Applied and Computational Harmonic Analysis

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1 App. Comput. Harmon. Ana Contents ists avaiabe at SciVerse ScienceDirect Appied and Computationa Harmonic Anaysis Pairs of dua periodic frames Oe Christensen a,,saysonggoh b a Technica University of Denmark, Department of Mathematics, Buiding 303, 800 yngby, Denmark b Department of Mathematics, Nationa University of Singapore, 10 Kent Ridge Crescent, Singapore 11960, Repubic of Singapore artice info abstract Artice history: Received 7 June 011 Revised 7 December 011 Accepted 30 December 011 Avaiabe onine 3 January 01 Communicated by Karheinz Gröchenig Keywords: Periodic frames Gabor frames Waveet frames Dua pairs of frames Trigonometric poynomias The time frequency anaysis of a signa is often performed via a series expansion arising from we-ocaized buiding bocks. Typicay, the buiding bocks are based on frames having either Gabor or waveet structure. In order to cacuate the coefficients in the series expansion, a dua frame is needed. The purpose of the present paper is to provide constructions of dua pairs of frames in the setting of the Hibert space of periodic functions 0, π. The frames constructed are given expicity as trigonometric poynomias, which aows for an efficient cacuation of the coefficients in the series expansions. The generaity of the setup covers periodic frames of various types, incuding nonstationary waveet systems, Gabor systems and certain hybrids of them. 01 Esevier Inc. A rights reserved. 1. Introduction Time frequency representations pay a fundamenta roe in many practica appications as they provide ocaized information of signas in time and frequency domains. Series representations in terms of frames capture such information at prescribed discrete points in the time frequency pane. Gabor frames and waveet frames are eading exampes of frames from the perspective of time frequency anaysis, and both of them have their respective strengths, see for instance [3,7 9, 15,0]. Technicay, the series expansion provided by a frame requires knowedge of a dua frame, either for the synthesis or the anaysis of the given signa. Therefore simutaneous constructions of a frame and a corresponding dua with desirabe properties is a key issue. In addition, many signas of practica interest can be considered as periodic. Apart from signas that are inherenty periodic, a signas resuting from experiments with a finite duration can in principe be modeed as periodic signas, see for exampe [18]. This motivates the current paper on periodic frames. The purpose of this paper is to construct expicity given frames and dua pairs of frames in 0, π, the Hibert space of π -periodic functions on R that are square-integrabe over 0, π. The frames wi be given as a coection of transates of a set of functions. Under suitabe conditions we wi aso derive expicit expressions for associated dua frames. As concrete exampes, we obtain frame constructions of Gabor type and waveet type, as we as a certain hybrid of these. The practica reevance of the resuts is expained in the context of signa processing. More detais on the premise of the paper wi appear ater in the introduction. An outine of the paper is as foows. In the rest of this introduction we present a few basic definitions and facts about frames. We aso give an exampe that motivates the theoretica resuts to foow. Then, in Section we present sufficient conditions for a sequence of transates of a coection of functions in 0, π to be a Besse sequence or a frame. In Section 3 we demonstrate how to expicity construct dua pairs of frames. These dua pairs are frames comprised * Corresponding author. E-mai addresses: Oe.Christensen@mat.dtu.dk O. Christensen, matgohss@nus.edu.sg S.S. Goh /$ see front matter 01 Esevier Inc. A rights reserved. doi: /j.acha

2 316 O. Christensen, S.S. Goh / App. Comput. Harmon. Ana of trigonometric poynomias, which faciitate efficient anaysis of periodic functions. The theoretica resuts are foowed by various concrete exampes, deaing with, e.g., Gabor anaysis, stationary as we as nonstationary waveet anaysis, and various hybrids of these. The generating functions of these trigonometric poynomia frames have desirabe properties such as being rea-vaued and symmetric, and possessing good time frequency ocaization. We now describe the setting of our study. et I denote a subset of the integers Z, et{ } be any countabe sequence of positive integers, and define associated transation operators T k acting on 0, π by T k f x := f x π. Note that composing T k with itsef eads to T k f x = f x π, Z, and so, T k f x = f x π = f x, i.e., T k equas the identity operator. For each k I we wi appy the operators T k to a function ψ k in 0, π; thus, we consider the coection of functions {T k ψ k},,...,k 1, where the index set is chosen to avoid repetitions. Note that this genera setup aows us to appy different shifts to the invoved functions ψ k. Our purpose is to consider frame properties for a coection of functions of the form {T k ψ k},,...,k 1 in 0, π, so we wi briefy reca a few standard resuts and facts about frames. We say that the coection {T k ψ k},,...,k 1 in 0, π forms a frame for 0, π if there exist positive constants A, B such that A f f, T k ψ k B f, f 0, π. 1.3 The constants A, B are caed bounds of the frame. If at east the second inequaity in 1.3 hods, then {T k ψ k},,...,k 1 is said to be a Besse sequence and B its bound. Two coections {T k ψ k},,...,k 1 and {T k ψ k },,...,k 1 in 0, π form a pair of dua frames if both coections are Besse sequences and f = f, T k ψ k T k ψ k, f 0, π. 1.4 It is we known that if {T k ψ k},,...,k 1 and {T k ψ k },,...,k 1 are dua frames, the roes of {T k ψ k},,...,k 1 and {T k ψ k },,...,k 1 can be interchanged in the anaysis and synthesis of f. The setup in the form of {T k ψ k},,...,k 1 encompasses periodic frames of various types. Different choices of { }, giving transation operators T k of different shifts as defined in 1.1, determine the frame systems on hand. In particuar, if I = Z, = D for some positive integer D and ψ k = e ik ψ 0,thenT k = T 0 for a k, and by 1., {T k ψ k} k Z,,...,D 1 can be written as {e π ik/d e ik T k ψ 0} k Z,,...,D 1 which is a Gabor system generated by ψ 0, up to the constant factors e π ik/d. On the other hand, if I = N {0} and = D k for some integer D, then T k amounts to shifting by π and D k {T k ψ k} k 0,,...,Dk 1 is a nonstationary waveet system. et us motivate the constructions to foow from the perspective of signa processing. Here, and in the rest of the paper, the Fourier coefficients for a function f 0, π are denoted by π 1 f n := f xe inx dx, n Z. π 0 et {T k ψ k},,...,k 1 and {T k ψ k },,...,k 1 be a pair of trigonometric poynomia dua frames, and f 0, π a signa to be anayzed and synthesized. Using the frame {T k ψ k},,...,k 1 for the anaysis of f,asin1.4, we compute the frame coefficient f, T k ψ k = f n T k ψ kn = f n ψ k ne π in/ 1.5 for k I, = 0,..., 1. This can be evauated efficienty as each T k ψ k is a trigonometric poynomia and so 1.5 isa finite sum. When expicit expressions for ψ k, k I, are avaiabe which is the case in this paper, the signa f can be readiy recovered from the reconstruction formua 1.4. Whie the frame coefficients f, T k ψ k, k I, = 0,..., 1, can be evauated efficienty, one aso needs to address whether they provide an effective time frequency anaysis of f. The foowing exampe highights some of the issues invoved in this.

