Probability measures on solenoids corresponding to fractal wavelets

The measure on generalized solenoids constructed using filters by Dutkay and Jorgensen is analyzed further by writing the solenoid as the product of a torus and a Cantor set. Using this decomposition, key differences are revealed between solenoid measures associated with classical filters in $\mathbb R^d$ and those associated with filters on inflated fractal sets. In particular, it is shown that the classical case produces atomic fiber measures, and as a result supports both suitably defined solenoid MSF wavelets and systems of imprimitivity for the corresponding wavelet representation of the generalized Baumslag-Solitar group. In contrast, the fiber measures for filters on inflated fractal spaces cannot be atomic, and thus can support neither MSF wavelets nor systems of imprimitivity.


Introduction
Let d be a positive integer and A be a diagonal d × d matrix whose diagonal entries N 1 , N 2 , · · · , N d are integers greater than 1. We write N = (N 1 , N 2 , · · · , N d ), and define β to be the induced map on T d given by β(z) = (z N 1 1 , z N 2 2 , · · · , z N d d ). In this context, the generalized solenoid S β is the inverse limit of T d under the map β. Methods of constructing probability measures on generalized solenoids via filter functions were first explored by Dutkay and Jorgensen in [12] and worked out explicitly for the solenoid given by d = 1, N 1 = 2 and the filter m(z) = 1+z 2 √ 2 corresponding to the inflated Cantor set wavelet by Dutkay in [9]. In a recent paper, Baggett, Larsen, Packer, Raeburn, and Ramsay used a slightly different construction to obtain probability measures on solenoids from more general filters on T d associated to integer dilation matrices [4]. The filters used in all of these papers, though not necessarily low-pass in the classical sense of the term [22], are required to be non-zero except on a set of measure 0, and bounded away from 0 in a neighborhood of the origin.
In Section 2, we will describe in detail the construction given in [4] of the probability measure τ on S β . Using the discussion in Chapters 3 and 4 of P. Jorgensen's book [18], we then examine the decomposition of τ over the d-torus T d , with fiber measures on the Cantor set. We show that for the measure on the solenoid built from classical filters in R d , the fiber measures are atomic, while for measures built from filters on inflated fractal sets, the fiber measures have no atoms.
In the remaining two sections, we explore the consequences of this distinction between the classical and fractal cases. First, in Section 3.1, we define the notion of a solenoid MSF (minimally supported frequency) wavelet using the Hilbert space L 2 (S β , τ ). This definition can be applied to both classical filters and filters on inflated fractals. Thus we are able to extend the concept of an MSF wavelet to the case of inflated fractal sets, where the standard Fourier transform is not available. The definition on the solenoid also allows us to compare the classical and fractal cases. We show that solenoid MSF wavelets exist if and only if the fiber measures on the Cantor set are almost everywhere atomic. Thus the difference between the nature of the fiber measures in the classical and fractal cases causes a difference in the existence of solenoid MSF wavelets.
Section 4 examines a further implication for the related representation of the generalized Baumslag-Solitar group on L 2 (S β , τ ). Let Q A = ∪ ∞ j=0 A −j (Z d ) ⊂ Q d represent the A-adic rationals in R d . The generalized Baumslag-Solitar group BS A is a semidirect product, with elements in Q A × Z and with group operation given by (q 1 , m 1 ) · (q 2 , m 2 ) = (q 1 + A −m 1 (q 2 ), m 1 + m 2 ), q 1 , q 2 ∈ Q A , m 1 , m 2 ∈ Z. In Section 4, we show that if ψ ∈ L 2 (S β , τ ) corresponds to a single wavelet for dilation and translation, then ψ is a solenoid MSF wavelet if and only if the wavelet subspaces {W j : j ∈ Z} for ψ form a system of imprimitivity for this representation, so that it is induced in the sense of Mackey from a representation of the discrete subgroup of A-adic rationals Q A . Thus a single wavelet on the solenoid is induced from the A-adic rationals if and only if the fiber measures are atomic. In the course of obtaining this result, we generalize a theorem of E. Weber [26] relating ordinary MSF wavelets in L 2 (R) to wavelet subspaces that are invariant under relevant translation operators.
The upshot of our study is to draw striking distinctions between solenoids associated with classical and fractal filters. The key difference is that the multiresolution analyses coming from the standard inflated fractal sets will not carry solenoid MSF wavelets in the sense defined here. This is not so surprising in light of recent work of Dutkay, D. Larson, and S. Silvestrov [13]. In contrast, classical multiresolution analyses carry solenoid MSF wavelets as well as standard Fourier MSF wavelets. In the classical case, the existence of MSF wavelets was an essential tool in proving the existence of single wavelets in L 2 (R d ). It is not yet known whether or not the inflated fractal Hilbert spaces and associated dilation and translation operators support a single wavelet. Another consequence of the existence of MSF wavelets in the classical case was the fact that the representation of the associated Baumslag-Solitar group is induced from the discrete subgroup Q A . We have shown in Section 4 that the connection between the existence of MSF wavelets and systems of imprimitivity for representations of Baumslag-Solitar carries over to the solenoid spaces. Thus, because we have solenoid MSF wavelets for solenoids built from classical filters, the representation of Baumslag Solitar on L 2 (S β , τ ) can also be seen to be induced from Q A in the classical filter case. However, for filters on inflated fractals, the representations on the associated solenoid space cannot be induced from the A-adic rationals.

