Zeros of the Zak transform of totally positive functions
aa r X i v : . [ m a t h . NA ] N ov Zeros of the Zak transform of totally positive functions ✩ Tobias Kloos
Faculty of Mathematics, TU Dortmund, D-44227 Dortmund
Abstract
We study the Zak transform of totally positive (TP) functions. We use theconvergence of the Zak transform of TP functions of finite type to prove thatthe Zak transforms of all TP functions without Gaussian factor in the Fouriertransform have only one zero in their fundamental domain of quasi-periodicity.Our proof is based on complex analysis, especially the Theorem of Hurwitz andsome real analytic arguments, where we use the connection of TP functions offinite type and exponential B-splines.
Keywords:
Gabor frame, Total positivity, Exponential B-spline, Zaktransform
Introduction
The Gabor transform provides an important tool for the analysis of a givensignal f : R → C in time and frequency. A window function g ∈ L ( R ) hastime-frequency shifts M ξ T y g ( x ) = e πiξx g ( x − y ) , ξ, y ∈ R . The Gabor transform of a square-integrable signal f is defined as S g f ( kα, lβ ) = h f, M lβ T kα g i , k, l ∈ Z , where the parameters ( kα, lβ ) ∈ α Z × β Z of the time-frequency shifts of g form a lattice in R , with lattice parameters α, β >
0. The family G ( g, α, β ) := { M lβ T kα g | k, l ∈ Z } is called a Gabor family. If there exist constants A, B >
0, which depend on g, α, β , such that for every f ∈ L ( R ) we have A k f k ≤ X k,l ∈ Z |h f, M lβ T kα g i| ≤ B k f k , (1) ✩ AMS subject classification 2000: 42C15,41A15,42C40,30E10
Email address: [email protected] (Tobias Kloos)
Preprint submitted to Elsevier August 25, 2018 he family is called a Gabor frame. In order to describe Gabor families of awindow function g , a very helpful tool is the Zak transformZ α g ( x, ω ) := X k ∈ Z g ( x − kα ) e πikαω , ( x, ω ) ∈ R . In Approximation Theory, the Zak transform was used by Schoenberg [14] inconnection with cardinal spline interpolation. For a polynomial B-spline B m of degree m −
1, Schoenberg called Z B the exponential Euler spline . The Zaktransform Z α g has the propertiesZ α g ( x, ω + α ) = Z α g ( x, ω ) , Z α g ( x + α, ω ) = e πiαω Z α g ( x, ω ) . (2)Therefore, its values in the lattice cell [0 , α ) × [0 , α ) define Z α g completely.A well-known result for Gabor families G ( g, α, β ), with α = 1, β = 1 /N , and N ∈ N , states that the values A opt = ess inf x,ω ∈ [0 , N − X j =0 | Z g ( x, ω + jN ) | , B opt = ess sup x,ω ∈ [0 , N − X j =0 | Z g ( x, ω + jN ) | , (3)are the optimal frame-bounds A, B in the inequality (1), whenever they arepositive and finite, see [3, page 981], [8]. For rational values of αβ , a connectionof the Zak transform Z α g with the frame-bounds of the Gabor family G ( g, α, β )was given by Zibulsky and Zeevi [18]. Therefore, the presence and the locationof zeros of Z α g is relevant for the existence of lower frame-bounds in (1).Moreover, the celebrated Balian-Low theorem [3] states that a Gabor familyat the critical density αβ = 1 cannot be a frame, if the window function g orits Fourier transform ˆ g ( ω ) = Z R g ( x ) e − πixω dx is continuous and in the Wiener space , W ( R ) := { g ∈ L ∞ ( R ) | k g k W = X n ∈ Z ess sup x ∈ [0 , | g ( x + n ) | < ∞} . The proof in [6] uses the topological argument, that every continuous functionwith the property (2) must have a zero in every lattice cell and, hence, A opt in (3) with N = 1 is zero. In [1] the connection of the Zak transform anddiscretized Gabor frames is summarized and it is pointed out, that the knowl-edge of the location of the zeros is sufficient for providing that a periodizedand sampled window function generates a discrete Gabor frame.Motivated by the results in [10], we show that every Zak transform of a to-tally positive function without a Gaussian factor in its Fourier transform hasexactly one zero in its fundamental domain, which appears at ω = . For this,in the first two sections we give a short introduction on totally positive func-tions, exponential B-splines and how they are related to each other. In sectionthree we prove the stated conjecture by using several convergence propertiesof totally positive functions of finite type.2 . Totally positive functions In this section, we want to introduce totally positive functions in Schoenberg’sterminology and give some remarkable properties. For more detailed informa-tion on total positivity in terms of functions and matrices see [9].