3 O. Christensen, S.S. Goh / App. Comput. Harmon. Ana Exampe 1.1. et H beafunctionin 0, π which is we ocaized in time for instance, a Dirichet kerne or a Fejér kerne, and consider the signa f 0, π defined by f x := cosn 1 xhx x 1 + cosn 1 xhx x + cosn xhx x 1 + cosn xhx x 1.6 for some distinct n 1,n N {0} and x 1, x [0, π. The signa f comprises four components of the form f rs x = cosn r xhx x s, a function that is ocaized in frequency around n =±n r and ocaized in time around x = x s. A trigonometric poynomia frame {T k ψ k},,...,k 1 woud resove the components of f if for each component f rs of f, we can find a frame eement T k ψ k that returns a arge vaue of f rs, T k ψ k, whie giving reativey sma vaues for the other components. In 1.6, if n 1 and n as we as x 1 and x are sufficienty far apart, then the anaysis of f provided by 1.5 woud successfuy resove the different components of f when ψ k and ψ k are transated by appropriate constant amounts in frequency and time domains respectivey. This is the typica setup of Gabor anaysis, see Exampe 3.1. If n 1 and n are arge vaues, then f is a high frequency signa which has rapid changes in time. In this case, for the anaysis 1.5, it is natura to empoy a finer shift T k ψ k given by a arger vaue of. On the other hand, if n 1 and n are sma vaues, then f is a ow frequency signa, and it suffices to take a smaer vaue of amounting to a coarser shift T k ψ k. This is precisey the fexibiity provided by waveet anaysis where = D k for some integer D, see Exampe 3.. In the typica waveet setup, the ength of the support of ψ k often grows rapidy in mutipes of D k as k increases. So when n 1 and n are arge vaues but near to each other, they may both end up in the same support of ψ k for some arge vaue of k. This can be avoided if the support of ψ k expands at a ess rapid rate as k increases. On the other hand, when x 1 and x in 1.6 are cose to each other, we shoud utiize a fine shift T k ψ k, given by a arge vaue of,toresovethe components ocaized at these points in time. The baancing of these desirabe features is incorporated into the construction of Exampe 3.3, which attempts to combine the strengths of Gabor anaysis and waveet anaysis. Previous work on periodic frames in the iterature incudes [13,14] in which periodic waveet frames are obtained from mutiresoution anayses, and [5] where pairs of obique duas are constructed for finite-dimensiona spaces of periodic functions. On the other hand, for the space R, expicit pairs of dua Gabor frames are constructed in [4,6] with corresponding resuts for dua waveet frames reported in [19] and [16,17]. Here we focus on the space 0, π, and we adapt, unify and further deveop these ideas to construct pairs of trigonometric poynomia frames. Some of our extensions are made possibe ony by the periodic setting on hand, i.e., corresponding resuts for R are not avaiabe. In contrast to [4,19], our approach is based on a genera nonstationary setup where different vaues of k may correspond to rather different functions ψ k and parameters. It aso does not assume the mutiresoution anaysis framework in [13, 14], which typicay takes = D k for some integer D.. Besse sequences and frames of the form {T k ψ k},,...,k 1 We first present a genera sufficient condition for a system of functions of the form {T k ψ k},,..., to be a Besse sequence or form a frame for 0, π. Simiar resuts are known for Gabor systems and waveet systems in R see [,3]. Whie our proof adapts appropriatey the main ideas in estabishing [3, Theorem 11..3] on waveet frames for R to the space 0, π, the nonstationary setting on hand gives a genera resut that is appicabe to periodic Gabor systems, periodic waveet systems, as we as other periodic systems of interest. This proof is provided in Appendix A. Theorem.1. Consider functions {ψ k } 0, π, et{ } be a sequence of positive integers, and assume that B := sup ψ k n ψ k n + q <..1 q Z Then {T k ψ k},,...,k 1 is a Besse sequence with bound B. If in addition, A := inf ψ k n ψ k n ψ k n + q > 0,. q Z\{0} then {T k ψ k},,...,k 1 is a frame for 0, π with bounds A, B. For = D k for some integer D, another sufficient condition for {T k ψ k} k 0,,...,D k 1 to be a Besse sequence can be found in [1, Theorem 4.1]. Whie Theorem.1 provides a condition for {T k ψ k},,...,k 1 to form a frame for 0, π, it does not contain information about how an appropriate dua frame can be obtained. In the next section we wi construct pairs of dua frames expicity, and in that context the Besse condition wi pay an important roe. For this reason we now state some