Construction of probability measures on solenoids from filter functions and their direct integral decompositions
Our context will be as follows: Let X ⊂ R d , and suppose X is invariant under under translations by {v : v ∈ Z d } and under multiplication by the expansive integer diagonal matrix A with a k,k = N k . Define N ≡ det A = N 1 N 2 · · · N d . Let µ be a σ-finite Borel measure on X that is invariant under integer translations. Suppose there is a positive constant K ≤ N such that µ(A(S)) = Kµ(S) for S a Borel subset of X. Define dilation and translation operators D and , A calculation shows that T v D = DT Av . In this paper we will consider two classes of such spaces. In the classical examples, we have X = R d with Lebesgue measure µ. In the other class of examples, X is built from We assume that this system satisfies the separation requirement given by the open set condition [16]. Following [10], X is then defined to be an inflated fractal and µ is defined to be Hausdorff measure of dimension log N (K) restricted to R, which coincides with the Hutchinson measure [16]. It will be useful to describe the classical example in the same terms as the fractal example by taking F to be [0, 1] d , K = N, and In both classes of examples there is a natural multiresolution (MRA) structure on L 2 (X, µ), which can be described as follows. We define a scaling function φ = χ F . Translates of φ are orthonormal, and we define the core subspace V 0 of the MRA to be the closure of their span. Setting V j = D j (V 0 ) it can be shown (see [4]) that ∪ j∈Z V j is dense in L 2 (X, µ) and ∩ j∈Z V j = {0}. The inclusion V j ⊂ V j+1 follows from the fact that With this notation, the refinement equation (1) above gives a low pass filter for dilation by A defined by m(z) We now recall the definition of the probability measure on the generalized solenoid S β that was developed in [12], [9], and [4]. As above, we write β(z) = (z N 1 1 , z N 2 2 , · · · , z N d d ). Proposition 2.1. (Proposition 6.2 of [4]) Denote by π n the canonical map of S β onto the nth copy of T d . Let m : T d → C be a Borel function such that the inverse image m −1 ({0}) has Haar measure equal to 0 that in addition satisfies Then there is a unique probability measure τ on S β such that for every f ∈ C(T d ), The inverse limit group S β carries a natural group automorphism induced by the shift, which is commonly studied in topological dynamics: Proof. It is an easy calculation that σ is a group homomorphism, and that its inverse is given by the formula above for σ −1 . Thus σ is a group automorphism, as desired.
Proposition 2.3. The measure τ is quasi-invariant with respect to σ. that is, τ (E) = 0 if and only if τ (σ(E)) = 0 if and only if τ (σ −1 (E)) = 0. The Radon-Nikodym derivatives are given as follows: Proof. It suffices to show that In fact it is enough to prove these equations for functions of the form f • π n , where f ∈ C(T d ). We prove the second equality, the first being proved analogously.
(Note that we made use of a generalization of the alternate form of the definition of τ given in [9,Proposition 4.2(i) In a broader context where β is the multiplicative dual of multiplication by an arbitrary expansive integer dilation matrix A, B. Brenken has noted in [7] that the corresponding solenoid S β can be expressed as a locally trivial principal Σ-bundle over T d for some 0-dimensional group Σ. In Corollary 2.2, A is the diagonal matrix with diagonal entries N 1 , N 2 , · · · , N d . In the case where d = 1, Σ is a variant of the Cantor set. Using another point of view, Jorgensen in [18] views S β as a bundle over T d whose fibers can be viewed as "random walks". Given a filter m on T d , for each z ∈ T d he is able to define a probability measure P z on the space of trees in the random walks. In this section, our aim is to combine the approaches of Brenken and Jorgensen and express the measure τ on S β as a direct integral measure over T d of Borel measures ν z each defined on Σ. The measures ν z correspond to the random walk measures P z of Jorgensen in a natural way.
We first observe that by using exact sequences of compact abelian groups, the fiber Σ over z ∈ T d can be describe in the following way: if n=0 : z n ∈ T d ; z n = β(z n+1 ), n ≥ 0}, then fixing z ∈ T d , the fiber π −1 0 ({z}) can be identified with the 0-dimensional group We can describe the map ρ explicitly in the case where d = 1 and A = (N). We define a Borel cross-section c : T → S β by in this case is given by Indeed, in this case, we can go further, and identify Σ β with the compact abelian group of N-adic integers, which is 0-dimensional, as follows. Let Z N denote the N-adic integer group, so that with the product topology coming from the discrete topology on the finite set {0, 1, · · · , N − 1} and group action induced by odometer addition, with the "carrying" operation taking place to the right. The map Φ : ∞ j=0 {0, 1, · · · , N − 1} = Z N → Σ β is given by Φ((a n ) ∞ n=0 ) = (z n ) ∞ n=0 , where z 0 = 1 and z n = e 2πi n−1 j=0 a j N j N n for n > 1.
Routine calculations show that in the above example, there is a Borel isomorphism Θ between the Cartesian product T × Σ β and the N-solenoid S β given by To generalize Example 2.4 to higher dimensions, we will use the notation e(t) ≡ e(t 1 , t 2 , · · · , t d ) = (e 2πit 1 , e 2πit 2 , · · · , e 2πit d ). Write Σ β for the kernel of the projection map π 0 : S β → T d , and define the cross section c : T d → S β by c(e(t)) = (e(A −n t)) ∞ n=0 . There is a Borel isomorphism Θ : Under the Borel isomorphism Θ, the shift map σ : S β → S β corresponds to a map on the Cartesian product T d × Σ β which we denote by A computation then shows that . where η 0 = 1 and c : T d → S β is the Borel cross-section discussed above.
