The dilation property for abstract Parseval wavelet systems

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1 The dilation property for abstract Parseval wavelet systems Bradley Currey and Azita Mayeli October 31, 2011 Abstract In this work we introduce a class of discrete groups called wavelet groups that are generated by a discrete group Γ 0 translations) and a cyclic group Γ 1 dilations), and whose unitary representations naturally give rise to a wide variety of wavelet systems generated by the pseudo-lattice Γ = Γ 1Γ 0. We prove a condition in order that a Parseval frame wavelet system generated by Γ can be dilated to an orthonormal basis that is also generated by Γ via a super-representation. For a subclass of groups where Γ 0 is Heisenberg, we show that this condition always holds, and we cite a number of familiar examples as applications. 1 Introduction Let G be a countable discrete group; by representation of G we shall mean a strongoperator continuous homomorphism π of G into the group of unitary operators on a given Hilbert space H π. Given a representation π of G, a countable subset Γ of G is a Parseval sampling set for π if there is ψ H π such that for any f H π f 2 = γ Γ f, πγ)ψ 2 1.1) A vector ψ is a Parseval frame vector for πγ) if 1.1) holds, and ψ is a wandering vector if πγ)ψ is an orthonormal basis for H π. A representation π is a superrepresentation of π if H π is a closed π-invariant subspace of H π and πx) = πx) Hπ holds for all x G. Definition 1.1. Let Γ be a Parseval sampling set for π with Parseval frame vector ψ. The system πγ)ψ has the G-dilation property if there is a super-representation π of π and a vector ψ H π such that πγ) ψ is an orthonormal basis for H π whose projection is πγ)ψ. In this case we say that ψ is a G-dilation of ψ. 1

2 2 We consider in this paper the following class of groups. Let Γ 0 be a discrete group and α : Γ 0 Γ 0 be a monomomorphism. The wavelet group associated with Γ 0 and α is the group G generated by Γ 0 and the infinite cyclic group Γ 1 = {u j : j Z}: G = Gα, Γ 0 ) := u, Γ 0 : uγu 1 = αγ), γ Γ 0. The subset Γ = Γ 1 Γ 0 where Γ 1 = {u j : j Z} will be called the standard pseudolattice in G. Representations of wavelet groups arise in a number of settings where it is natural to consider wavelet systems. The combination of integer translations and 2-dilations on the line is naturally associated with the Baumslag-Solitar group BS1, 2), where Γ 0 = Z and αk) = 2k. More generally, if N is any homogeneous group with dilations and Γ 0 is a lattice in N, one can pick a dilation α that determines a monomorphism of Γ 0, and consider wavelet systems in L 2 N) consisting of translations by elements of Γ 0 and unitary dilations D α on L 2 N). We are interested in the following two questions: Question 1.2. Let G be a wavelet group, π a representation of G, and ψ H such that πγ)ψ is a Parseval frame. When does πγ)ψ have the G-dilation property? Question 1.3. If the answer to the preceding question is yes, then what is the precise structure of a super-representation and a super-wavelet? Both questions have been studied in [5] when πγ)ψ is a standard 2-wavelet system G = BS1, 2).) Section 2 of this paper is based upon a generalization of methods from [5], using positive-definite maps to obtain a representation theoretic sufficient condition on the group G in order that every Parseval frame wavelet system πγ)ψ has the G-dilation property. In Section 3 we prove that our condition holds for several classes of wavelet groups. Finally, we describe several examples. 2 The G-dilation property. A map K : X X C is called a positive definite map if for all finite sequences {γ 1, γ 2,..., γ k } in X and {ξ 1, ξ 2,..., ξ k } C, Kγ i, γ j )ξ i ξ j 0. 1 i,j k If X = G is a group, then following [1], we say that K : G G C is a group positive definite map if K is a positive definite map and Ksx, sy) = Kx, y) holds for all s, x and y in G. By [1, Theorem 2.8], every group positive definite map has the form K ρ,η x, y) = ρx)η, ρy)η