4 318 O. Christensen, S.S. Goh / App. Comput. Harmon. Ana easiy accessibe conditions, based on Theorem.1, for{t k ψ k},,...k 1 to be a Besse sequence. Here, and in the rest of the paper, we put S k := supp ψ k,.3 i.e., S k is the set of a n Z for which ψ k n 0. We wi subsequenty assume that ψ k is a trigonometric poynomia, meaning that S k is a finite set. Coroary.1. Consider functions {ψ k } 0, π and a sequence { } of positive integers, and assume the foowing: i For each k I, the set S k is contained in an interva of ength stricty ess than J for some J N; ii There exists a constant C > 0 such that ψ k n C, k I, n Z; k iii There exists a number K N such that each n Z beongs to at most K of the sets S k. Then {T k ψ k},,...,k 1 is a Besse sequence with bound K C J 1. Proof. Fix n Z. Assume that n S k for k {k 1, k,...,k μ }, where k 1,...,k μ I; by assumption μ K.Then ψ k n ψ k n + q = q Z J 1 k {k 1,...,kμ} q= J+1 K J 1 sup ψ k n, KC J 1. ψ k n ψ k n + q The resut now foows from Theorem.1. We wi now use Coroary.1 to check the Besse condition for some specia casses of functions. In Section 3 we return to these exampes and construct dua pairs of frames. Our first exampe eads to a system of functions with Gabor structure. Exampe.1. et g be a nontrivia, bounded, and rea-vaued function on R with support in an interva [M, N] for some M, N Z, M < N. Forthesequence{ } in Coroary.1, takei = Z and = D for some positive integer D. Wedefinea famiy of trigonometric poynomias ψ k x = ψ k ne inx, k Z,.4 by ψ k n := gn k D, n Z..5 Then for k Z, = 0,...,D 1, using 1.,.4 and.5, a cacuation gives T k ψ kx = e π ik/d e ikx ψ 0 x π = e π ik/d e ikx T k D ψ 0x..6 Thus, up to the constant factors e π ik/d, we are deaing with a Gabor system generated by the function ψ 0. Checking the conditions of Coroary.1, we see that i ceary hods because by.5, the set S k is contained in an interva of ength N M which is independent of k. Sinceg is bounded and = D, ii is aso satisfied. Finay, it foows from.5 that iii hods with K = N M + 1. Hence, by Coroary.1, {T k ψ k} k Z,,...,D 1 is a Besse sequence. In Exampe.1 which eads to Gabor frames, the size of the support of ψ k is independent of k. The next exampe shows how to construct Besse sequences consisting of trigonometric poynomias such that the support of ψ k grows exponentiay with k, which sets the scene for waveet frames.

5 O. Christensen, S.S. Goh / App. Comput. Harmon. Ana Exampe.. et g be a nontrivia, bounded, and rea-vaued function on R with support in an interva [M, N] for some M, N R, M < N. eti = N {0}, and consider the sequence { } k 0 given by = D k for some integer D. Choose any integer k 0 > 0 and define a famiy of trigonometric poynomias ψ k x = ψ k ne inx, k = 0, 1,...,.7 by ψ k 0 := 1 D k δ k,k 0, and, for n 0, 0, if k = 0,...,k 0 1, gog D n ψ k n :=, if k = k 0, D k n gog D D k+k +gog n D 0 D k k 0, if k > k 0. D k.8 Then the functions ψ k are rea-vaued and symmetric on Z; and this gives generators ψ k that are rea-vaued and symmetric. In order to check condition i in Coroary.1, it is ceary enough to consider k > k 0. Then it is ony possibe that ψ k n 0if n n M og D D k+k N or M og D N, 0 D k k.9 0 i.e., if D M k+k 0 n D N k+k 0 or D M+k k 0 n D N+k k Thus the support of ψ k is contained in an interva of ength D N+k k 0 D N+k k 0 = D N k 0 D k which is stricty ess than JD k with J = D N k The condition ii in Coroary.1 is triviay satisfied. On the other hand, given any n Z\{0} it foows from.9 that it is ony possibe that ψ k n 0if M og D n k + k0 N or M og D n k k0 N. This hods for at most N M + 1vauesofk, so condition iii in Coroary.1 is satisfied. Thus {T k ψ k} k 0,,...,D k 1 is a Besse sequence. In our third exampe, supp ψ k is the union of two disjoint sets of constant cardinaity that move farther apart as k increases. This provides the setup for constructing pairs of dua trigonometric frames which are hybrids of Gabor and waveet systems. Exampe.3. et g be a nontrivia, bounded, and rea-vaued function on R with support in an interva [M, N] for some M, N Z, M < N. With the index set I = N {0}, et{ } k 0 be a sequence of positive integers. We now fix some integer k 0 > 0 and define a famiy of trigonometric poynomias ψ k, k = 0, 1,...,oftheform.7 by 0, if k = 0,...,k 0 1, ψ k n := gn k, if k = k 0, gn k+k 0 +gn+k k 0 k, if k > k 0, for n Z. The assumption iii in Coroary.1 is satisfied with K = N M + 1. Due to the assumption that g is bounded, we can choose a constant C > 0 such that gx C for a x R; by the choice of ψ k n in.11 this impies that ψ k n C, k 0, n Z, i.e., condition ii in Coroary.1 is satisfied. Finay, note that if ony one of the terms gn k + k 0 and gn + k k 0 appeared in.11 fork > k 0, then the set S k woud be contained in a transate of the interva [M, N] of ength N M which is stricty ess than J with J = N M + 1. Thus, by Coroary.1 we can concude that if ψ k is modified to just contain one of these terms for k > k 0, then we have a Besse sequence. This means that the sequence {T k ψ k} k 0,,...,k 1 generated by our ψ k as defined in.11 can be considered as a sum of two Besse sequences, and therefore it is a Besse sequence itsef..11

