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Reproducing formulas for generalized translation invariantsystems on locally compact abelian groups

Mads Sielemann Jakobsen, Jakob Lemvig

February 27, 2015

Abstract: In this paper we connect the well established discrete frame theoryof generalized shift invariant systems to a continuous frame theory. To do so, welet j , j J , be a countable family of closed, co-compact subgroups of a sec-ond countable locally compact abelian group G and study systems of the formjJ{gj,p( )}j ,pPj with generators gj,p in L2(G) and with each Pj beinga countable or an uncountable index set. We refer to systems of this form asgeneralized translation invariant (GTI) systems. Many of the familiar transforms,e.g., the wavelet, shearlet and Gabor transform, both their discrete and continu-ous variants, are GTI systems. Under a technical local integrability condition(-LIC) we characterize when GTI systems constitute tight and dual frames thatyield reproducing formulas for L2(G). This generalizes results on generalized shiftinvariant systems, where each Pj is assumed to be countable and each j is a uni-form lattice in G, to the case of uncountably many generators and (not necessarilydiscrete) closed, co-compact subgroups. Furthermore, even in the case of uniformlattices j , our characterizations improve known results since the class of GTIsystems satisfying the -LIC is strictly larger than the class of GTI systems sat-isfying the previously used local integrability condition. As an application of ourcharacterization results, we obtain new characterizations of translation invariantcontinuous frames and Gabor frames for L2(G). In addition, we will see that theadmissibility conditions for the continuous and discrete wavelet and Gabor trans-form in L2(Rn) are special cases of the same general characterizing equations.

1 IntroductionIn harmonic analysis one is often interested in determining conditions on generators of functionsystems, e.g., Gabor and wavelet systems, that allow for reconstruction of any function in agiven class of functions from its associated transform via a reproducing formula. The work ofHernndez, Labate, andWeiss [30] and of Ron and Shen [46] on generalized shift invariant systemsin L2(Rn) presented a unified theory for many of the familiar discrete transforms, most notablythe Gabor and the wavelet transform. The generalized shift invariant systems are collections offunctions of the form jJ {Tgj}j , where J is a countable index set, T denotes translationby , j a full-rank lattice in Rn, and {gj}jJ a subset of L2(Rn). Here, the word shift is

2010 Mathematics Subject Classification. Primary: 42C15, 43A32, 43A70, Secondary: 43A60, 46C05.Key words and phrases. continuous frame, dual frames, dual generators, g-frame, Gabor frame, generalized

shift invariant system, generalized translation invariant system, LCA group, Parseval frame, wavelet frameTechnical University of Denmark, Department of Applied Mathematics and Computer Science, Matematik-

torvet 303B, 2800 Kgs. Lyngby, Denmark, E-mail: msja@dtu.dkTechnical University of Denmark, Department of Applied Mathematics and Computer Science, Matematik-

torvet 303B, 2800 Kgs. Lyngby, Denmark, E-mail: jakle@dtu.dk

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msja@dtu.dkjakle@dtu.dk

Jakobsen, Lemvig Reproducing formulas for GTI systems on LCA groups

used since the translations are discrete and the word generalized since the shift lattices jare allowed to change with the parameter j J . The main result of Hernndez, Labate, andWeiss [30] is a characterization, by so-called t-equations, of all functions gj that give rise toisometric transforms, called Parseval frames in frame theory.

The goal of this work is to connect the discrete transform theory of generalized shift invari-ant systems to a continuous/integral transform theory. In doing so, the scope of the unifiedapproach started in [30, 46] will be vastly extended. What more is, this new theory will coverintermediate steps, the semi-continuous transforms, and we will do so in a very general set-ting of square integrable functions on locally compact abelian groups. In particular, we recoverthe usual characterization results for discrete and continuous Gabor and wavelet systems asspecial cases. For discrete wavelets in L2(R) with dyadic dilation, this result was obtained in1995, independently by Gripenberg [23] and Wang [48], and it can be stated as follows. Definethe translation operator Tbf(x) = f(x b) and dilation operator Daf(x) = |a|1/2 f(x/a) forb R, a 6= 0. The discrete wavelet system {T2jkD2j}j,kZ generated by L2(R) is indeeda generalized shift invariant system with J = Z, j = 2jZ, and gj = D2j. Now, the linearoperator Wd defined by

Wd : L2(R) `2(Z2), Wdf(j, k) = f, T2jkD2j

is isometric if, and only if, for all jZ 2

jZ, the following t-equations hold:

t :=

jZ :2jZ

(2j)(2j( + )) = ,0 for a.e. R, (1.1)

where R denotes the Fourier domain. In the language of frame theory, we say that generators L2(R) of discrete Parseval wavelet frames have been characterized by t-equations.