Definition 1.1 (Totally positive (TP) function, [12]) . A measurable, non-constant function g : R → R is called totally positive (TP), if for every N ∈ N and two sets of real numbers x < x < . . . < x N , y < y < . . . < y N , the corresponding matrix A = (cid:0) g ( x j − y k ) (cid:1) Nj,k =1 has a non-negative determinant.The most popular examples of TP functions are the exponential functions e ax , a ∈ R \ { } , which are not integrable, the one- and two-sided exponen-tials e − bx χ [0 , ∞ ) ( x ), e − b | x | , b >
0, and the Gaussian e − x , which are in L ( R ).Schoenberg gave a characterization of TP functions by their two-sided Laplacetransforms and specified the subclass of integrable TP functions as follows. Theorem 1.2 ([12], [13]) . A function g : R → R , which is not an exponential g ( x ) = Ce ax with C, a ∈ R , is a TP function, if and only if its two-sided Laplacetransform exists in a strip S = { s ∈ C | α < Re s < β } with −∞ ≤ α < β ≤ ∞ and is given by ( L g )( s ) = Z ∞−∞ g ( t ) e − st dt = Cs − n e γs − δs ∞ Y ν =1 e a − ν s a − ν s , where n ∈ N and C, γ, δ, a ν are real parameters with C > , γ ≥ , a ν = 0 , < γ + ∞ X ν =1 (cid:16) a ν (cid:17) < ∞ . Moreover, g is integrable and TP, if and only if its Fourier transform is givenby ˆ g ( ω ) = Z ∞−∞ g ( t ) e − πitω dt = Ce − γω e − πiδω ∞ Y ν =1 e πia − ν ω πia − ν ω , with the same conditions on C, γ, δ, a ν as above. Unless otherwise specified, we will consider integrable TP functions with γ = 0and distinguish the infinite and finite type. Thus, by disregarding scaling andshifting, we focus on functions g, g n ∈ L ( R ), given by their Fourier transformsˆ g ( ω ) = ∞ Y ν =1 e πi ωaν πi ωa ν , ˆ g n ( ω ) = n Y ν =1 e πi ωaν πi ωa ν , n ∈ N , (4)3here ( a ν ) ν ∈ N ⊂ R \ { } and P ∞ ν =1 a − ν < ∞ . By applying the inverse Fouriertransform to (4), we get the following expressions in the time domain g ( x ) = ∞ ∗ ν =1 | a ν | e − a ν ( x + aν ) χ [0 , ∞ ) (cid:0) sign( a ν )( x + a ν ) (cid:1) , (5) g n ( x ) = n ∗ ν =1 | a ν | e − a ν ( x + aν ) χ [0 , ∞ ) (cid:0) sign( a ν )( x + a ν ) (cid:1) . Following Schoenberg, we define the reciprocals of their Laplace transforms asΨ or Ψ n ,Ψ( s ) = ∞ Y ν =1 (1 + a − ν s ) e − a − ν s , Ψ n ( s ) = n Y ν =1 (1 + a − ν s ) e − a − ν s , ∞ X ν =1 a − ν < ∞ . (6)Obviously, these are entire functions with zeros only on the real line. Byusing their Laplace transforms, Hirschman and Widder in [5, Theorem 4a] andSchoenberg in [13] show independently from each other, thatlim n →∞ g n ( x ) = g ( x ) , uniformly on R . Since we want to extend the result of the supposed conver-gence of TP functions to convergence of their Zak transforms in Section 3, weneed to adapt the proof by Hirschman and Widder. We start with a detailedinspection of the Laplace transforms. Since | − (1 − z ) e z | ≤ | z | , for all z ∈ C with | z | ≤
1, it is easy to see, that ∞ X ν =1 (cid:12)(cid:12)(cid:12) − (1 + a − ν s ) e − a − ν s (cid:12)(cid:12)(cid:12) , s ∈ C , converges locally uniformly in C and so does ∞ Y ν =1 (1 + a − ν s ) e − a − ν s . Moreover the following was proved.