6 30 O. Christensen, S.S. Goh / App. Comput. Harmon. Ana Dua pairs of trigonometric poynomia frames The purpose of this section is to provide a construction of pairs of dua trigonometric poynomia frames {T k ψ k},,...,k 1, {T k ψ k },,...,k 1 for 0, π. The approach is very genera and it can be taiored to give, among others, periodic Gabor frames and periodic waveet frames. In the entire section, we wi assume that the index set I is either I = N {0} or I = Z. Asin.3, we et S k denote the support of ψ k. Theorem 3.1. et {ψ k } be a coection of trigonometric poynomias with rea-vaued Fourier coefficients, and consider any sequence { } of positive integers. Assume that the foowing conditions are satisfied: i There exists a constant P N such that S k S k+ν = for ν Panda; ii The coection {T k ψ k},,...,k 1 forms a Besse sequence; iii For any n Z, 1 = ψ k n; iv For any k I, { ρ k := max n m n S k, m For k I, et ψ k be defined by ψ k n := ψ k n + k P 1 P 1 ν=0 S k+ν } <. k+ν ψ k+ν n, n Z. 3.1 If {T k ψ k },,...,k 1 is aso a Besse sequence, then the functions {T k ψ k},,...,k 1 and {T k ψ k },,...,k 1 form a pair of dua frames for 0, π. Proof. Fix n Z. It foows from assumption i that n S k can ony hod for finitey many k I. Choose k n as the smaest integer such that n S kn ; then, if n S k for some k, wehave k {k n,k n + 1,...,k n + P 1}. Using iii, a standard but rather invoved argument via induction on P shows that kn +P 1 1 = ψ k n k=k n [ = n ψ kn n ψ kn n + n +1 [ + n +1 ψ kn +1n kn ψ kn +1n + n + kn kn +P 1ψ kn +P 1n [ ψ kn +P 1n ]. ψ kn +1n + + n +P 1 kn ] ψ kn +P 1n ψ kn +n + + n +P 1 kn +1 ] ψ kn +P 1n Coecting the terms via finite sums and adding zeros eads to [ ] 1 = n ψ kn n ψ kn n + P 1 kn +ν ψ kn +νn kn [ ] P 1 + n +1 ψ kn +1n ψ kn +1n + kn +1+νψ kn +1+νn + kn +1 [ ] + P 1 kn +P 1ψ kn +P 1n ψ kn +P 1n + kn +P 1+νψ kn +P 1+νn. kn +P 1 For any k I, ψ k n is defined by 3.1; using that ψ k n = 0fork / {k n,...,k n + P 1}, the above cacuation shows that

7 O. Christensen, S.S. Goh / App. Comput. Harmon. Ana k n +P 1 1 = ψ k n ψ k n = ψ k n ψ k n. 3. k=k n Empoying the notation in A. and A.3 in the proof of Theorem.1 and a simiar one with h, T k ψ k = k 1 r=0 β k re π ir/, we have that for a trigonometric poynomias f, h, f, T k ψ k Then appying A.3 gives f, T k ψ k k 1 h, T ψ k k = = h, T k ψ k = k r=0 α k je π ij/ α k j β k r p Z r=0 β k re π ir/ e π i j r/ = f j + k p ψ k j + p q Z α k j β k j. ĥ j + q ψ k j + q = j + k p ψ k j + p ĥ j + p + q ψ k j + p + q p Z f q Z = n ψ k n ĥn + q ψ k n + q f q Z [ = f n ψ k nĥn ψ k n + ] f n ψ k nĥn + q ψ k n + q. 3.3 q Z\{0} In this expression the second term q Z\{0} f n ψ k nĥn + q ψ k n + q actuay vanishes for a k I. Ifn / S k this is trivia; and if n S k, then for any q Z \{0} we have n n + q = q, which by assumption iv impies that n + q / P 1 ν=0 S k+ν and thus n + q / supp ψ k. Hence, considering the sum over k I of the terms in 3.3 and using 3., we arrive at k 1 f, T k ψ k h, T ψ k k = n ψ k nĥn ψ k n = f nĥn = f, h. f From here, since both the coections {T k ψ k},,...,k 1 and {T ψ k k },,...,k 1 are Besse sequences, a standard duaity argument impies that they form a pair of dua frames for 0, π. Note that in order to appy Theorem 3.1, we need to check the assumptions i to iv as we as that the functions {T k ψ k },,...,k 1 form a Besse sequence. In the foowing coroaries, we provide severa ways of satisfying this, either in terms of conditions on the sequence { } or by appropriate conditions that impy i to iv and the Besse condition simutaneousy. Coroary 3.1. In the specia case where = D for some positive integer D, the assumptions i to iv in Theorem 3.1 produce a dua frame {T k ψ k },,...,D 1 from the functions ψ k defined by P 1 ψ k n := ψ k n + ψ k+ν n, n Z. 3.4 Proof. Note that, as a finite inear combination of Besse sequences, {T ψ k k },,...,D 1 resut foows from Theorem 3.1. is a Besse sequence. Thus the Coroary 3.. In the specia case where I = N {0} and = D k for some integer D, the assumptions i to iv in Theorem 3.1 yied a dua frame {T k ψ k } k 0,,...,D k 1 generated by the functions ψ k given by P 1 ψ k n := ψ k n + D ν ψ k+ν n, n Z. 3.5