Caldern [6] discovered in 1964 that any function L2(R) satisfying the Caldern admis-sibility condition

R\{0}

|(a)|2

|a|da = 1 for a.e. R (1.2)

leads to reproducing formulas for the continuous wavelet transform. To be precise, the linearoperator Wc defined by

Wc : L2(R) L2

(R\{0} R, dadb

a2

), Wcf(a, b) = f, TbDa

is isometric if, and only if, the Caldern admissibility condition holds. We will see that theCaldern admissibility condition is nothing but the t-equation (there is only one!) for thecontinuous wavelet system. Similar results hold for the Gabor case; here the continuous transformis usually called the short-time Fourier transform. Actually, the theory is not only applicableto the Gabor and wavelet setting, but to a very large class of systems of functions includingshearlet and wave packet systems, which we shall call generalized translation invariant systems.We refer the reader to the classical texts [12,14,27] and the recent book [38] for introductions tothe specific cases of Gabor, wavelet, shearlet and wave packet analysis.

In [36], Kutyniok and Labate generalized the results of Hernndez, Labate, and Weiss togeneralized shift invariant systems jJ {Tgj}j in L

2(G), where G is a second countablelocally compact abelian group and j is a family of uniform lattices (i.e., j is a discrete subgroupand the quotient group G/j is compact) indexed by a countable set J . The main goal of thepresent paper is to develop the corresponding theory for semi-continuous and continuous framesin L2(G). In order to achieve this, we will allow non-discrete translation groups j , and we

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Jakobsen, Lemvig Reproducing formulas for GTI systems on LCA groups

will allow for each translation group to have uncountable many generators, indexed by someindex set Pj , j J . We say that the corresponding family jJ {Tgj,p}j ,pPj in L

2(G) is ageneralized translation invariant system. To be precise, we will, for each j J , take Pj to bea -finite measure space with measure Pj and j to be closed, co-compact (i.e., the quotientgroup G/j is compact) subgroups. We mention that any locally compact abelian group has aco-compact subgroup, namely the group itself. On the other hand, there exist groups that do notcontain uniform lattices, e.g., the p-adic numbers. Thus, the theory of generalized translationinvariant systems is applicable to a larger class of locally compact abelian groups than the theoryof generalized shift invariant systems.

The two wavelet cases described above fit our framework. The discrete wavelet system canbe written as jZ{T(D2j)}2jZ, so we see that Pj is a singleton and Pj a weighted countingmeasure for each j J = Z, and that there are countably many different (discrete) j . For thecontinuous wavelet system on the form {T(Dp)}R,pR\{0}, we have that J is a singleton,e.g., {j0} since there is only one translation subgroup j0 = R. On the other hand, here Pj0is uncountable and Pj0 a weighted Lebesgue measure. We stress that our setup can handlecountable many (distinct) j and countable many Pj , each being uncountable.

The characterization results in [30, 36] rely on a technical condition on the generators andthe translation lattices, the so-called local integrability condition. This condition is straightfor-ward to formulate for generalized translation invariant systems, however, we will replace it by astrictly weaker condition, termed local integrability condition. Therefore, even for generalizedshift invariant systems in the euclidean setting, our work extends the characterization resultsby Hernndez, Labate, and Weiss [30]. Under the local integrability condition, we show inTheorem 3.5 that jJ {Tgj,p}j ,pPj is a Parseval frame for L

2(G), that is, the associatedtransform is isometric if, and only if,

t :=

jJ :j

Pj

gj,p()gj,p( + ) dPj (p) = ,0 a.e. G

for every jJj , where j ={ G : (x) = 0 for all x j

}denotes the annihilator of

j . Now, returning to the two main examples of this introduction, the discrete and continuouswavelet transform, we see why the number of the t-equations in (1.1) and (1.2) are so different.In the discrete case the corresponding union of the annihilators of the translation groups isjZ2jZ, while in the continuous case the annihilator of R is simply {0}, which corresponds toonly one t-equation ( = 0).

Finally, as Kutyniok and Labate [36] restrict their attention to Parseval frames, there arecurrently no characterization results available for dual (discrete) frames in the setting of locallycompact abelian groups. Hence, one additional objective of this paper is to prove characterizingequations for dual generalized translati