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 Γ0 (translations) and a cyclic group Γ1 (dilations), and whose unitary representations naturally give rise to a wide variety of wavelet systems generated by the pseudo-lattice Γ = Γ1 Γ0 . 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 Γ0 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 Hπ . Given a representation π of G, a countable subset Γ of G is a Parseval sampling set for π if there is ψ ∈ Hπ such that for any f ∈ Hπ X kf k2 = | hf, π(γ)ψi |2 (1.1) γ∈Γ
A vector ψ is a Parseval frame vector for π(Γ) if (1.1) holds, and ψ is a wandering vector if π(Γ)ψ is an orthonormal basis for Hπ . A representation π ˜ is a superrepresentation of π if Hπ is a closed π ˜ -invariant subspace of Hπ˜ and π(x) = π ˜ (x)|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 π(Γ)ψ has the G-dilation property if there is a super-representation π ˜ of π and a vector ψ˜ ∈ Hπ˜ such that π ˜ (Γ)ψ˜ is an orthonormal basis for Hπ˜ whose projection is π(Γ)ψ. In this case we say that ψ˜ is a G-dilation of ψ. 1
2
We consider in this paper the following class of groups. Let Γ0 be a discrete group and α : Γ0 → Γ0 be a monomomorphism. The wavelet group associated with Γ0 and α is the group G generated by Γ0 and the infinite cyclic group Γ1 = {uj : j ∈ Z}: G = G(α, Γ0 ) := hu, Γ0 : uγu−1 = α(γ), ∀ γ ∈ Γ0 i. The subset Γ = Γ1 Γ0 where Γ1 = {uj : 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 BS(1, 2), where Γ0 = Z and α(k) = 2k. More generally, if N is any homogeneous group with dilations and Γ0 is a lattice in N , one can pick a dilation α that determines a monomorphism of Γ0 , and consider wavelet systems in L2 (N ) consisting of translations by elements of Γ0 and unitary dilations Dα on L2 (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 π(Γ)ψ is a Parseval frame. When does π(Γ)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 π(Γ)ψ is a standard 2-wavelet system (G = BS(1, 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 π(Γ)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 {γ1 , γ2 , . . . , γk } in X and {ξ1 , ξ2 , . . . , ξk } ⊂ C, X K(γi , γj )ξi 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 K(sx, sy) = K(x, y) holds for all s, x and y in G. By [1, Theorem 2.8], every group positive definite map has the form Kρ,η (x, y) = hρ(x)η, ρ(y)ηi
3
where ρ is a representation of G and η is a cyclic vector for ρ. Let Γ0 be a discrete group, α a monomorphism of Γ0 , and ρ a representation of Γ0 . 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 = Γ1 Γ0 the standard pseudo-lattice in G and suppose that every representation of Γ0 has an α-root. Let K : Γ × Γ → C be a positive definite mapping such that for any j, j 0 ∈ Γ1 , γ, γ 0 ∈ Γ0 , the relations K((j + 1, γ), (j 0 + 1, γ 0 )) = K((j, γ), (j 0 , γ 0 )), γ0 ∈ Γ0 , 0
K((j, α−j (γ0 )γ), (j 0 , α−j (γ0 )γ 0 )) = K((j, γ), (j 0 , γ 0 ))
for , γ0 ∈ Γ0 , j 6 0,
(2.1)
both hold. Then K is the restriction of a group positive definite map Kτ,ψ . More explicitly there exists a Hilbert space H, and unitary operators D and representation T of Γ0 on with DT (γ)D−1 = T α(γ), and a vector ψ ∈ H such that H = span{Dj T (γ)ψ} and 0
K((j, γ), (j 0 , γ 0 )) = hDj T (γ)ψ, Dj T (γ 0 )ψi. 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{v(j, γ) : (j, γ) ∈ Γ}, and K((j, γ), (j 0 , γ 0 )) = hv(j, γ), v(j 0 , γ 0 )i holds for all (j, γ) and (j 0 , γ 0 ) belonging to Γ. Define the operator D : H → H by Dv(j, γ) = v(j + 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 Hn = span{v(j, γ) : (j, γ) ∈ Γ, j ≤ n}. Note that DHn = Hn+1 and Hn ⊂ Hn+1 . Set Kn = Hn Hn−1 , n ≥ 0. For γ0 ∈ Γ0 , define the operator T0 (γ0 ) on H0 by T0 (γ0 ) v(j, γ) = v(j, α−j (γ0 )γ) and again extending to all of H0 ; the second relation in (2.1) shows that γ 7→ T0 (γ) is a (unitary) representation of Γ0 . Since the subspace K0 is invariant under T0 , we can define the representation ρ1 of Γ0 acting in K1 by ρ1 (γ) = DT0 (γ)D−1 . Now by our hypothesis, ρ1 has an α-root T1 : T1 acts in K1 and satisfies T1 ◦ α = ρ1 .