8 3 O. Christensen, S.S. Goh / App. Comput. Harmon. Ana Proof. For ν = 1,...,P 1, observe that {T k ψ k+ν} k 0,,...,D k 1 is a Besse sequence. This is because D k 1 f, T k ψ D k 1 k+ν = f, Tk+ν Dν ψ k+ν k=0 k=0 D k+ν 1 f, Tk+ν ψ k+ν k=0 D κ 1 f, Tκ ψ κ, κ=0 D κ 1 = f, Tκ ψ κ κ=ν and {T k ψ k} k 0,,...,D k 1 is a Besse sequence. The rest of the proof is the same as in the proof of Coroary 3.1. Coroaries 3.1 and 3. dea with specia vaues of. For the genera case, our strategy is first to construct ψ k that satisfy the assumptions in Coroary.1 and then consider the extra assumptions in Theorem 3.1. As observed in the foowing coroary, for genera vaues of, it is possibe to impose a stronger condition in one of these assumptions and hereby ensure that {T k ψ k },,...,k 1 automaticay is a Besse sequence. Coroary 3.3. With {ψ k } and { } as in Theorem 3.1, suppose that the assumptions ii and iii in Coroary.1 and i and iii in Theorem 3.1 hod. In addition, assume that for any k I, { } P 1 σ k := max n m n,m S k+ν <. 3.6 ν=0 For ψ k defined by 3.1, the coections {T k ψ k},,...,k 1 and {T k ψ k },,...,k 1 form a pair of dua frames for 0, π. Proof. Note that the assumption 3.6 impies condition i in Coroary.1 as we as condition iv in Theorem 3.1. Then by Coroary.1, {T k ψ k},,...,k 1 is a Besse sequence. The resut wi foow from Theorem 3.1 once we estabish that {T k ψ k },,...,k 1 is aso a Besse sequence. To this end, we appy Coroary.1 to the functions { ψ k }.Indeed,for, wedefine Sk := supp ψ k.then3.1 impies that Sk P 1 ν=0 S k+ν, and by 3.6, Sk is contained in an interva of ength stricty ess than. Using condition ii in Coroary.1, weobtainfrom3.1 that ψ k n ψ k n + k P 1 k+ν ψ k+ν n P 1C k, n Z. Now, for a fixed n Z, condition iii in Coroary.1 impies that n S k ony for k {k 1, k,...,k μ }, where k 1,...,k μ I and μ K.Takeanyk {k 1,...,k μ }. By i in Theorem 3.1, S k P 1 ν=0 S κ+ν = for κ k + P or κ k P + 1. Thus S k intersects at most k + P 1 k P = 3P setsoftheform P 1 ν=0 S κ+ν.since Sκ P 1 ν=0 S κ+ν, it foows that S k intersects at most 3P ofthesets Sκ. This in turn shows that n ies in at most K 3P of the sets Sκ as μ K. Hence we concude from Coroary.1 that {T ψ k k },,...,k 1 is a Besse sequence. With these resuts in pace, we are ready to construct various genera casses of dua pairs of trigonometric poynomia frames. As the foowing exampes wi demonstrate, a key issue turns out to be various partition of unity conditions. Exampe 3.1. We continue the anaysis of the setup in Exampe.1 with some minor adjustments in order to adapt to the assumptions of Theorem 3.1. First, we further assume that k Z gk = 1. This can aways be achieved by mutipying g with a nonzero constant, provided that k Z gk 0. Note that this assumption impies that gx k = 1, x Z. 3.7 k Z Second, we aso assume that the number D and the ength of the interva [M, N] are reated by D > N M. 3.8 Checking conditions i to iv in Theorem 3.1, wenotethatfork Z and ν 1, S k {M + k,...,n + k} and S k+ν {M + k + ν,...,n + k + ν}. ThenS k S k+ν = if N + k < M + k + ν, i.e., if ν > N M. So i hods with P = N M + 1. Condition ii has aready been estabished in Exampe.1.