We let Just as in the above example, we can identify Σ β with Z d A . It is not difficult to show that the map Θ then can be described by A j (a j ))), · · · ). while σ can be described by , then σ −1 is giving by (6) σ −1 (e(t), (a j ) ∞ j=0 ) = (e(A(t)), (s(t), a 0 , a 1 , · · · )). Using the map Θ, we obtain the following result concerning the decomposition of the measure τ of Proposition 2.1. We remark here that while we have been able to make an explicit decomposition, of the measure, there is a general theorem that could have been applied. References to earlier work in this direction and a proof of a suitable general theorem can be found in Lemma 4.4 of [14].
Let τ be the Borel measure on S β constructed using the filter m. Using the Borel isomorphism Θ : T d × Z d A → S β defined in the above paragraph, the measure τ = τ • Θ corresponds to a direct integral Proof. We have already seen how the measure τ is defined on the solenoid S β as follows: If f ∈ C(T d ) and k ∈ N are fixed, then It follows that with respect to the Borel isomorphism Θ : It follows directly from Proposition 2.3 and the definition of Θ that the measures τ and τ • σ −1 are equivalent, and that the Radon-Nikodym derivative is given by We now investigate conditions on the filter m : T d → C under which any of the measures ν z : z ∈ T d will be atomic. This has been investigated within the framework of random walks in Chapters 2 and 3 of [18]; here, we take a slightly different approach.
Lemma 2.6. Let m : T d → C be a filter for dilation by the matrix A as defined in Proposition 2.5. For each z ∈ T d , let ν z be the measure on Z d A constructed in Proposition 2.5. Then ν z has atoms if and only if there is a sequence (a j ) ∞ j=0 ∈ Z d A such that for each n ∈ N, m(e(A −n (t)) · e(A −n ( n−1 i=0 A i (a i )))) = 0, and moreover such that the sequence converges to 1 at a sufficiently rapid rate.
it follows that writing z = e(t) with t ∈ [0, 1) d , we have Thus in order to have ν z ({(a j ) ∞ j=0 }) > 0 it is necessary and sufficient that the infinite product converge to a positive number. This will happen if and only if A i (a i )))) = 0 ∀j ∈ N and the terms 1 )))| converge to 1 at a sufficiently rapid rate as j → ∞. Example 2.7. We discuss the conditions of Lemma 2.6 further in the case where d = 1, A = (N), and m is low-pass in the classical sense of [22]. Thus, suppose that m(1) = √ N , m is Lipschitz continuous in a neighborhood of z = 1, and m satisfies Cohen's condition, so that it is non-zero in a large enough neighborhood of 1. Let us also assume that m is a Laurent polynomial in z ∈ T, (i.e., is a so-called "FIR" filter). The discussion which follows has its roots in the description of "cycles" given by Dutkay and Jorgensen in Section 2 of [11].
Since m is a Laurent polynomial, so is |m| 2 , and it follows that the set Z m will be finite. Following Lemma 2.6, suppose (a Since lim j→∞ e(t/N j ) = 1, by continuity we have But in order for lim j→∞ |m(e( )} must be becoming arbitrarily close to elements in Z m . Given the fact that # Z m is finite, and m is a Laurent polynomial and continuous on the unit circle, this dictates restrictive conditions on the choice of the (a i ) ∞ i=0 . In exploring these conditions, we will make use of the key observation that One way the conditions can be met is if {e( exists, the sequence is Cauchy, and an easy argument shows that there must exist J > 0 and a ∈ {0, 1, · · · , N − 1} such that for all j > J, a j = a.
A second way to achieve lim j→∞ |m(e( bounce around very close to a variety of points z ∈ Z m as k gets larger and larger. When Z m is finite, this can only happen under special circumstances. Suppose that for some fixed positive integer l ≥ 1, and {d 1 , d 2 , · · · , d l } ∈ {0, 1, · · · , N − 1}, we have the repeating N-adic expression {0, 1, · · · , N − 1} that is an atom as follows: Fix a non-negative integer J, and choose a 0 , a 1 , a 2 , · · · , a Jl ∈ {0, 1, · · · , N − 1} arbitrarily, and for all j ≥ 0 and i = 1, · · · , l, let It will then be the case that as k → ∞, will tend to one of the l values e(x 0 ), e(x 1 ), · · · , e(x l−1 ) ∈ Z m . It might seem that any choice {d 1 , d 2 , · · · , d l } will do in this construction, for l as large as we want. But this is not the case, because as noted above, the accumulation points e(x 0 ), e(x 1 ), · · · , e(x l−1 ) all need to be in Z m . Moreover if one has made such a Thus the only possible elements of Z m that can possibly give rise to sequences having atoms are numbers modulus 1 of the form e(r) where r is a rational number in [0, 1) lying in We leave to the interested reader the verification that if d > 1 with dilation factors N 1 , N 2 , · · · , N d , the only possible elements of Z m that can possibly give rise to sequences having atoms are numbers modulus 1 of the form e(r) where r is a d-tuple of rational numbers in [0, 1) d that are elements of More generally, it is intriguing that the existence of atoms is independent of the fiber z = e(t) ∈ T d chosen. However, if an atom does exist in the fiber, the value of the fiber measure ν z on the atom will depend on z.