3 3 where ρ is a representation of G and η is a cyclic vector for ρ. Let Γ 0 be a discrete group, α a monomorphism of Γ 0, and ρ a representation of Γ 0. We say that a representation T is an α-root of ρ if T α = ρ. This notion naturally arises in the problem of the extension of a positive definite map on a subset of a group G, to a group positive definite map on G. In the following abstract version of [5, Theorem 2.1], we use this notion to formulate a sufficient condition for such an extension to exist in the case of wavelet groups. Proposition 2.1. Let G be a wavelet group, Γ = Γ 1 Γ 0 the standard pseudo-lattice in G and suppose that every representation of Γ 0 has an α-root. Let K : Γ Γ C be a positive definite mapping such that for any j, j Γ 1, γ, γ Γ 0, the relations Kj + 1, γ), j + 1, γ )) = Kj, γ), j, γ )), γ 0 Γ 0, Kj, α j γ 0 )γ), j, α j γ 0 )γ )) = Kj, γ), j, γ )) for, γ 0 Γ 0, j 0, 2.1) both hold. Then K is the restriction of a group positive definite map K τ,ψ. More explicitly there exists a Hilbert space H, and unitary operators D and representation T of Γ 0 on with DT γ)d 1 = T αγ), and a vector ψ H such that H = span{d j T γ)ψ} and Kj, γ), j, γ )) = D j T γ)ψ, D j T γ )ψ. Proof. By a theorem attributed to Kolmogorov see for example [1]), we have a Hilbert space H and a mapping v : Γ H, such that H = span{vj, γ) : j, γ) Γ}, and Kj, γ), j, γ )) = vj, γ), vj, γ ) holds for all j, γ) and j, γ ) belonging to Γ. Define the operator D : H H by Dvj, γ) = vj +1, γ) and by extending to all of H by linearity and density as usual. The first of the relations 2.1) shows that D is unitary. For each n = 1, 0, 1, 2,..., set H n = span{vj, γ) : j, γ) Γ, j n}. Note that DH n = H n+1 and H n H n+1. Set K n = H n H n 1, n 0. For γ 0 Γ 0, define the operator T 0 γ 0 ) on H 0 by T 0 γ 0 ) vj, γ) ) = vj, α j γ 0 )γ) and again extending to all of H 0 ; the second relation in 2.1) shows that γ T 0 γ) is a unitary) representation of Γ 0. Since the subspace K 0 is invariant under T 0, we can define the representation ρ 1 of Γ 0 acting in K 1 by ρ 1 γ) = DT 0 γ)d 1. Now by our hypothesis, ρ 1 has an α-root T 1 : T 1 acts in K 1 and satisfies T 1 α = ρ 1.

4 4 Now the representation γ ρ 2 γ) = DT 1 γ)d 1 of Γ 0 acting in K 2 has an α-root T 2 acting in K 2. Continuing in this way, we obtain, for each positive integer n, a representation T n of Γ 0 acting in K n, so that T n α = DT n 1 D 1. Again in the preceding, T 0 is restricted to K 0.) Now write H = H 0 n 1 K n ) and define the representation T of Γ 0 by T = T 0 n 1 T n. Next we must verify the relation DT γ)d 1 = T αγ) ). Fix γ 0 Γ 0 ; for vj, γ) with j 0, DT0 γ 0 )D 1) vj, γ) ) = DT 0 γ 0 ) ) vj 1, γ) ) = D vj 1, α j+1 γ 0 )γ) ) = T 0 αγ0 ) ) vj, γ) ) and hence the relation DT 0 γ)d 1 = T 0 αγ) ) holds on H0. Now for v H, write v = n 0 v n. We have DT γ)d 1 v 0 = T αγ) ) v 0 and for n 1, DT γ)d 1 v n = DT n 1 γ)d 1 v n = T n αγ) ) vn so DT γ)d 1 v = n 0 DT γ)d 1 v n = n 0 T n αγ) ) vn = T αγ) ). It follows that the mapping τ defined by τu) = D and τγ) = T γ) is a representation of G. Finally, take ψ = v0, 0). Then vj, γ) = D j v0, γ) = D j T γ)v0, 0) = D j T γ)ψ so ψ is cyclic for τ. Hence the group positive definite map defined for all x, y G by K τ,ψ x, y) = τx)ψ, τy)ψ is an extension of K. We combine the preceding with general results also from [5] to obtain our condition for the G-dilation property. Theorem 2.2. Let G be a wavelet group, Γ = Γ 1 Γ 0 the standard pseudo-lattice in G and suppose that every representation of Γ 0 has an α-root. Let π be any representation of G. Then every Parseval frame wavelet system πγ)ψ has the G-dilation property. Proof. Let Γ = Γ 1 Γ 0 G as above and recall that we write u j γ = j, γ). Define Kj, γ), j, γ )) = δ j,j δ γ,γ πj, γ)ψ, πj, γ )ψ.