Lemma 1.3 ([5, Theorem 2b]) . For any number τ ∈ R \ { } and any integer p , there exists a constant M p > with (cid:12)(cid:12)(cid:12)(cid:12) ω + iτ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ M p | τ | − p , for all | τ | ≥ | τ | , and for all n ≥ p (cid:12)(cid:12)(cid:12)(cid:12) n ( ω + iτ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ M p | τ | − p , for all | τ | ≥ | τ | , uniformly for ω in any compact interval of existence. Theorem 1.4.
Let g be a TP function of infinite type, g n the TP function oftype n as in (4) and let a := min ν ∈ N | a ν | . Then for ≤ σ < a lim n →∞ | g ( x ) − g n ( x ) | e σ | x | = 0 , uniformly for x ∈ R .Proof. Since the Laplace transforms of the given functions exist and are holo-morphic in the strip with − a < Re s < a , the inverse transforms provide for0 ≤ σ < a and positive x | g ( x ) − g n ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12) πi Z − σ + i ∞− σ − i ∞ (cid:18) s ) − n ( s ) (cid:19) e xs ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ e − σx π Z R (cid:12)(cid:12)(cid:12)(cid:12) iτ − σ ) − n ( iτ − σ ) (cid:12)(cid:12)(cid:12)(cid:12) dτ. To show the convergence of the last integral, we split it into two integrals overthe interval I = [ − R, R ] and R \ I , for R >
0. For any given ε > n ∈ N , such that for every n ≥ n the first part yields Z R − R (cid:12)(cid:12)(cid:12)(cid:12) iτ − σ ) − n ( iτ − σ ) (cid:12)(cid:12)(cid:12)(cid:12)| {z } −→ n →∞ ) uniformly in [ − R,R ] dτ ≤ εR. With Lemma 1.3, for any integer p and every n > p the remaining integral canbe estimated by Z R \ I (cid:12)(cid:12)(cid:12)(cid:12) iτ − σ ) − n ( iτ − σ ) (cid:12)(cid:12)(cid:12)(cid:12) dτ ≤ M p Z ∞ R τ − p dτ = 4 M p p − R − p +1 , where M p is a constant independent of n . Now, choosing p ≥ R arbitrar-ily large completes the proof. The case of negative x is given analogously byintegrating over the parallel line to the imaginary axis, which includes σ .Note that the stated proof shows the same convergence properties for all thederivatives g ( r ) n → g ( r ) , by including a factor s r in the integrand and choosing n ∈ N sufficiently large.Next, we take a closer look at the finite type, which is very interesting forimplementing and computational usage of TP functions and also may reveal aconnection to exponential B-splines. Since there are only finitely many expo-nential terms in (4), which realizes a shift by P nν =1 a ν in the time domain, wedisregard them for the moment. Expression (5) then simplifies to g n ( x ) = n ∗ ν =1 | a ν | e − a ν x χ [0 , ∞ ) (cid:0) sign( a ν ) x (cid:1) . b , . . . , b r , b i = b j , are distinct and µ , . . . , µ r are the associated multiplicities,such that { a , . . . , a n } = { b , . . . , b | {z } µ , . . . , b r , . . . , b r | {z } µ r } , this notation implies, that g n ( x ) = X b i > p b i ( x ) e − b i x , x ≥ , X b i < p b i ( x ) e − b i x , x ≤ , (7)where p b i are polynomials of degree µ i −
1. To be more precise, by using divideddifferences, St¨ockler [16] gives the following closed form for these functions g n ( x ) = ( − n − sign( x ) n Y ν =1 a ν ! [ a , . . . , a n | e − x · χ [0 , ∞ ) ( x · )] , x = 0 ,g n ( x ) = ( − n − n Y ν =1 a ν ! [ a , . . . , a n | χ [0 , ∞ ) ( · )] , x = 0 . With this form and the well-known identity[ a , . . . , a n | f ] = r X i =1 µ r X j =1 c i,j f ( j − ( a i ) , c i,j ∈ R , c i,µ i = 0 , of divided differences, see e.g. [15], it is possible to compute the polynomialsin (7). In the case of distinct weights, the formula reduces to[ a , . . . , a n | f ] = n X i =1 n Y j =1 j = i ( a i − a j ) − f ( a i ) , which provides an explicit form for g n , under these assumptions. In [1] Bannert,Gr¨ochenig and St¨ockler also computed this explicit form of the coefficients, byusing the partial fraction decomposition of ˆ g n , instead of divided differences.All in all, these representations provide a good way for implementation andcomputing these functions for usage in time-frequency analysis. Moreover itreveals the close resemblance to exponential B-splines, which will be definedin the next section.