4 Now the representation γ 7→ ρ2 (γ) = DT1 (γ)D−1 of Γ0 acting in K2 has an α-root T2 acting in K2 . Continuing in this way, we obtain, for each positive integer n, a representation Tn of Γ0 acting in Kn , so that Tn ◦ α = DTn−1 D−1 . (Again in the preceding, T0 is restricted to K0 .) Now write M H = H0 (⊕n≥1 Kn ) L and define the representation T of Γ0 by T = T0 ⊕n≥1 Tn . Next we must verify the relation DT (γ)D−1 = T α(γ) . Fix γ0 ∈ Γ0 ; for v(j, γ) with j ≤ 0, DT0 (γ0 )D−1 v(j, γ) = DT0 (γ0 ) v(j − 1, γ) = D v(j − 1, α−j+1 (γ0 )γ) = T0 α(γ0 ) v(j, γ) and hence the relation DT0 (γ)D−1 = T0 α(γ) holds on H0 . Now for v ∈ H, write P v = n≥0 vn . We have DT (γ)D−1 v0 = T α(γ) v0 and for n ≥ 1, DT (γ)D−1 vn = DTn−1 (γ)D−1 vn = Tn α(γ) vn so DT (γ)D−1 v =
X n≥0
DT (γ)D−1 vn =
X
Tn α(γ) vn = T α(γ) .
n≥0
It follows that the mapping τ defined by τ (u) = D and τ (γ) = T (γ) is a representation of G. Finally, take ψ = v(0, 0). Then v(j, γ) = Dj v(0, γ) = Dj T (γ)v(0, 0) = Dj T (γ)ψ so ψ is cyclic for τ . Hence the group positive definite map defined for all x, y ∈ G by Kτ,ψ (x, y) = hτ (x)ψ, τ (y)ψi 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 = Γ1 Γ0 the standard pseudo-lattice in G and suppose that every representation of Γ0 has an α-root. Let π be any representation of G. Then every Parseval frame wavelet system π(Γ)ψ has the G-dilation property. Proof. Let Γ = Γ1 Γ0 ⊂ G as above and recall that we write uj γ = (j, γ). Define K((j, γ), (j 0 , γ 0 )) = δj,j 0 δγ,γ 0 − hπ(j, γ)ψ, π(j 0 , γ 0 )ψi.
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Observe that δj+1,j 0 +1 = δj,j 0 and δj,j 0 δ(a−j γ0 )γ,(a−j0 γ0 )γ 0 = δj,j 0 δγ,γ 0 , and that in the group G, u−j γ0 uj = a−j γ0 holds for j, γ0 ∈ Γ0 . Hence K((j + 1, γ), (j 0 + 1, γ 0 )) = δj+1,j 0 +1 δγ,γ 0 − hπ(j + 1, γ)ψ, π(j 0 + 1, γ 0 )ψi = δj,j 0 δγ,γ 0 − hDπ(j, γ)ψ, Dπ(j 0 , γ 0 )ψi = K((j, γ), j 0 , γ 0 )) and for j ≤ 0, K((j, (α−j γ0 )γ),(j 0 , (α−j γ0 )γ 0 )) 0
= δj,j 0 δ(α−j (γ0 )γ,α−j0 (γ0 )γ 0 − hπ(jα−j (γ0 )γ)ψ, π(j 0 α−j (γ0 )γ 0 )ψi = δj,j 0 δγ,γ 0 − hπ(γ0 jγ)ψ, π(γ0 j 0 γ 0 )ψi = δj,j 0 δγ,γ 0 − hπ(γ0 )π(jγ)ψ, π(γ0 )π(j 0 γ 0 )ψi = δj,j 0 δγ,γ 0 − hπ(jγ)ψ, π(j 0 γ 0 )ψi = K((j, γ), (j 0 , γ 0 )). 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 = Kτ,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 π ˜ (x)ψ = π(x)ψ.
We illustrate the preceding with the example G = BS(1, 2). Here Γ0 is the infinite cyclic group {tk }, and α(t) = t2 . 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 L2 (R) by π(u)φ(x) = φ(2−1 x)2−1/2 and π(t)φ(x) = φ(x − 1). With the standard pseudo-lattice {uj tk : j, k ∈ Z}, the above theorem yields that every Parseval wavelet frame of the form π(Γ)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 G(α, Γ0 ): that every representation of Γ0 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.
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Question 2.3. For what groups G(α, Γ0 ) does the α-root property hold? In the following section we describe two families of groups G(α, Γ0 ) 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 π(Γ)ψ are known to exist. We begin with the case where Γ0 is a finitely-generated, torsion-free abelian group. Example 3.1. (A-wavelet system) Let Γ0 be the free abelian group generated by a a a t1 , t2 , · · · , tn , and let α(tj ) = t1 1j t2 2j · · · tnnj where A = [ai,j ] ∈ GL(n, Z). We claim that the wavelet group G(α, Γ0 ) has the α-root property. Let ρ be any representation of Γ0 , and write A−1 = [bi,j ]. Since the bi,j are rational, the Borel functional calculus obtains operators Vi,j , 1 ≤ i, j ≤ n such that Vi,j = ρ(t1 )bi,j . Define T (tj ), 1 ≤ j ≤ n by T (tj ) = V1,j V2,j · · · Vn,j . An easy computation shows that T ◦ α = ρ. Next we consider wavelet groups where Γ0 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 = U V U −1 V −1 satisfies U W = 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] ⊂ cent(A). and hence W ∈ cent(A). 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 a0 , b0 , c0 with a0 b0 = c0 and c0 < c, we have 0
0
0
0
0
W c = U a V b U −a V −b . If a > 1, then we have W (a−1)b = U a−1 V b U −a+1 V −b .