9 O. Christensen, S.S. Goh / App. Comput. Harmon. Ana Using.5 and 3.7, for every n Z, D ψ k n = gn k = 1, k Z k Z which is condition iii. In view of P 1 ν=0 S k+ν {M + k,...,n + k + P 1}, we see that ρ k in iv satisfies ρ k N + k + P 1 M + k = N M<D, where 3.8 gives the fina inequaity. Hence, we may appy Coroary 3.1 to construct a pair of dua periodic Gabor frames. In particuar, for ψ k defined from its Fourier coefficients ψ k as in.4, it foows from a cacuation via 3.4 and.5 that N M 1 gn + gn ν e inx = e ikx ψ 0 x. D ψ k x = e ikx So for k Z, = 0,...,D 1, using 1., T k ψ k x = e π ik/d e ikx ψ 0 x π D = e π ik/d e ikx T k ψ 0 x, 3.9 which aso has the Gabor structure. As.6 and 3.9 have the same constant factors e π ik/d, the frame expansion 1.4 reduces to the Gabor expansion f = D 1 f, e ik T k ψ 0 e ik T ψ k 0, k Z f 0, π. Note that our construction of dua Gabor frames for 0, π in Exampe 3.1 originates from the frequency domain, whie the approach for Gabor systems in R in [4] takes pace in the time domain. As the frequency domain for 0, π is the integers Z, we ony require condition 3.7, which, as we have discussed, can be satisfied after a simpe modification of the function g. On the other hand, the construction for R requires a partition of unity over the rea ine R that is more compicated to satisfy. Next, we use Coroary 3. to construct pairs of dua periodic waveet frames. Exampe 3.. We continue the anaysis of the functions ψ k in Exampe., with the extra assumption that g satisfies the partition of unity condition gx k = 1, x R. k Z We wi aso assume that k 0 is a positive integer satisfying k 0 > N + og D D N M + 1. The reason for this choice of k 0 wi be reveaed ater in the exampe. Note that in contrast to the situation in Exampe 3.1, see 3.7, we now need the partition of unity to hod for a x R. This is more restrictive, but it is satisfied, e.g., for any B-spine or any scaing function. We wi verify conditions i to iv in Theorem 3.1 and then appy Coroary 3.. First note that.10 shows that ψ k might consist of two bumps on the positive axis and two bumps on the negative axis. If N k + k 0 < 0, i.e., if k > N + k 0, there wi be ony one bump on each of the positive axis and the negative axis. We now check condition i in Theorem 3.1. To this end, et denote the foor function, and et P = N M + 1. If k > k 0,thenforν P > N M, wehaved N k+ν+k 0 < D M k+k 0 and D N+k k 0 < D M+k+ν k 0. Appying both.10 directy as we as.10 withk + ν in pace of k, we see that S k S k+ν =. In addition, by.8, ψ k0 n 0onyifD M n D N. For ν P > N M, wehaved N ν < D M and D N < D M+ν, which show that S k0 S k0 +ν =.AsS k = for k = 0,...,k 0 1, condition i in Theorem 3.1 hods for a k 0. Condition ii in Theorem 3.1 has aready been verified in Exampe.. Using that og D n D κ = og D n κ, the definition of the functions ψ k shows that for n 0, D k ψ k n = g og D n + g ogd n + k k0 + g og D n k k k=0 k=k 0 +1 = κ Z g og D n κ = 1. Thus condition iii in Theorem 3.1 is satisfied for n 0. Ceary, it is satisfied for n = 0 as we.

10 34 O. Christensen, S.S. Goh / App. Comput. Harmon. Ana In order to verify condition iv in Theorem 3.1, we note that the argest number in P 1 ν=0 S k+ν is not more than D N+k+P 1 k 0 = D N+k+ N M+1 1 k 0 D N+k+N M k 0 = D N M+k k 0. The minima number in S k is not ess than D N+k k 0.Thus ρ k D N M+k k 0 D N+k k 0 = D k D N M k 0 + D N k 0. For condition iv in Theorem 3.1 to hod, it suffices to have D N M k 0 + D N k 0 < 1 which is equivaent to 3.10, as assumed in this exampe. Hence, it foows from Coroary 3. that {T k ψ k} k 0,,...,D k 1 and {T ψ k k } k 0,,...,D k 1 form a pair of dua waveet frames for 0, π. Concrete constructions based on the above can easiy be reaized by taking the function g to be a centered B-spine supported on [ N, N]. As an iustration, we set 1 x + 3 x + 9 8, if 3 x 1, x gx := B 3 x =, if 1 x 1, 1 x 3 x + 9 8, if 1 x 3, 0, otherwise. Then g is nonnegative, bounded, rea-vaued, and supported on [ 3, 3 ], i.e., M = 3 and N = 3. We take the diation factor D to be. Thus we can et P = N M + 1 =4, and take k 0 > N + og N M + 1 = 3 + og 9, e.g., k 0 = 5. With this choice, ψ k ony has one bump on each of the positive and negative axes if k > N + k 0 = 13. Fig. 1 showsthepotsofψ k and ψ k defined by 3.11 and.8 ford = andk = 8, and the corresponding ψ k and ψ k from 3.5. Note that Exampe 3. shows that Coroary 3. can be used to construct waveet frames {T k ψ k} k 0,,...,D k 1 for 0, π with ony one generator ψ k for each eve k. This is in contrast to the approaches based on mutiresoution anayses in [13,14]: typicay, for a diation factor D, a periodic waveet frame constructed from a mutiresoution anaysis woud have at east D 1 generators at each eve. Returning to more genera vaues of, we now empoy Coroary 3.3 to obtain expicit pairs of periodic frames that are hybrids of Gabor and waveet types. Exampe 3.3. We continue further the setup in Exampe.3 with a few minor modifications. As in Exampe 3.1, if k Z gk 0, by mutipying g with a nonzero constant, we can ensure that 3.7 hods. In addition, we assume that for an appropriate integer k 0 > 0, 3N M + k <, k k This is possibe, e.g., if we assume that k as k. We now verify the conditions in Coroary 3.3. As conditions ii and iii in Coroary.1 have aready been verified in Exampe.3, to appy Coroary 3.3, it remains to check conditions i and iii in Theorem 3.1 and the inequaity 3.6. Note that supp ψ k0 {M,...,N}, andfork > k 0, supp ψ k { M k k 0,...,N k k 0 } {M + k k 0,...,N + k k 0 } { } M k k 0,...,N + k k It foows that for ν 1, S k+ν { M k + ν k 0,...,N k + ν k 0 } {M + k + ν k 0,...,N + k + ν k 0 }. Consequenty, S k S k+ν = if N + k k 0 < M + k + ν k 0 and N k + ν k 0 <M k k 0, i.e., if ν N M + 1. So condition i in Theorem 3.1 is satisfied with P = N M + 1. Aso, for each n Z, ψ k n = gn + gn k + k0 + gn + k k 0 = gn κ = 1, k=0 k=k 0 +1 κ Z where 3.7 is appied in the ast equaity. Thus condition iii in Theorem 3.1 is satisfied. In order to check 3.6, we note that the argest eement in P 1 ν=0 S k+ν is at most N + k + N M k 0 and that the minima eement in P 1 ν=0 S k+ν is at east M k + N M k 0.Thus,forσ k in 3.6, we have σ k N + k + N M k 0 M k + N M k 0 = 3N M + k k 0 3N M + k. As a resut, choosing k 0 such that 3.1 hods, the inequaity 3.6 is satisfied. Hence, by Coroary 3.3, this setup eads to pairs of dua periodic frames that are hybrids of Gabor frames and waveet frames. 3.11