2. Exponential B-splines
We want to give a short summary of exponential B-splines and list some im-portant features of them. For a more detailed introduction, see e.g. [15], [11]and [10].For a set of weights Λ = ( η , . . . , η | {z } µ , . . . , η r , . . . , η r | {z } µ r ) , η , . . . , η r ∈ R , and each η j repeated with multiplicity µ j ∈ N , the space U m , given by U m = span (cid:0) e η x , xe η x , . . . , x µ − e η x , . . . , e η r x , . . . , x µ r − e η r x (cid:1) , forms an extended complete Tschebycheff (ECT) space. This means, thatthere exists a basis { u , . . . , u m } ∈ U m , such thatdet (cid:18) M (cid:18) u , . . . , u ℓ t , . . . , t ℓ (cid:19)(cid:19) > ≤ ℓ ≤ m and t ≤ . . . ≤ t ℓ ∈ R . The matrix M (cid:18) u , . . . , u ℓ t , . . . , t ℓ (cid:19) is the collocation matrix of Hermite interpolation, if some nodes t j coincide.In our case this basis is given by the exponentials.For ECT spaces in general it is an important result in spline interpolationtheory, that there exist functions, which are compactly supported, piecewisein these spaces and sufficiently smooth. These properties are fulfilled by B-splines. In our special case, the associated splines are defined as follows. Definition 2.1.
Let Λ = ( λ , . . . , λ m ) ∈ R m , λ = 0, and let exponentialweight functions w j := e ( λ j − λ j − ) x , j = 1 , . . . , m, be given. Then, with proper normalization, the exponential B-spline (EB-spline) B Λ , with knots 0 , , . . . , m , is given by the convolution of the functions e λ j ( · ) χ [0 , , B Λ = e λ ( · ) χ [0 , ∗ e λ ( · ) χ [0 , ∗ . . . ∗ e λ m ( · ) χ [0 , , (8)so its Fourier transform is given byˆ B Λ ( ω ) = m Y j =1 e λ j − πiω − λ j − πiω . By the definition, it is easy to see, that B Λ ∈ C m − ( R ) and supp B Λ = [0 , m ].Moreover, B Λ | ( j,j +1) ∈ U m , 0 ≤ j ≤ m −
1, so they are piecewise exponentialsums B Λ ( x + k −
1) = r X j =1 p ( k ) j ( x ) e η j x , x ∈ [0 , , ≤ k ≤ m, (9)with real polynomials p ( k ) j of degree µ j − L j f = ddx (cid:18) fw j (cid:19) , L j = L j · · · L , j = 1 , . . . , m,
7s in [15, page 365] and take note, that U m is the kernel of L m . Furthermore,these operators can also be written as L j = e − λ j x j Y k =1 (cid:18) ddx − λ k id (cid:19) . Up to normalization, L B Λ denotes the first reduced EB-spline and is given by B { λ − λ ,...,λ m − λ } = e ( λ − λ )( · ) χ [0 , ∗ . . . ∗ e ( λ m − λ )( · ) χ [0 , .
3. The Zak transform of totally positive functions
The Zak transform is an important tool in Gabor analysis which is commonlyapplied in spline theory. For a parameter α > f : R → C itis defined by Z α f ( x, ω ) := X k ∈ Z f ( x + αk ) e − πikαω , whenever this series exists. In the following we need the properties below ofthis transform. More facts can be found in [4]. Lemma 3.1.