We expand further on the decomposition of the measure τ in the case of a classical low-pass filter, and recall Theorem 6.6 of [4]. Reviewing the set-up for this theorem, let m : T d → C be a low-pass filter for dilation by N in the sense of [22], that is, suppose that m is Lipschitz continuous at 1, is non-zero in a sufficiently large neighborhood of 1, and satisfies We also assume m be a Laurent polynomial in the variables (z 1 , z 2 , · · · z d ) of z. Let A be the diagonal matrix with diagonal entries N 1 , N 2 , · · · , N d , and let N = d i=1 N i . It is then a standard construction due to Mallat and Meyer that defining the Fourier transform of the scaling function φ ∈ L 2 (R d ) by this product converges pointwise and in L 2 (R d ). Theorem 6.6 of [4] states that if we define the winding line map w : then for any f ∈ L 2 (S β , τ ), This result states that τ is supported on the image of w in S β , that is, on the winding line L w , the image of R under w in the N-solenoid. In this case, the structure of the fiber measures ν z can be deduced exactly.
As a result of this analysis, we have deduced the following result: Proposition 2.8. Let m : T d → C be a classical low-pass filter, and let τ be the Borel probability measure on the solenoid S β associated to m by Proposition 2.1. Decompose τ as dτ = T d dν z dz as in Proposition 2.5. Then for almost every Because of this, τ (S β \L w ) = 0. It follows that for almost all z ∈ T, ν z (π −1 0 ({z})\L w ) = 0. Thus for almost all z ∈ T, the nature of the measure ν z on π −1 0 ({z}) can be deduced from its behavior on the intersection π −1 0 ({z}) ∩ E w . Now an arbitrary element (e(x), e(A −1 (x)), e(A −2 (t)), · · · , e(A −n (x)), · · · , ) ∈ L w will be an element of π −1 0 ({z}) for z = e(t), t ∈ [0, 1) d if and only if e((x) = z which will occur if and only if x = t + k for some k ∈ Z d , where t is the unique element in [0, 1) d such that e(t) = z. We note that each k ∈ Z d gives rise to a different element of E w . Thus {w(t + k) : k ∈ Z d } = L w ∩ π −1 0 ({z}) and for almost all z, ν z is supported on this countable set.
We deduce from the earlier portion of the proof that ν z ({(a 0 , a 1 , · · · , a j k −1 , 0, 0, · · · , )}) = = lim Similarly, if −k is a negative integer, one can use the N-adic expansion of −k with only a finite number of digits not equal to N − 1 to show that for It follows that setting W z = π −1 0 ({z} ∩ L w , we have that W z is countable, and since if we let E z consist of the set of those points in W z having positive measure, E z is both countable and made up of atoms with , it follows that we must have ν z (π −1 0 ({z}\E z ) = 0, for almost all z ∈ T d . Example 2.9. Consider the classical Haar filter m(z) = 1+z+z 2 √ 3 defined on T corresponding to dilation by N = 3. We have seen in the proposition above that the measure ν z corresponding to the fiber will have an atom at (a 0 , a 1 , a 2 , · · · , ) if and only if there exists J such that we either have a j = 0f oreveryj > J, or we have a j = 2 for every j > J. This means that any atom (a 0 , a 1 , a 2 , · · · , ) ∈ ∞ j=0 {0, 1, 2} ≡ Σ 3 corresponds to the natural embedding of the integers in Σ 3 . Thus, if we have either a j = 0 for all j > J or a j = 2 for all j > J, and z = e 2πit , the value of the fiber measure ν z ({(a 0 , a 1 , a 2 , · · · )}) will be equal to | φ(t + n)| 2 , where (a 0 , a 1 , a 2 , · · · ) corresponds to the integer n ∈ Z.
Example 2.10. Consider the non-classical filter m(z) = 1+z 2 √ 2 associated to the inflated Cantor set wavelet in [10]. We note that m is Lipschitz continuous, and an easy application of the triangle inequality shows that |m(z)| ≤ √ 2 < √ 3 for all z ∈ T. It follows that for all z ∈ T, the measure ν z never has atoms. This is true as well to the non-classical filter m(z, w) = 1+z+w 2 associated to the inflated Sierpinski gasket set wavelet in [8]. By the triangle inequality, we have |m(z, w)| ≤ 3 2 . The dilation on the inflated Sierpinski gasket set comes from the 2 × 2 matrix 2 0 0 2 , and the square root of the determinant of this matrix is 2. Again, we have |m(z,w)| 2 ≤ 3 4 < 1 for all (z, w) ∈ T 2 . It follows that if one considers the measure τ on the inverse limit of T 2 coming from the map (z, w) → (z 2 , w 2 ), the fiber measures on the generalized Sierpinski gasket sets corresponding to a fixed (z, w) ∈ T 2 will not have atoms. Here N = |det(A)|, where A is a diagonal expansive matrix. Let τ be the Borel measure on S β = lim ← − (T, z → z N ) constructed using the filter m in Proposition 2.1. We recall from Corollary 5.8 of [9] and as modified by Corollary 6.5 of [4], that if the filter m gives rise to a multi-resolution analysis, either on an inflated fractal space (X, µ) or on R d , there is a "modified" Fourier transform F : L 2 (X, µ) → L 2 (S β , τ ), where (X, µ) represents the measure space carrying the original multiresolution analysis; recall that X ⊂ R d and might be all of R d with µ equal to Lebesgue measure, or might be an inflated fractal set with µ singular with respect to Lebesgue measure.