5 5 Observe that δ j+1,j +1 = δ j,j and δ j,j δ a j γ 0)γ,a j γ 0)γ = δ j,j δ γ,γ, and that in the group G, u j γ 0 u j = a j γ 0 holds for j, γ 0 Γ 0. Hence Kj + 1, γ), j + 1, γ )) = δ j+1,j +1δ γ,γ πj + 1, γ)ψ, πj + 1, γ )ψ and for j 0, Kj, α j γ 0 )γ),j, α j γ 0 )γ )) = δ j,j δ γ,γ Dπj, γ)ψ, Dπj, γ )ψ = Kj, γ), j, γ )) = δ j,j δ α j γ 0)γ,α j γ 0)γ πjα j γ 0 )γ)ψ, πj α j γ 0 )γ )ψ = δ j,j δ γ,γ πγ 0 jγ)ψ, πγ 0 j γ )ψ = δ j,j δ γ,γ πγ 0 )πjγ)ψ, πγ 0 )πj γ )ψ = δ j,j δ γ,γ πjγ)ψ, πj γ )ψ = Kj, γ), j, γ )). The calculations show that the map K satisfies the both conditions 2.1). By Proposition 2.1 we can conclude that K is a positive definite map and hence there exists a representation τ of G with Hilbert space K and η K such that K = K τ,η on Γ Γ. Then [5, Lemma 2.5, proof of Theorem 2.6] π τ is a super-representation of π acting in H K) for which ψ = ψ η is a G-dilation vector for ψ and πx)ψ = πx)ψ. We illustrate the preceding with the example G = BS1, 2). Here Γ 0 is the infinite cyclic group {t k }, and αt) = t 2. Here the α-root property is a simple consequence of the Borel functional calculus: for every unitary operator T on a Hilbert space H, there is a unitary operator S such that S 2 = T. Let π be the representation of G acting in L 2 R) by πu)φx) = φ2 1 x)2 1/2 and πt)φx) = φx 1). With the standard pseudo-lattice {u j t k : j, k Z}, the above theorem yields that every Parseval wavelet frame of the form πγ)ψ has the G-dilation property. See [5] for further analysis of dilations in this case. Observe that in the above theorem, the pivotal condition concerns only the structure of the group Gα, Γ 0 ): that every representation of Γ 0 has an α-root. If this condition holds then we say that G has the α-root property. Of course, the α-root property is a strong one and it is far from clear how to give a general answer to the following.

6 6 Question 2.3. For what groups Gα, Γ 0 ) does the α-root property hold? In the following section we describe two families of groups Gα, Γ 0 ) for which the α-root property does in fact hold. 3 Examples In this section we list several exampes of wavelet groups that possess the α-root property. We also give examples of representations π for which Parseval frames of the form πγ)ψ are known to exist. We begin with the case where Γ 0 is a finitely-generated, torsion-free abelian group. Example 3.1. A-wavelet system) Let Γ 0 be the free abelian group generated by t 1, t 2,, t n, and let αt j ) = t a1j 1 t a2j 2 t anj n where A = [a i,j ] GLn, Z). We claim that the wavelet group Gα, Γ 0 ) has the α-root property. Let ρ be any representation of Γ 0, and write A 1 = [b i,j ]. Since the b i,j are rational, the Borel functional calculus obtains operators V i,j, 1 i, j n such that V i,j = ρt 1 ) bi,j. Define T t j ), 1 j n by T t j ) = V 1,j V 2,j V n,j. An easy computation shows that T α = ρ. Next we consider wavelet groups where Γ 0 is not abelian. Lemma 3.2. Let A, B, and C be unitary operators on a Hilbert space H satisfying AB = CBA, AC = CA, BC = CB, and let a, b and c be positive integers such that c = ab. Suppose that U and V are unitary operators belonging to the von-neumann algebra generated by A and B, and satisfying U a = A and V b = B. Then the element W = UV U 1 V 1 satisfies UW = W U, V W = W V, and W c = C. Proof. Let A be the von Neumann algebra generated by A and B. Since ABA 1 B 1 lies in the center of A, then [A, A] centa). and hence W centa). t remains to show that W c = C. To prove this, we proceed by induction on c = ab: if c = 1 then a = b = 1 and there is nothing to prove. Suppose that c > 1 and that for any a, b, c with a b = c and c < c, we have If a > 1, then we have W c = U a V b U a V b. W a 1)b = U a 1 V b U a+1 V b.