Let f be an element of the Wiener space W ( R ) .a) Z α f ( x, ω ) is bounded in R , and if f is continuous, then Z α f is contin-uous.b) For every n ∈ Z , we have the identities for periodicity Z α f ( x, ω + nα ) = Z α f ( x, ω ) and quasi-periodicity Z α f ( x + nα, ω ) = e πinαω Z α f ( x, ω ) . c) If ˆ f ∈ W ( R ) as well, then α · Z α f ( x, ω ) = e πixω Z /α ˆ f ( ω, − x ) . d) Let f α = f ( α · ) be the scaled function of f . Then Z α f ( x, ω ) = Z f α ( xα , αω ) . Because of the last property and since the considered function spaces are scal-ing invariant, we restrict ourselves to the case α = 1 and therefore write Z f ,instead of Z f . We also use the following connection of the Zak transform ofTP functions of finite type and EB-splines, which was found recently in [10].8 heorem 3.2 ([10, Theorem 3.4]) . Let g ∈ L ( R ) be a TP function of finitetype, defined by its Fourier transform ˆ g ( ω ) = m Y ν =1 (1 + 2 πi ωa ν ) − , where a , . . . , a m ∈ R \ { } . With λ ν := − a ν and B Λ defined as in (8), we have Z g ( x, ω ) = m Y ν =1 a ν − e − ( a ν +2 πiω ) Z B Λ ( x, ω ) , ( x, ω ) ∈ [0 , × [0 , . Note that this factorization of the Zak transform of TP functions of finite typeholds on R by quasi-periodicity of the transform.Next, we complexify the argument ω of the Zak transform and show the con-vergence of the transforms of TP functions of finite type to the transforms ofthe related TP functions of infinite type. Definition 3.3.
The complexified Zak transform of a function f ∈ L ( R ) isgiven by Z f ( x, s ) = X k ∈ Z f ( x + k ) e − πiks = X k ∈ Z f ( x + k ) e − πikω e πkτ , where x, ω, τ ∈ R , s = ω + iτ ∈ C are chosen, such that the series converges.Schoenberg proved in [13], that integrable TP functions ˜ g decay exponentially,lim x →±∞ e xs ˜ g ( x ) = 0 , − a < s < a . Hence, their complexified Zak transforms exist for all | τ | < a π . Theorem 3.4.
Let g be a TP function of infinite type and g n the TP functionof type n as in (4). Then for a fixed x ∈ [0 , the transforms Z g n ( x , · ) and Z g ( x , · ) are holomorphic in the strip S ξ = { s ∈ C | | Im( s ) | ≤ ξ } , whenever ≤ ξ < a π , and lim n →∞ | Z g ( x, s ) − Z g n ( x, s ) | = 0 , uniformly for all s = ω + iτ in the strip S ξ , ≤ ξ < a π , and all x ∈ [0 , .Proof. The holomorphism of Z g n ( x , · ) of TP functions of finite type easilyfollows from Theorem 3.2, which provides an expression as a finite sum ofexponentials, multiplied with a function with singularities outside of S ξ . Theholomorphism in the case of TP functions of infinite type is given by theuniform convergence, which we prove next.9et x ∈ [0 ,
1) and c ∈ R with 2 πξ < πc < a . By Theorem 1.4, for a given ε >
0, there exists n ∈ N , such that for every n ≥ n | Z g ( x, s ) − Z g n ( x, s ) | ≤ X k ∈ Z | g ( x + k ) − g n ( x + k ) | e π | kτ | ≤ ε X k ∈ Z e − πc | x + k | e π | kτ | ≤ ε ∞ X k =1 e πk ( | τ |− c ) + e − πcx + e πc − X k = −∞ e πk ( c −| τ | ) ! ≤ ε (cid:0) e πc (cid:1) ∞ X k =0 e πk ( | τ |− c ) ≤ ε (cid:0) e πc (cid:1) e π ( c −| τ | ) π ( c − | τ | ) . This proves the second part of the Theorem and implies, that the Zak transformof a TP function Z g ( x , · ) of infinite type is the uniform limit of holomorphicfunctions, in S ξ , which completes the first part of the proof.Next, we will use this convergence property to show, by arguments of com-plex analysis, that the transforms have exactly one zero in their fundamentaldomain. For this we will use the Theorem of Hurwitz. Lemma 3.5 (Hurwitz) . Let D ⊂ C be a domain and ( f n ) be a sequence ofholomorphic functions in D , which converges locally uniformly to a function f .If every f n has at most k zeros in D , then f has at most k zeros or f ( z ) = 0 ,for all z ∈ D . The following result in [10] shows, that the Zak transform of TP functions offinite type m ≥ Theorem 3.6 ([10, Corollary 3.5]) . Let g n be a totally positive function offinite type n ≥ . Then there exists ˜ x ∈ [0 , , such that Z g n (˜ x, ) = 0 , and Z g n ( x, ω ) = 0 for all ( x, ω ) ∈ [0 , \ { (˜ x, ) } . Next, we extend this result and show, that the complexified Zak transform hasno zero in [0 , × { s = ω + iτ | | ω | < , | τ | < a π } . Then Lemma 3.5 impliesthat this property holds for TP functions of infinite type. The main idea ofthe proof is counting sign changes and using the quasi-periodicity to constructa contradiction of having a zero in the given domain. Following [2], we saythat a function f : [ a, b ] → R has at least p strong sign changes, if there existsa nondecreasing sequence ( τ j ) ≤ j ≤ p in [ a, b ] with f ( τ ) = 0 and, in case p ≥ f ( τ j − ) f ( τ j ) < j = 1 , . . . , p . The supremum of the number of strongsign changes of f is denoted by S − ( f ). Similarly, we define the total numberof sign changes S − ( c ) of a sequence of real numbers c = ( c k ) ≤ k ≤ N . Theorem 3.7.