For F : L 2 (X, µ) → L 2 (S β , τ ), it was established in Theorem 6.4 of [4] that for any f ∈ L 2 (S β , τ ), and every v ∈ Z d , Our aim in this section is to use our previous decomposition of probability measures on solenoids to set possible conditions under which L 2 (X, µ) will contain within it a single unit vector ψ such that {D j T v (ψ) : v ∈ Z d , j ∈ Z} is an orthonormal basis for L 2 (X, µ). It will be desirable that such a single wavelet ψ should have as many properties of MSF (minimally supported frequency) wavelets in L 2 (R d ) as possible. Setting ψ = F (ψ), the "single wavelet" condition is equivalent to finding a vector ψ ∈ L 2 (S β , τ ) such that We model our study after the study of MSF wavelets in L 2 (R d ).
Definition 3.1. Let (X, µ) be an inflated fractal set, and let ψ be a unit vector in L 2 (X, µ) that is a single orthonormal wavelet in L 2 (X, µ) for translation by Z d and dilation by the diagonal dilation matrix A. We say that ψ is a generalized MSF wavelet if ψ is equal to λ((z n ))χ E , for E a Borel subset of S β such that τ (σ j (E) ∩ σ k (E)) = 0 for j = k and τ (S β \(∪ j∈Z σ j (E)) = 0, and some function λ : S β → C.
Definition 3.1 is deceivingly simple. For example, let (R, µ) be the inflated fractal set contained in R created from the Cantor set. It is an open question as to whether or not MSF-wavelets, or even single wavelets, occur in L 2 (R). Remark 3.2. Formula 6.7 in Theorem 6.6 of [4] shows us that even when (X, µ) = (R d , Lebesgue), and m is a classical low-pass filter, and we compute F (ψ) for ψ an ordinary MSF wavelet in L 2 (R d ), we cannot expect that the function λ will be a constant function.
In the rest of this section, we investigate a few properties that E ⊂ S β and λ : S β → C would need to have in order that F * (λ · χ E ) be an MSF single wavelet in L 2 (X).
First, given an MSF wavelet ψ ∈ L 2 (X, µ), the family {T v (ψ) : v ∈ Z d } is an orthonormal set in L 2 (X, µ). It follows that , τ ), and from this we deduce that the family . We now wish to apply the knowledge of τ obtained in the previous section to this question. Note that χ E • Θ = χ Θ −1 (E) , so our first aim will be to come up with conditions on a subset E ′ ⊂ T d × Z d A and (by abuse of notation) λ : The following is just a restatement of the notion of orthonormality.
if and only if h(z) ≡ 1 as an element of L 2 (T d ).
Proof. If h(z) ≡ 1 as an element of L 2 (T d ), then by the theory of Fourier series, we know that T d h(z)dz = 1, a.e. z ∈ T d and for every v ∈ Z d \{0}, T d z v h(z)dz = 0. For v, v ′ ∈ Z d , the inner product of z v λ · χ E ′ and z v ′ λ · χ E ′ , which we can write as . From this result, we obtain the following corollary, in the case where λ depends only on the variable z : A satisfies the conditions of Corollary 3.4, it will be the case that The following proposition gives necessary and sufficient conditions on E ′ to have if and only if upon setting z = e(t), for every m ∈ Z d and j ∈ N ∪ {0}, , (a n ))χ E ′ ((e(t), (a 0 , a 1 , · · · , a n , · · · )) In particular, if we fix m ∈ Z d and j ∈ N ∪ {0}, the function g m,j (e(t), (a 0 , a 1 , · · · , a n , · · · , )) = [e(A −j (t))] m [e( (t), (a 0 , a 1 , · · · , a n , · · · )) We now consider χ = A −j (m) as an element of S β . For each f ∈ L 2 (S β , τ ), Let E = Θ(E ′ ). Then to say that (t), (a 0 , a 1 , · · · , a n , · · · )) ∈ span{z v λ·χ By our measure-theoretic isomorphism between (S β , τ ) and Corollary 3.4 and Proposition 3.5 immediately give us the following result.
We can rewrite condition (3) in Corollary 3.6 to obtain the following characterization of generalized MSF wavelets.
) ∈ E ′ } has a single atom with respect to ν z , that is, for almost all z, there exists (a j ) ∞ j=0 ∈ E ′ z such that Proof. Assume that λ · χ E ′ is a generalized MSF wavelet. By Corollary 3.6, we can assume that conditions (1), (2) and (3) hold for E ′ . This setting is a special case of a theorem of G. W. Mackey (Theorem 2.11 of [21]) about a direct integral of Hilbert spaces relative to a measure that has a decomposition over a quotient space. Mackey's theorem says that the original direct integral is naturally isomorphic to the direct integral over the quotient space of the direct integrals over the fibers. In our case, the original Hilbert spaces are all equal to C, and the measure τ is decomposed over T d relative to the Haar measure on T, with fiber measures ν z as in Proposition 2.4. For each z let H z be L 2 (E ′ , ν z ). Then Mackey's theorem assures us that the spaces H z form an integrable bundle and that in such a way that the class of a bounded Borel function f in L 2 (τ ) corresponds to the class of the section of the bundle over T d whose value at each z is the class of f in H z . This works because the bounded Borel functions are square-integrable relative to every finite measure, providing a dense subspace in every one of the Hilbert spaces involved.
In particular, the function χ E ′ corresponds in this particular way to a section ψ of the Hilbert bundle over T d whose fibers are the spaces H z . Multiplying ψ by Laurent polynomials in z times λ(z) gives a subspace of the sections of the bundle that is L 2dense because the corresponding functions on E ′ are dense. Hence, for almost every z the one-dimensional space spanned by ψ(z) is dense in H z . Thus almost every H z is one dimensional, which implies that ν z is a point mass for almost every z.