11 O. Christensen, S.S. Goh / App. Comput. Harmon. Ana Fig. 1. Pots of ψ k top eft and ψ k top right defined by 3.11 and.8 ford =, k 0 = 5andk = 8, and the corresponding ψ k bottom eft and ψ k bottomrightfrom3.5 asinexampe3.. In a specia case the setup in Exampe 3.3 can be taiored to a construction of rea-vaued and symmetric frame generators ψ k. Exampe 3.4. Empoying the setup in Exampe 3.3, we impose the additiona assumptions that g is continuous, symmetric, and supported on [ N, N] for some N 1. This is in ine with our aim of constructing frame generators ψ k that are rea-vaued and symmetric, which amounts to the requirement that the functions ψ k are rea-vaued and symmetric on Z. For a fixed positive integer k 0, defining ψ k n as in.11, we now check that for every k it hods that ψ k n is reavaued and that ψ k n = ψ k n for a n Z. We note that for k = k 0, these statements are obvious because g x = gx. Next, consider any k > k 0. Then, for any n Z, ψ k n = g n k + k 0 + g n + k k 0 = gn + k k 0 + gn k + k 0 = ψ k n. k k Since g is continuous and supported on [ N, N], wehaveg N = gn = 0. Thus supp ψ k0 { N + 1,...,N 1}, and it foows from 3.13 that for k > k 0,

12 36 O. Christensen, S.S. Goh / App. Comput. Harmon. Ana supp ψ k { N + 1 k k 0,...,N 1 + k k 0 }. Consequenty, we can take P in Theorem 3.1 as P = N 1 N = N 1. As an iustration of this construction, for any positive integer N, wetake { tan π 4N gx := cos π x, if N x N, N 0, otherwise Then g is nonnegative, continuous, rea-vaued, symmetric, and supported on [ N, N]. By a straightforward cacuation, N k= N cos kπ N = N 1 k= N+1 cos kπ N = cot π 4N, which shows that k Z gk = 1. Thus, 3.7 hods. For the sequence {} k 0,weput = k. Then we can find an appropriate k 0 for which 6N 1 + k < k, k k This gives 3.1. Based on the function g in 3.14, the trigonometric poynomia ψ k obtained from.11 achieves optima timeocaization in the foowing sense. In [1], time-ocaization of a function f in 0, π with f n f n is measured by its anguar variance defined by θ f = f n f n f n + 1 f n f n + 1 It is shown in [10,11] that among a h 0, π with supp ĥ ={ N + 1,...,N 1}, θ h θ ψ k0, i.e., the minimum anguar variance is attained by ψ k0. Using this fact and the definition 3.16, a cacuation then shows that for each k k 0 + N, ψ k gives the minimum anguar variance among a f whose Fourier coefficients are of the form f n = ĥn k + k 0 + ĥn + k k 0, n Z, where h 0, π with supp ĥ ={ N + 1,...,N 1}. Taking N = 5, 3.15 hodsforak 7, so we choose k 0 = 7. Fig. showsthepotsofψ k and ψ k defined by 3.14 and.11 forn = 5 and k = 1, and the corresponding ψ k and ψ k from 3.1. We end the paper with a comment on appication of the constructed dua periodic frames to practica probems. Certain appications invove threshoding, such as during denoising and deconvoution, or require visuaization of the time frequency pane. In these instances, the frame coefficients f, T k ψ k given by 1.5 have to be compared over a k I and = 0,..., 1. However, our periodic frames {T k ψ k},,...,k 1 are nonstationary; in particuar, the norm T k ψ k, which equas ψ k, changes as k varies. Thus, in order to obtain a meaningfu anaysis of f, the frame coefficients f, T k ψ k need to be normaized during the processing. The caibration can be achieved via dividing f, T k ψ k by ψ k. Note that this is just a practica measure for processing in appications. After the required anaysis, we woud sti appy 1.4 tothe origina coefficients f, T k ψ k for the synthesis of f. Acknowedgments The first-named author wants to thank the Waveets and Information Processing Programme at the Nationa University of Singapore for support and hospitaity during visits in 008 and 010. The second-named author thanks the Department of Mathematics at the Technica University of Denmark for support and hospitaity during a visit in 008. Both authors thank the referees for usefu suggestions. Appendix A. Proof of Theorem.1 To show that {T k ψ k},,...,k 1 isabessesequencewithboundb, it suffices to estabish that the inequaity k 1 f, T k ψ k B f

13 O. Christensen, S.S. Goh / App. Comput. Harmon. Ana Fig.. Pots of ψ k top eft and ψ k top right defined by 3.14 and.11 for = k, N = 5, k 0 = 7andk = 1, and the corresponding ψ k bottom eft and ψ k bottomrightfrom3.1 asinexampe3.4. hods for a f in a dense subspace of 0, π; we wi consider the subspace formed by a trigonometric poynomias. By the same arguments as in the first part of the proof of [1, Theorem 4.1], k 1 f, T k ψ k = f j + k p ψ k j + p. A.1 p Z Indeed, as we have seen aready in 1.5, we can write f, T k ψ k = f n ψ k ne π in/ = α k je π ij/, A. where α k is the -periodic sequence defined by