Let ˜ g be a continuous TP function, defined by (4), includingthe infinite type. Then for the complexified Zak transform, it holds that Z˜ g ( x, s ) = 0 , for x ∈ [0 , and s ∈ ( − , ) × i ( − a π , a π ) . roof. With Theorem 3.2, the finite case ˜ g = g n , n ∈ N , results by provingthe same property for the Zak transform of the associated EB-spline B Λ . Wewill show this by following the proof in [10]. Again we use the notationΛ = ( η , . . . , η , . . . , η r , . . . , η r )with multiplicities µ j of the pairwise distinct weights η j . For a fixed s ∈ ( − , ) × ( − a π , a π ), we consider the complex valued function h := Z B Λ ( · , s ).By (9), we obtain for x ∈ [0 , h ( x ) = n − X k =0 B Λ ( x + k ) e − πiks = n X k =1 r X j =1 p ( k ) j ( x ) e η j x e − πiks = r X j =1 n X k =1 p ( k ) j ( x ) e − πiks | {z } := q j ( x ) e η j x , (10)where q j are complex polynomials of degree σ j − ≤ µ j −
1. Some of the q j arenonzero, since the shifts of EB-splines are locally linearly independent. Thismeans that, if Re( h ) vanishes on an interval ( a, b ) ∈ R , then all coefficientsRe( e − πiks ) with supp B Λ ( · + k ) ∩ ( a, b ) = ∅ vanish as well. Since | Re( s ) | < ,no consecutive coefficients of Re( h ) vanish simultaneously. Therefore there isno non-empty interval where Re( h ) is identically zero. We let σ j = 0 if q j = 0and define γ = σ + . . . + σ r . Without loss of generality, we can assume σ ≥ q ( x ) e η x in h | [0 , is nonzero. We use the identity e η j x ddx (cid:0) e − η j x h ( x ) (cid:1) = q ′ j ( x ) e η j x + X k = j (( η k − η j ) q k ( x ) + q ′ k ( x ))) e η k x . Writing D j for the differential operator on the left hand side and D := D σ − Q rj =2 D σ j j , we obtain D h ( x ) = be η x , x ∈ (0 , , with a nonzero constant b ∈ C . The quasi-periodicity of h leads directly to D h ( x ) = be η ( x − k ) e πiks , x ∈ ( k, k + 1) , for all k ∈ Z .Now we assume that there exists ˜ x ∈ [0 ,
1) with Z B Λ (˜ x, s ) = 0. By quasi-periodicity of h = Z B Λ ( · , s ), the function f = Re h vanishes at all points ˜ x + k , k ∈ Z , and these points are isolated zeros of f by local linear independenceagain. This guarantees that f has at least N ∈ N isolated zeros in [0 , N ]. Notethat D is a differential operator of order γ − ≤ m −
1. Since f ∈ C m − ( R ),with f ( m − absolutely continuous, we obtain by Rolle’s theorem that S − ( D f ) ≥ N − γ + 1 on [0 , N ] . (11)11owever, on each interval [ k, k + 1) with k ∈ Z and s = ω + iτ , ω, τ ∈ R , thesign of D f is fixed bysign ( D f )( x ) = sign Re (cid:0) b e πik ( ω + iτ ) (cid:1) = sign Re (cid:0) b e πikω (cid:1) , x ∈ [ k, k + 1) . This implies S − ( D f ) ≤ N | ω | on [0 , N ] , which is a contradiction to (11) for | ω | < and large N . This completes theproof for TP functions of finite type (and their associated EB-splines).The Zak transform of TP functions of infinite type is not identically zero on( − , ) × i ( − a π , a π ), because g is positive and Z g ( x,
0) = P k ∈ Z g ( x + k ) > g n ( x, · )without any zeros, which completes the proof.This Theorem especially implies, that Z g has no zero in [0 , × ( − , ). Since itis well-known, that Zak transforms of continuous functions do have a zero, weare left to show, that they have exactly one zero in their fundamental domain,whenever g is a TP function of infinite type without a Gaussian factor. Forthis, we need some preparations. Lemma 3.8.