Conversely, if (1) and (2) hold and almost every ν z is a point mass, we have that almost every H z is one dimensional. Thus the space of multiples of ψ by Laurent polynomials times λ is dense in the space of L 2 sections. This implies that the same multiples of χ E ′ are dense in L 2 ( τ ). Thus, condition (3) holds, and by Corollary 3.6 we have that λ · χ E ′ is a generalized MSF wavelet.
As an immediate consequence of the above theorem, we see that the MRA corresponding to the inflated Cantor fractal set and the inflated Sierpinski fractal set cannot have generalized MSF single wavelets: Corollary 3.8. Let L 2 (X, µ) be a measure space carrying a translation by Z d and dilation by the diagonal dilation matrix A, and suppose that L 2 (X, µ) has a MRA corresponding to a single scaling function φ and a finite wavelet family , with corresponding continuous"low-pass" filter function m : T d → C. If |m(z)| < √ N, ∀z ∈ T d , then L 2 (X, µ) does not have a MSF wavelet corresponding to the translation and dilation operators.
Proof. Since |m(z)| < √ N, ∀ z ∈ T d , the fiber measures ν z will never have atoms, by Lemma 2.6. By Theorem 3.7, we see that L 2 (X, µ) will not have MSF wavelets.
Remark 3.9. Corollary 3.8 does not rule out the existence of a single wavelet in the inflated fractal spaces corresponding to the Cantor set or Sierpinski gasket. It just states that if a single wavelet does exist, it cannot be a generalized MSF wavelet as defined above.
4. The wavelet representation of the Baumslag-Solitar group, the measure τ, on S β , and induced representations We now relate the existence of generalized MSF wavelets to certain properties of the associated representation of the Baumslag-Solitar group. Recall that for N ≥ 2, the classical Baumslag-Solitar group BS N has two generators a, b, satisfying the relations The discrete group BS N can also be written as a semidirect product Q N ⋊ θ Z where θ(q) = Nq, for q ∈ Q N , the N-adic rationals. We generalize the Baumslag-Solitar group to diagonal d × d dilation matrices A with diagonal entries N 1 , N 2 , · · · , N d all integers ≥ 2. Recall we consider the A-adic rationals Q A = ∪ ∞ j=0 A −j [Q d ], and defining θ : Q A → Q A by θ(q) = A(q), we can parametrize Q A ⋊ θ Z as the set of pairs {(q, m) : q ∈ Q A , m ∈ Z}, where the group operation is given by Denote this group by BS A . If X ⊂ R d is a subset invariant under integral translation and dilation by A, and µ is a σ-finite Borel measure on X that is invariant under translation by the integers and quasi-invariant via the constant K > 0 under dilation by A, so that µ(A(S)) = Kµ(S) for all Borel subsets S of X, there is a related representation of BS A on L 2 (X, µ), called a wavelet representation of BS A , defined on L 2 (X, µ) by We remark that for d = 1, the generator a ∈ BS N corresponds to (1, 0) and the generator b corresponds to (0, −1) ∈ Q N ⋊ θ Z.
Let m be a quadrature mirror filter on T d with respect to dilation by A such that the Haar measure of m −1 ({0}) is equal to 0. Usually the filter m will be a polynomial filter coming from the self-similarity relation satisfied by the generating subset of X ⊂ R d . In earlier sections, we studied the structure of the measure τ on S β such that the wavelet representation of BS A on L 2 (X, µ) is equivalent to the representation on L 2 (S β , τ ) defined by ). For simplicity of notation we have used W to denote both the (equivalent) representations of BS A rks on L 2 (X, µ) and L 2 (S β , τ ).
We now suppose that a wavelet representation of the above form, either on L 2 (S β , τ ) or on L 2 (X, µ), has a single orthonormal wavelet, i.e. suppose there exists ψ ∈ L 2 (X, µ) such that {D j T v (ψ) : j ∈ Z, v ∈ Z d , } is an orthonormal basis for L 2 (X, µ). Our aim is to characterize when ψ is a generalized MSF frequency wavelet in terms of properties of the associated wavelet representation.
The following theorem is related to a theorem from Eric Weber's 1999 CU Ph.D. thesis [26], which gave necessary and sufficient conditions for a single wavelet for dilation by N in L 2 (R) to be an MSF wavelet, in terms of the invariance of the wavelet subspace W 0 ⊂ L 2 (R) under translation by R. In our more general case, the space X ⊂ R d does not necessary carry an action by translation by all of R d , but only by the group Q A . As before, F : L 2 (X, µ) → L 2 (S β , τ ) represents the generalized Fourier transform of Dutkay from the Hilbert space associated to the (possibly fractal) space X to the Hilbert space associated to the solenoid.