7 7 Observe that U commutes with V b U a+1 V b : indeed, by definition of W, UV b = W b V b U so UV b = W b V b U, from which the observation follows. Hence W ab = W a 1)b W b = U a 1 V b U a+1 V b) UV b U 1 V b) = U a 1 V b U a+1 V b) U V b U 1 V b) = U a 1 U V b U a+1 V b) V b U 1 V b) = U a V b U a V b. If a = 1 then b > 1 and the proof is similar. We now introduce our class of wavelet systems. Let n be a positive integer, and let t 1, t 2,..., t n, and z ij, 1 i, j n satisfy the relations for all i, j and k: Observe that the relation z ji = z 1 ij t i t j = z i,j t j t i, and z ij t k = t k z ij. follows from the above. The group F n = t 1, t 2,... t n, z ij, 1 i, j n is the free, two-step discrete) nilpotent group generated by the n elements t k, 1 k n. Theorem 3.3. Define α : F n F n by αt k ) = t a k k and αz ij ) = z aiaj ij α k are integers. Then Gα, F n ) has the α-root property. where the Proof. Let ρ be any representation of F n acting in H and put A k = ρt k ) and C ij = ρz ij ), 1 i, j, k n. Let A be the von-neumann algebra generated by {A 1,..., A n }. By the Borel functional calculus, for each k we have U k A such that U a k k = A k. Now for each i and j put W ij = U i U j U 1 i U 1 j. By Lemma 3.2, W ij is central and satisfies W aiaj ij = C i,j. Put T t k ) = U k and T z ij ) = W ij, 1 i, j, k n. Since T z ij ) = T t i )T t j )T t i ) 1 T t j ) 1 holds for all i and j, then T is a representation of F n. Since and T αt k )) = T t a k k ) = T t k) a k = A k = ρt k ), T αz ij )) = T z aiaj ij ) = T z ij ) aiaj = C ij = ρz ij ), then T α = ρ. Thus G has the α-root property. When n = 2, then F n is Heisenberg. We cite several examples of representations of Gα, F 2 ) where it is known that Parseval wavelet frames exist.

8 8 Example 3.4. Let π be the representation of Γ acting in L 2 R 2 ) by t 1 e 2πiλ I, t 2 M and t 3 T where I is the identity operator, and M and T are the operators on H = L 2 R 2 ) given by Mfλ, t) = e 2πiλt fλ, t), T fλ, t) = fλ, t 1). Now define πu)fλ, t) = f4λ, 2 1 t)2 3/2. The systems πγ)ψ are Fourier transform of wavelet systems of multiplicity one subspaces of L 2 H) for the Heisenberg group H, and large classes of Parseval wavelet frames have been found in [3]. Example 3.5 Shearlet system). Let π be the representation of G given by u D, t 1 T 1, t 2 T 2, and t 3 M, where D, T 1, T 2, M are the unitary operators on L 2 R 2 ) defined by Dfx) = a 3/2 fa 2 x 1, a 1 x 2 ) Mfx) = fx 1 x 2, x 2 ) T 1 fx) = fx 1 1, x 2 ) T 2 fx) = fx 1, x 2 1). Finally, we note that Lemma 3.2 can be used to prove that other groups have the α-root property as well. Example 3.6. Let Γ 0 = t 1, t 2, t 3, t 4, t 5 : t 5 t 4 = t 4 t 5 t 2, t 5 t 3 = t 3 t 5 t 1, and {t 1, t 2, t 3, t 4 } commute. Note that Γ 0 is the B-integer lattice in the two-step CSCN group whose Lie algebra has basis B = {X 1, X 2,..., X 5 } with [X 5, X 4 ] = X 2 and [X 5, X 3 ] = X 1 X 1,..., X 4 commute.) Let a and b be integers and define α : Γ 0 Γ 0 by αt 5 ) = t a 5, αt ) = t b k, k = 3, 4 and for k = 1, 2, αt k) = t a+b k. Then G = Gα, Γ 0 ) is a wavelet group. One example of a representation of G is the following. Let π : G U L 2 R 4 ) ) be given by u D, t k T k, k = 1, 2, 3, 4, and t 5 M, where T k is the translation operator T k fx) = fx 1,..., x k 1,... x 4 ) and D, M are defined by Dfx) = a 3/2 fab) 1 x 1, ab) 1 x 2, a 1 x 3, a 1 x 4 ) Mfx) = fx 1 x 3, x 2 x 4, x 3, x 4 ) Then π, L 2 R 4 )) is a representation of G. As far as is known to these authors, Parseval frames of the form πγ)ψ have yet to be studied. References [1] D.E. Dotkay, Positive definite maps, representations, and frames [2] B. Currey, Decomposition and multiplicities for the quasiregular representation of algebraic solvable Lie groups, Jour. Lie Th., to appear

9 9 [3] B. Currey, A. Mayeli Gabor fields and wavelet sets for the Heisenberg group, Mon. Math., to appear. [4] B. Currey, A. Mayeli A density condition for interpolation on the Heisenberg group, preprint. [5] D.E.. Dotkay, D. Han, G. Picioraga, Q. Sun, Orthonormal dilations of Parseval wavelets, Math. Ann ), [6] D.E. Dotkay, Positive definite maps, representations, and frames [7] D. Han, D. Larsen, Frames, Bases, and Group Representations, Mem. Amer. Math. Soc. 147, No ) Bradley Currey, Department of Mathematics and Computer Science, Saint Louis University, St. Louis, MO address: curreybn@slu.edu Azita Mayeli, Mathematics Department, New York City College of Technology, CUNY, Brookly, 11201, USA address: amayeli@citytech.cuny.edu