Let ˜ g be a continuous TP function, defined by (4), including theinfinite type. Then Z˜ g ( · , ) is real, -periodic and not identically zero.Proof. By the definition it is easy to see, thatZ˜ g ( x, ) = X k ∈ Z ˜ g ( x + k ) e − πik = X k ∈ Z ˜ g ( x + k )( − k is a real valued function. Since it is quasi-periodic,Z˜ g ( x + n, ) = e πin Z˜ g ( x, ) = ( − n Z˜ g ( x, ) , obviously Z˜ g ( · , ) is a 2-periodic function. Moreover, the Inversion Formulafor Zak transforms, Z Z˜ g ( x, ω ) e − πixω dx = Z X k ∈ Z ˜ g ( x − k ) e πiω ( k − x ) ! dx = X k ∈ Z Z ˜ g ( x − k ) e − πiω ( x − k ) dx = ˆ˜ g ( ω ) , implies that Z˜ g ( · , ) cannot be identically zero.12 emma 3.9. Let η , . . . , η r ∈ R be some pairwise distinct numbers, q , . . . , q r some real nonzero polynomials of degree σ j − . Let γ = σ + . . . + σ r and con-sider a real valued function h ∈ C γ − ( R ) , where h ( γ − is absolutely continuous, h ( x + 1) = − h ( x ) and h ( x ) = r X j =1 q j ( x ) e η j x , x ∈ [0 , . Then there exists x ∈ R , such that h is monotone on each interval [ x + k, x + k + 1) , k ∈ Z .Proof. Analogously to the proof of Theorem 3.7, consider the differential op-erator D = D σ − Q rj =2 D σ j j . Due to continuity and periodicity, h ( x + 2) = − h ( x + 1) = h ( x ), the range of h is bounded and so h has at least one maxi-mum at a point x (and one minimum) in the interval [0 ,
2) of a period. If h is not monotonously decreasing on [ x , x + 1), the 2-periodicity implies, thatthere is a number c ∈ R , such that the function h c := h ( · ) + c has at least fourzeros in [0 , D to this function we get D h c ( x ) = be η x + r Y j =2 ( − η j ) σ j ( − η ) σ − c | {z } =:˜ c , b ∈ R , x ∈ (0 , , and since h ( x + 1) = − h ( x ), we have D h c ( x ) = ( − k be η x + ˜ c, x ∈ ( k, k + 1) , k ∈ Z . For large N ∈ N we get a contradiction, by counting sign changes again:4 N − γ + 1 ≤ S − ( D h c ) ≤ N on [0 , N ] . This shows, that h is monotone on every interval [ x + k, x + k + 1), k ∈ Z ,where x + k are the extrema of h . Corollary 3.10.
Let
Λ = ( η , . . . , η , . . . , η r , . . . , η r ) ∈ R m , η j = η i , and B Λ bethe associated EB-spline as defined in (8). Then there exist numbers x , y ∈ R ,such that Z B Λ ( · , ) is monotone on [ x + k, x + k + 1) and the derivative D n Z B Λ ( · , ) , ≤ n ≤ r , is monotone on [ y + k, y + k + 1) , for all k ∈ Z .Proof. With the notations as in equation (10) these functions can be writtenas a sum of exponentials with polynomial coefficients,Z B Λ ( x, ) = r X j =1 q j ( x ) e η j x ,D n Z B Λ ( x, ) = D n r X j =1 q j e η j · !! ( x ) = e η n x ddx e − η n x r X j =1 q j ( x ) e η j x ! = q ′ n ( x ) e η n x + X j = n (( η j − η n ) q k ( x ) + q ′ k ( x ))) e η k x . Therefore the Corollary follows directly from Lemma 3.8 and Lemma 3.9.13 orollary 3.11.