Proof. i) implies ii): Suppose that ψ = F −1 (g) is a generalized MSF wavelet in the sense of Definition 3.1. This means that g((z n ) ∞ n=0 ) = λ(z 0 )χ E ((z n ) ∞ n=0 ) where the sets {σ j (E) : j ∈ Z} are pairwise disjoint up to sets of τ -measure 0 and τ (S β \ ∪ j∈Z σ j (E)) = 0, with λ : To say that f ∈ W 0 is equivalent to saying that f vanishes almost everywhere off of E. But if f vanishes almost everywhere off of E, it is clear that In a similar fashion, assuming that i) holds, f ∈ W j if and only if f vanishes τ almost everywhere off of σ j (E). But if f vanishes τ almost everywhere off of σ j (E), T q (f ) = (z j ) m · f will vanish τ -almost everywhere off of σ j (E). Thus W j is invariant under T α , and since q ∈ Q A was chosen arbitrarily, we see that W j is invariant under T q , ∀ q ∈ Q A .
ii) implies i): Let ψ = F −1 (g) be an orthonormal wavelet in L 2 (X, ν), so that {T k (ψ) : k ∈ Z} forms an orthonormal set. Then { T k (g) : k ∈ Z} is an orthonormal set in L 2 (S β , τ ), and by hypothesis, By the formula giving the operators T v , for v ∈ Z d , it is certainly true that To show that span{ T v (g) : v ∈ Z d } = L 2 (E, τ ), it suffices to show that whenever f ∈ L 2 (E, τ ) is a compactly supported simple function with support F lying in E, then f ∈ span{ T v (g) : v ∈ Z d }. Since for any fixed j ∈ Z, we know that W j is invariant under Since finite linear combinations of characters are norm-dense in C(S β ), we obtain that p((z n )) · g((z n )) ∈ span{ T v (g) : v ∈ Z d }, for any continuous function p defined on S β . From this we deduce that if q ∈ L ∞ (S β , τ ) is an essentially bounded function defined on S β , then q · g is in span{ T v (g) : v ∈ Z d }. Let u ∈ L ∞ (S β , τ ) be the unique function of modulus 1 defined on S β such that u · g = |g|. It follows that |g| ∈ span{ T v (g) : v ∈ Z d }.
For n = 1, let n }, and define F n = F ∩ E n . Note that the sets {F n } are pairwise disjoint and their union is equal to F. Let K = sup|f ((z n ))|. Fix ǫ > 0, and find N such that |g| , which is strictly less than N on ∪ n≤N . Let f N = f g · χ ∪ n≤N Fn . Then since f is bounded, f 0 is bounded as well, and by our earlier remarks, f N · g is an element of span{ T k (g) : k ∈ Z}. One calculates f N · g = f · χ ∪ n≤N Fn . We see that f N · g is equal to f on ∪ n≤N F n , so that Since the set of all such f 's is dense in L 2 (E, τ ), we obtain that L 2 (E, τ ) ⊂ span{ T v (g) : v ∈ Z d }. It follows that and g = λ · χ E , for λ ∈ L 2 (S β , τ ) so that F −1 (g) is a generalized MSF wavelet, as desired.
Next we will show that in the case where one can find a single wavelet ψ ∈ L 2 (X, µ), whether or not ψ can be classified as a generalized MSF wavelet is characterized by whether the wavelet subspaces determined by ψ give the wavelet representation the structure of an induced representation. First, we consider the case where there exists a subset C of S β = Q A such that Ind BS A Q A ⊕ C γdτ is equivalent to the wavelet representation W. In this case, the Imprimitivity Theorem of G. Mackey shows that if W is induced from a representation U of Q A on the Hilbert space L, there must be a projection-valued measure from BS A /(Q A ) to L 2 (S β , τ ) such that W −1 (q,k) P E W (q,k) = P (q,k) −1 ·E .
Since BS A /(Q A ) ∼ = Z, any projection valued measure on BS N amounts to a orthogonal decomposition of L 2 (S β , τ ) into infinitely many closed subspaces: In particular, setting P {n} = P Wn := P n , so that P n represents the orthogonal projection of L 2 (S β , τ ) onto W n , Mackey's Theorem gives the necessary conditions This is equivalent to the statement that the subspace W 0 is invariant under the translations T q , ∀q ∈ Q N , or, what is the same thing, the subspaces W j are invariant under all integer translations. Since W −1 (0,−1) P 0 W (0,−1) = P 1 , and D = W −1 (0,−1) , we obtain D(W 0 ) = W 1 . It follows that if the representation of BS A above is induced from a representation U of the subgroup Q A in such a way that there exists g ∈ W 0 with { T v (g) : v ∈ Z d } being an orthonormal basis for W 0 , then { D j T v (g)} will be an orthonormal basis for W j , for each j ∈ Z. By the induced representation assumption, L 2 (S β , τ ) = ⊕ j∈Z W j . If g exists, it would follow that g would be an orthonormal wavelet in L 2 (S β , τ ) for dilation by D and translation by { T v : v ∈ Z d . We will now show that if any wavelet of this type exists, it must be a generalized MSF wavelet; i.e., there will exist E ⊂ S β with {σ j (E) : j ∈ Z} pairwise disjoint up to sets of τ -measure 0 with τ (S β \ ∪ j∈Z σ j (E)) = 0, and a function λ : T d → C such that T d |λ(z)| 2 ν z (E z )dz = 1 and g((z n ) ∞ n=0 ) = λ(z 0 )χ E ((z n ) ∞ n=0 ).