7 Observe that U commutes with V b U −a+1 V −b : indeed, by definition of W , U V b = W b V b U so U V −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 U V 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 t1 , t2 , . . . , tn , and zij , 1 ≤ i, j ≤ n satisfy the relations for all i, j and k: ti tj = zi,j tj ti ,
and zij tk = tk zij .
−1 Observe that the relation zji = zij follows from the above. The group
Fn = ht1 , t2 , . . . tn , zij , 1 ≤ i, j ≤ ni is the free, two-step (discrete) nilpotent group generated by the n elements tk , 1 ≤ k ≤ n. a aj
Theorem 3.3. Define α : Fn → Fn by α(tk ) = takk and α(zij ) = ziji αk are integers. Then G(α, Fn ) has the α-root property.
where the
Proof. Let ρ be any representation of Fn acting in H and put Ak = ρ(tk ) and Cij = ρ(zij ), 1 ≤ i, j, k ≤ n. Let A be the von-Neumann algebra generated by {A1 , . . . , An }. By the Borel functional calculus, for each k we have Uk ∈ A such that Ukak = Ak . Now for each i and j put Wij = Ui Uj Ui−1 Uj−1 . By Lemma 3.2, Wij is a a central and satisfies Wiji j = Ci,j . Put T (tk ) = Uk and T (zij ) = Wij , 1 ≤ i, j, k ≤ n. Since T (zij ) = T (ti )T (tj )T (ti )−1 T (tj )−1 holds for all i and j, then T is a representation of Fn . Since T (α(tk )) = T (takk ) = T (tk )ak = Ak = ρ(tk ), and a a
T (α(zij )) = T (ziji j ) = T (zij )ai aj = Cij = ρ(zij ), then T ◦ α = ρ. Thus G has the α-root property.
When n = 2, then Fn is Heisenberg. We cite several examples of representations of G(α, F2 ) where it is known that Parseval wavelet frames exist.
8 Example 3.4. Let π be the representation of Γ acting in L2 (R2 ) by t1 7→ e2πiλ I, t2 7→ M and t3 → T where I is the identity operator, and M and T are the operators on H = L2 (R2 ) given by M f (λ, t) = e−2πiλt f (λ, t), T f (λ, t) = f (λ, t − 1). Now define π(u)f (λ, t) = f (4λ, 2−1 t)23/2 . The systems π(Γ)ψ are Fourier transform of wavelet systems of multiplicity one subspaces of L2 (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 7→ D, t1 7→ T1 , t2 7→ T2 , and t3 7→ M , where D, T1 , T2 , M are the unitary operators on L2 (R2 ) defined by Df (x) = a−3/2 f (a−2 x1 , a−1 x2 )
M f (x) = f (x1 − x2 , x2 )
T1 f (x) = f (x1 − 1, x2 )
T2 f (x) = f (x1 , x2 − 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 Γ0 = ht1 , t2 , t3 , t4 , t5 : t5 t4 = t4 t5 t2 , t5 t3 = t3 t5 t1 , and {t1 , t2 , t3 , t4 } commutei. Note that Γ0 is the B-integer lattice in the two-step CSCN group whose Lie algebra has basis B = {X1 , X2 , . . . , X5 } with [X5 , X4 ] = X2 and [X5 , X3 ] = X1 (X1 , . . . , X4 commute.) Let a and b be integers and define α : Γ0 → Γ0 by α(t5 ) = ta5 , α(t) = tbk , k = 3, 4 and for k = 1, 2, α(tk ) = ta+b k . Then G = G(α, Γ0 ) is a wavelet group. One example of a representation of G is the following. Let π : G → U L2 (R4 ) be given by u 7→ D, tk 7→ Tk , k = 1, 2, 3, 4, and t5 7→ M , where Tk is the translation operator Tk f (x) = f (x1 , . . . , xk − 1, . . . x4 ) and D, M are defined by Df (x) = a−3/2 f ((ab)−1 x1 , (ab)−1 x2 , a−1 x3 , a−1 x4 ) M f (x) = f (x1 − x3 , x2 − x4 , x3 , x4 ) Then (π, L2 (R4 )) is a representation of G. As far as is known to these authors, Parseval frames of the form π(Γ)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
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[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. 341 (2008), 483–515 [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. 697 (2000)
Bradley Currey , Department of Mathematics and Computer Science, Saint Louis University, St. Louis, MO 63103 E-mail address:
[email protected]
Azita Mayeli, Mathematics Department, New York City College of Technology, CUNY, Brookly, 11201, USA E-mail address:
[email protected]