14 38 O. Christensen, S.S. Goh / App. Comput. Harmon. Ana α k j := p Z f j + k p ψ k j + p, j = 0,..., 1; A.3 now the expression in A.1 is obtained by appying the inverse finite Fourier transform to α k, foowed by an appication of Parseva s identity. Now it foows from A.1 that k 1 f, T k ψ k 1 k = j + k p ψ k j + p f j + k q ψ k j + q p Z f q Z = p Z f j + k p ψ k j + p q Z f j + k q ψ k j + q. A.4 Note that with the exception of the sum over k, a the above summations are finite sums as f is a trigonometric poynomia. The -periodicity of α k impies that for every j = 0,..., 1 and p Z, α k j + p = α k j = q Z f j + k q ψ k j + q; thus A.4 can be rewritten as k 1 f, T k ψ k = p Z = f j + k p ψ k j + p q Z f n ψ k n q Z f n + k q ψ k n + q. f j + k p + q ψ k j + k p + q We sha now interchange the infinite sum over k with some of the other sums in A.5. This can be justified by repacing a the terms in A.5 by their absoute vaues and foowing the arguments beow. Interchanging the sums in A.5, we have A.5 k 1 f, T k ψ k = f n ψ k n f n + k q ψ k n + q q Z = f n ψ k n f n ψ k n + = f n ψ k n + R, q Z\{0} f n ψ k n f n + k q ψ k n + q A.6 where R := f n ψ k n f n + k q ψ k n + q. q Z\{0} Appying the Cauchy Schwarz inequaity twice, we have R f n f n + k q ψ k n ψ k n + q q Z\{0} f n ψ k n ψ k n + q 1/ f n + k q ψ k n ψ k n + q 1/ q Z\{0} f n ψ k n ψ k n + q 1/ f n + k q ψ k n ψ k n + q 1/. A.7 q Z\{0} q Z\{0} Observethatbythesubstitutionm = n + q for each fixed q Z\{0}, f n + k q ψ k n ψ k n + q = f m ψ k m q ψ k m q Z\{0} q Z\{0} m Z = q Z\{0} m Z f m ψ k m ψ k m + q ;

15 O. Christensen, S.S. Goh / App. Comput. Harmon. Ana therefore A.7 impies that R f n ψ k n ψ k n + q. q Z\{0} A.8 Appying this estimate to A.6, we obtain k 1 f, T k ψ k f n ψ k n + f n = f n ψ k n ψ k n + q. q Z Hence, it foows from.1 that k 1 f, T k ψ k B f n = B f, q Z\{0} proving that {T k ψ k},,...,k 1 is a Besse sequence with bound B. If. aso hods, et again f be a trigonometric poynomia. Then by A.6 and A.8, ψ k n ψ k n + q k 1 f, T k ψ k f n ψ k n f n ψ k n ψ k n + q q Z\{0} = f n ψ k n ψ k n ψ k n + q q Z\{0} A f n = A f. Since this hods for a trigonometric poynomias f, we concude that {T k ψ k},,...,k 1 is a frame for 0, π with bounds A, B. References [1] E. Breitenberger, Uncertainty measures and uncertainty reations for ange observabes, Found. Phys [] P.G. Casazza, O. Christensen, Wey Heisenberg frames for subspaces of R, Proc. Amer. Math. Soc [3] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, 003. [4] O. Christensen, Pairs of dua Gabor frames with compact support and desired frequency ocaization, App. Comput. Harmon. Ana [5] O. Christensen, S.S. Goh, Pairs of obique duas in spaces of periodic functions, Adv. Comput. Math [6] O. Christensen, R.Y. Kim, On dua Gabor frame pairs generated by poynomias, J. Fourier Ana. App [7] I. Daubechies, Ten ectures on Waveets, SIAM, Phiadephia, 199. [8] H.G. Feichtinger, T. Strohmer Eds., Gabor Anaysis and Agorithms: Theory and Appications, Birkhäuser, Boston, [9] H.G. Feichtinger, T. Strohmer Eds., Advances in Gabor Anaysis, Birkhäuser, Boston, 00. [10] S.S. Goh, T.N.T. Goodman, Inequaities on time-concentrated or frequency-concentrated functions, Adv. Comput. Math [11] S.S. Goh, C.A. Micchei, Uncertainty principes in Hibert spaces, J. Fourier Ana. App [1] S.S. Goh, K.M. Teo, An agorithm for constructing mutidimensiona biorthogona periodic mutiwaveets, Proc. Edinb. Math. Soc [13] S.S. Goh, K.M. Teo, Waveet frames and shift-invariant subspaces of periodic functions, App. Comput. Harmon. Ana [14] S.S. Goh, K.M. Teo, Extension principes for tight waveet frames of periodic functions, App. Comput. Harmon. Ana [15] K. Gröchenig, Foundations of Time Frequency Anaysis, Birkhäuser, Boston, 001. [16] I. Kim, Gabor frames with trigonometric spine windows, PhD thesis, University of Iinois at Urbana Champaign, 011. [17] R.S. augesen, Gabor dua spine windows, App. Comput. Harmon. Ana [18] E.A. ee, P. Varaiya, Structure and Interpretation of Signas and Systems, Addison Wesey, Boston, 003. [19] J. emvig, Constructing pairs of dua bandimited frameets with desired time ocaization, Adv. Comput. Math [0] S. Maat, A Waveet Tour of Signa Processing, second ed., Academic Press, San Diego, 1999.