Theorem 4.2. Let ψ ∈ L 2 (X, µ) be a single wavelet with associated unitary dilation and translation operators D and {T v : ∈ Z d }, i.e. suppose that {D j T v (ψ) : j ∈ Z, v ∈ Z d } is an orthonormal basis for L 2 (X, µ). Let F : L 2 (X, µ) → L 2 (S β , τ ) be the generalized Fourier transform of Dutkay corresponding to a multiresolution analysis coming from a self-similar space L 2 (X, µ). Then, the following are equivalent: i) ψ is a generalized MSF wavelet.
ii) The wavelet subspaces W j = span{D j T v (ψ)} are the closed subspace corresponding to a system of imprimitivity Proof. Assume that condition i) holds. We know that the wavelet representation generated by D and {T v : v ∈ Z d } on L 2 (X, µ) is unitarily equivalent to the representation on L 2 (S β , τ ) generated by D and { T v : v ∈ Z d }. As in the beginning of this section, we denote this latter representation by W. Theorem 4.1, condition i) implies that for each j ∈ Z, W j is invariant under the operators { T v : v ∈ Z d }. From the discussion given prior to the statement of the Theorem, in order that the wavelet representation W : BS A → U(L 2 (S β , τ )) be induced from a representation of Q A , we need to construct a projectionvalued measure {P n : n ∈ Z} onto L 2 (S β , τ ), such that for every (q, n) ∈ Q A ⋊ θ Z and every S ⊂ Z ≡ Q A ⋊ θ Z/Z, W (q,−k) P 0 (f ) = P k W (q,−k) f, ∀k ∈ Z, q ∈ Q A .
By condition i), F (ψ) = λ · χ E , where E ⊂ S β is a Borel set such that the sets {σ j (E) : j ∈ Z} pairwise disjoint up to sets of τ -measure 0 and τ (S β \[∪ j∈Z σ j (E)]) = 0. Setting W j = L 2 (σ −j (E), τ ), we claim that defining P j = P W j , we obtain a projectionvalued measure on Z satisfying the desired imprimitivity conditions. First we note that W j ⊥ W k for j = k, so that the projections {P j : j ∈ Z} are mutually orthogonal. Secondly, it's clear that ⊕ j∈Z L 2 (σ j (E), τ ) = L 2 (S β , τ ). Therefore our definition provides a projection-valued measure from Z into projections on L 2 (S β , τ ). We now show that this projection-valued measure satisfies the requirements stated in Equation 7.
To verify Equation 7, it is enough to show that W (q, k)P j = P W j−k W (q, k), ∀(q, k) ∈ Q A ⋊ θ Z, ∀j ∈ Z.
Recall that given unitary operators D and T q , q ∈ Q A on L 2 (S β , τ ), as defined above, W (q, n) is defined by W (q, n); = T q · D n . we obtain the wavelet representation of the Baumslag-Solitar group BS A .
We calculate: On the other hand, P j−k W (q, k)f ((z n )) = χ σ j−k (E) · T q D k f ((z n )) We have established the equality W (q, k)P j = P j−k W (q, k) and it follows that Equation 7 is satisfied, so that W is a representation that is induced from a representation on Q A , with imprimitivity structure provided by the pairwise orthogonal wavelet subspaces. We thus have established condition (ii). Assume now that condition (ii) holds. Then Equation 7 is satisfied with respect to the wavelet subspaces {W j : j ∈ Z}. In particular, Equation 7 is satisfied with respect to W 0 = span{ T k (F (ψ) : k ∈ Z}. so that W (q, 0)P 0 = P 0 W (q, 0), ∀ q ∈ Q A , or, what is the same thing, W 0 is invariant under the translation operators T q , ∀ q ∈ Q A . Note that D j T q D −j = T A −j (q) , ∀ j ∈ Z, and all q ∈ Q A , so that T A −j (q) D j = D j T q , ∀ j ∈ Z, and all q ∈ Q A .
Since W j = D j (W 0 , ) for any h ∈ W j , we can find f ∈ W 0 with h = D j (f ). Then if r ∈ Q A , T r (h) = T r D j (f ) = D j T A j (r) (f ).
Since we have shown W 0 is invariant under the translation operators T β , ∀ β ∈ Q N , T A j (r) (f ) ∈ W 0 . But then D j T A j (r) (f ) ∈ D j (W 0 ) = W j . It follows that for each fixed j ∈ Z, W j is invariant under T r , ∀ r ∈ Q A . Now suppose that either of the equivalent conditions of Theorem 4.2 are satisfied, and E is the subset of (S β , τ ) serving as the candidate for a "wavelet set". For each (z n ) ∈ E, by Pontryagin duality (z n ) is a character on Q A so that the pairing q, (z n ) defines a one-dimensional unitary representation of Q A . If we take the direct integral of these representations of Q A , we obtain the direct integral representation ⊕ E (z n )dτ of Q A on L 2 (E, τ ), given by the obvious formula L(q)f ((z n ) ∞ n=0 ) = q, (z n ) f ((z n ) ∞ n=0 ). Recall that the process of inducing and taking direct integrals commutes, so that A computation similar to that given in Theorem 2.1 of [20] gives the following Corollary 4.3. Let g be an orthonormal wavelet in L 2 (S β , τ ) corresponding to dilation by D and "translation" by the operators { T v : v ∈ Z d }, and suppose that g corresponds to a generalized MSF wavelet, so that g = λ · χ E where the sets {σ k (E) : k ∈ Z} tile S β with respect to the measure τ. Then the wavelet representation W of BS A on L 2 (S β , τ ) is unitarily equivalent the direct integral of induced representations ⊕ E [Ind BS A Q A (z n ) ∞ n=0 ]dτ. Proof. The proof is very similar to that given in Theorem 2.1 of [20]. The most technical part is to construct a Hilbert space isomorphism between L 2 (S β , τ ) and L 2 (E × Z, τ ×count) that intertwines the wavelet representation W and the induced representation in the desired fashion. We leave details to the reader. Remark 4.4. Thus, a single wavelet ψ ∈ L 2 (X, µ) can be used to construct orthogonal subspaces that regulate the induction of a wavelet representation from a representation of the normal subgroup Q N if and only if ψ is a generalized MSF wavelet. Moreover, the induced representation involved will be a direct integral of monomial representations, as in [20]