. For n, the Zak transform Z λ N n is continuous, while Z λ N 1 is piecewise continuous on [0,1) with discontinuities along at most finitely many lines of the form t = const. The next result shows that, for > 3, the Zak transform Z 1/N n (x,ν) of the B-spline N n for (x,ν) (0,1] is zero along 1 parallel lines of length 1 n F(). Proposition 6. Let n N and > 3. Assume that F() 1 n. (i) If F() 0, then k Z 1/ N n (x, ) = 0 for x [m+nf(),m+1],k Z\Z. (3.4) m Z (ii) If F() 0, then k Z 1/ N n (x, ) = 0 for x [m,m+1+nf()],k Z\Z. (3.5) m Z Proof. We will only prove (i) as the proof of (ii) is similar. So assume that F() 0. Let ν 0 = s/ for some s Z\Z. We rewrite the Zak transform of N n as follows: Z 1/ N n (x,ν 0 ) = k Z = 1 l=0 N n ( 1 (x+k))e πik s r Z date/time:6-apr-018/5:43 6 of 10 N n ( 1 (x+r +l))e πil s. (3.6)

7 We claim that there exists a constant d R such that for every l {0,1,..., 1}: [m+nf(),m+1]. r Z N n ( 1 (x+r +l)) = d for x m Z Fix l {0,1,..., 1} for a moment. Since 1 ( ) (x+r+l) = (x+l)/+r, it follows from Lemma 5 that r Z N n( 1 (x+r +l)) is constant for x+l [ ] m+nf( ),m+1, m Z that is, for x ( m Z [m+nf(),(m+1)] Taking the intersection over l {0, 1,..., 1} completes the proof of the claim. Hence, for x m Z [m+nf(c),m+1], we can continue (3.6): Z 1/ N n (x,ν 0 ) = 1 l=0 ) de πil s = 0. l. Note that method of the aove proof relies crucially on the assumption > 3 which guarantees. 3.3 A family of counterexamples The main result of this note is the following family of ostructions for the frame property of G(B n,a,). Theorem 7. Let n N, let a > 0, > 3, and let p,q N, where gcd(p,q) = 1. If a = p q and F() 1 nq, then G(B n,a,) is not a frame for L (R). Proof. Let n N and let a = p, where > 3 and F() 1. Assume that F() 0. q nq Takex 0 = 1/q andν 0 = 1/. Using the quasi-periodicity of the Zak transform, the first coordinate of the vectors { φ Nn l (x 0,ν 0 ) } q can e written (up to a scalar multiplication) l=0 as: Z 1/ N n ( l,ν q 0) l = 1,,...,q. (3.7) It follows from Proposition 6 thatz 1/ N n (x,ν 0 ) is zero forx [nf(),1]. SincenF() 1, q it follows that the first coordinate of all the vectors in { φ Nn l (x 0,ν 0 ) } q is zero, hence { l=0 φ N n l (x 0,ν 0 ) } q is not a frame. By Theorem 4 and Remark, this shows that G(N l=0 n,a,) is not a frame for L (R). The proof of the case F() < 0 is similar. date/time:6-apr-018/5:43 7 of 10

8 Remark 3. The range F() 1 in Theorem 7 might not e optimal in the sense that nq the non-frame property of G(N n,a,) along the hyperola a = p might extend eyond q the range F() 1. Indeed, in the proof of Theorem 7, we show that the Ziulski-Zeevi nq matrix Φ Bn (x,ν) contains a row of zeros for some choice of (x,ν), ut to have a non-frame property of G(N n,a,), one just needs that Φ Bn (x,ν) is not of full rank. However, for p = 1, that is, whenφ Bn (x,ν) is a row vector, we elieve the range is optimal. This elieve is supported y the fact the F() 1 is optimal for the first B-spline N q 1, see [5, Theorem.]. Note that any sufficiently nice function g that generates a partition of unity and that yields a partly partition of unity under perturation of the integer lattice as in Lemma 5 for c 1 will have the a family of ostructions for the frame property of G(g,a,) as in Theorem A different type of counterexamples The following result shows that the exist other counterexamples, not covered y those presented in Theorem 7. The hyperolic curves from Theorem 7 together with the horizontal lines =,3,... form path-connected sets around =,3,... where G(B n,a,) is not a frame. In the following counterexamples we have [ 7, 8 3 3] which is far from the two path-connected sets at = and = 3, see Figure 1. Theorem 8. If a = 5 6 and [ 7 3, 8 3], then the Gaor system G(B,a,) is not a frame for L (R). Proof. Since a = 5, the Ziulski-Zeevi type matrix 6 ΨB (x,ν) for the Gaor system G(B,a,) is of size 5 6. At (x,ν) = (0,0) it reads: [ ] Ψ B (0,0) = 1 B (6an+al+k/) n Z. k=0,...,4 l=0,...,5 The (a,)-values in question can e expressed as 1/ = 6a/5 for a [ 5, ]. We consider the entries in Ψ B (0,0) as 6a-periodizations of B at sampling locations al+k/, that is, at al+ 6ak (mod 6a). The nd and 5th row of 5 ΨB (0,0) are sampled at locations and k = 1 : 6 11 a, 5 5 a, 14 5 a, 9 5 a, 4 5 a, 1 a, mod 6a 5 k = 4 : 6 5 a, 1 5 a, 4 5 a, 9 14 a, 5 5 a, 11 a, mod 6a, 5 respectively. Note that the6a-periodizations ofb at any sampling location in the interval [ 14 5 a, 14 5 a] mod 6a only has one nonzero term. Therefore, y definition of B, we directly see that R R 5 = [ 0 a a 0 a a ] where R i denotes the ith row of Ψ B (0,0). In the same way, we see that R 3 R 4 = [ 0 a a 0 a a ] date/time:6-apr-018/5:43 8 of 10