Let g be a TP function of infinite type as defined in (4).Then there exists x ∈ [0 , , such that Z g ( · , ) is monotone on every interval [ x + k, x + k + 1) , k ∈ Z , of length one.Proof. Because of continuity and periodicity the range of f := Z g ( · , ) isbounded and there is at least one maximum (and one minimum) in the in-terval [0 , x ∈ [0 , x + 1 a minimum). If f is not monotonously decreasing on [ x , x + 1),there exist δ > z < z < z ∈ [ x + δ, x + 1 − δ ], such that f ismonotone on [ x , x + δ ) and f ( z ) < f ( z ) , f ( z ) < f ( z ) . With ε := · min { f ( x ) − f ( z ) , f ( z ) − f ( z ) , f ( z ) − f ( z ) } the uniformconvergence of the Zak transforms of TP functions implies, that there is anumber n ∈ N , such that for every n ≥ n the function f n := Z g n ( · , ) fulfills f n ( x ) > f n ( z ) , f n ( z ) > f n ( z ) , f n ( z ) > f n ( z ) . Therefore there is also no interval of length one, where f n is monotone. Be-cause of Theorem 3.2, the same property holds for the Zak transform of theassociated EB-spline Z B Λ ( · , ) = C · Z g n ( · , ), C ∈ R , which is a contradictionto Lemma 3.10.Now we can prove the last part of our main result. Theorem 3.12.
Let g be a TP function of infinite type as defined in (4).Then there exists ˜ x ∈ [0 , , such that Z g (˜ x, ) = 0 , and Z g ( x, ) = 0 for all x ∈ [0 , \ { ˜ x } .Proof. It was first pointed out in [17] that if the Zak transform Z g is continuous,then it has a zero in [0 , × [0 , g is real, we know from the results in[7], that there exists ˜ x ∈ [0 , g (˜ x, ) = 0. Now, we assume thatthere exist at least two zeros in an interval of monotonicity [ x , x + 1), whereZ g ( · , ) has a maximum at x again. If there are some isolated zeros or morethan one zero intervals, Z g ( · , ) can not be monotone, which is a contradictionto Corollary 3.11. Therefore we suppose, that there is one single zero interval[ z , z ] ⊂ [ x , x + 1). First we assume that a <
0, whereˆ g ( ω ) = ∞ Y ν =1 e πi ωaν πi ωa ν , and we let f := Z g ( · , ). Since f is monotonously decreasing on [ x , x + 1),the mean-value theorem implies, that for 0 < ε < − a − there is 0 < θ < ε with | f ′ ( z + θ ) | = | f ( z + ε ) | ε > | f ( z + θ ) | ε > − a | f ( z + θ ) | .
14n the other hand, f ′ ( x + 1) = 0, f ( x + 1) < D f = (cid:18) ddx + a id (cid:19) f = f ′ + a f (12)fulfills D f ( x ) < D f ( z ) = 0, D f ( z + θ ) < D f ( x + 1) >
0, whichimplies, that D f is not monotone on any interval of length one. Analogouslyto the proof of Corollary 3.11, the convergence property of Theorem 3.4 implies,that there is a number n ∈ N , such that for every n ≥ n the derivative D Z g n ( · , ) also is not monotone on any interval of length one. Hence, becauseof Theorem 3.2, the same holds for the Zak transform of the associated EB-spline D Z B Λ ( · , ), which is a contradiction to Lemma 3.10 again. Consideringthe interval [ x , z ], instead of [ z , x + 1], the case a > P K : L ( R ) → L ( T K ) and the sampling operator S : L ( R ) → ℓ ( Z ), P K g = X k ∈ Z g ( · − kK ) , Sg = ( g ( k )) k ∈ Z , the following holds. Corollary 3.13 (cf. [1, Theorem 8]) . Let ˜ g be a continuous TP function,defined by (4), including the infinite type. Assume α = M ∈ N and let β =1 /M and k ∈ N such that K/M ∈ N . • If K/M is odd, then G ( P K ˜ g, α, β ) is a Gabor frame for L ( T k ) . • If K/M is odd, then G ( P K S ˜ g, α, β ) is a Gabor frame for C K .In addition, assume that ˜ g is even, which means, that { a ν | a ν > } = {− a ν | a ν < } . • If M is odd, then G ( S ˜ g, α, β ) is a Gabor frame for L ( T k ) . • If M is odd, then G ( P K S ˜ g, α, β ) is a Gabor frame for C K .Proof. For the proof see [1, Theorem 8].
Acknowledgement
We are very grateful to J. St¨ockler for some discussions about the subjects inthis article. eferenceseferences