9 for a [ 5, ]. We see that Ψ B (0,0) does not have full rank, implying that the smallest singular values is zero. By Theorem 4, the lower frame ound of G(B,a, 5 ) is zero for 6a a [ 5, ]. Theorem 8 raises a numer of questions. A first question is whether the non-frame property along a = 5 in extends eyond [ 7, ]. The numerical computations in Figure suggest that [ 7, 8 3 3] in Theorem 8 is, at the least, very close to eing optimal. We remark that the method of proof for Theorem 8 reaks down for / [ 7, 8 3 3] ecause these (a,)-values no longer guarantee sampling locations in [ 14a, 14a] mod 6a, which 5 5 causes some of the 6a-periodizations of B to have more than one nonzero term. A second A A a (a) Estimate of A along the horizontal line (a, 5 ) for a [ 0, 5] in the (a,)-plane () Estimate of A along the hyperola ( 5 6,) for [.,.8] in the (a,)-plane. Figure : Numerical approximations (using Matla) of the lower frame ound for G(B,a,) near the point (a,) = ( 1 3, 5 ) (marked with red). question is whether this type of counterexamples is singular. We do not elieve that this is the case. In the spirit of [8], although not as old, we make a new conjecture. Our conjecture is ased on Theorem 8 and exact symolic calculations in Maple of the Ziulski-Zeevi representation associated with G(B,a,). Conjecture 1. The Gaor system G(B,a 0, 0 ) is not a frame for a 0 = 1 m+1, 0 = k +1, k,m N, k > m, a 0 0 < 1. (3.8) Furthermore, the Gaor system G(B,a,) is not a frame along the hyperolas [ ] k +1 a = (m+1), for k m k m 0 a 0, 0 +a 0, (3.9) for every a 0 and 0 defined y (3.8). The conjecture is verified for the case m = 1 and k = y Theorem 8. Acknowledgments. The authors would like to thank Ole Christensen for posing the prolem of characterizing the frame set of B-splines in a talk at the Technical University of Denmark on Feruary 3, 015, that initiated the work presented in this paper. The authors would also like to thank Karlheinz Gröchenig for comments improving the presentation of the paper. date/time:6-apr-018/5:43 9 of 10

10 References [1] A. Aldroui and K. Gröchenig. Beurling-Landau-type theorems for non-uniform sampling in shift invariant spline spaces. J. Fourier Anal. Appl., 6(1):93 103, 000. [] O. Christensen. An introduction to frames and Riesz ases. Applied and Numerical Harmonic Analysis. Birkhäuser Boston Inc., Boston, MA, 003. [3] O. Christensen. Six (seven) prolems in frame theory. In A. I. Zayed and G. Schmeisser, editors, New Perspectives on Approximation and Sampling Theory, Applied and Numerical Harmonic Analysis, pages Springer International Pulishing, 014. [4] O. Christensen, H. O. Kim, and R. Y. Kim. On Gaor frames generated y signchanging windows and B-splines. Appl. Comput. Harmon. Anal., to appear. [5] X.-R. Dai and Q. Sun. The ac prolem for Gaor systems. Memoirs of American Mathematical Society, to appear. [6] I. Dauechies, A. Grossmann, and Y. Meyer. Painless nonorthogonal expansions. J. Math. Phys., 7(5): , [7] K. Gröchenig. Foundations of time-frequency analysis. Applied and Numerical Harmonic Analysis. Birkhäuser Boston, Inc., Boston, MA, 001. [8] K. Gröchenig. The mystery of Gaor frames. J. Fourier Anal. Appl., 0(4): , 014. [9] M. S. Jakosen and J. Lemvig. Co-compact gaor systems on locally compact aelian groups. J. Fourier Anal. Appl., appeared online May 1, 015, to appear in print, DOI: /s [10] A. J. E. M. Janssen. Zak transforms with few zeros and the tie. In Advances in Gaor analysis, Appl. Numer. Harmon. Anal., pages Birkhäuser Boston, Boston, MA, 003. [11] T. Kloos and J. Stöckler. Zak transforms and Gaor frames of totally positive functions and exponential B-splines. J. Approx. Theory, 184:09 37, 014. [1] Y. Lyuarskii and P. G. Nes. Gaor frames with rational density. Appl. Comput. Harmon. Anal., 34(3): , 013. date/time:6-apr-018/5:43 10 of 10