Orthonormal bases generated by Cuntz algebras
aa r X i v : . [ m a t h . F A ] D ec ORTHONORMAL BASES GENERATED BY CUNTZ ALGEBRAS
DORIN ERVIN DUTKAY, GABRIEL PICIOROAGA, AND MYUNG-SIN SONG
Abstract.
We show how some orthonormal bases can be generated by representations of theCuntz algebra; these include Fourier bases on fractal measures, generalized Walsh bases on the unitinterval and piecewise exponential bases on the middle third Cantor set.
Contents
1. Introduction 12. QMF bases and representations of the Cuntz algebra 33. Orthonormal bases generated by Cuntz algebras 83.1. Piecewise exponential bases on fractals 93.2. Walsh bases 11References 151.
Introduction
The Cuntz algebra O N , [Cun77] is the C ∗ -algebra generated by N isometries S i , i = 0 , . . . , N − S ∗ i S j = δ ij , i, j = 0 , . . . , N − , N − X i =0 S i S ∗ i = I. The Cuntz algebras are ubiquitous in analysis, but we draw our inspiration from wavelet theory.The role played by the Cuntz algebras in wavelet theory was described in the work of Bratteliand Jorgensen [BJ02a, BJ02b, BEJ00, BJ97]. Orthonormal wavelet bases are constructed fromvarious choices of quadrature mirror filters (QMF) (see [Dau92]). These filters are in one-to-onecorrespondence with certain representations of the Cuntz algebra. In section 2, we will show howthe ideas of Bratteli and Jorgensen carry over without too much difficulty in a more general settingassociated to some non-linear dynamics. We describe here this setting and give some examples.
Definition 1.1.
Let X be a compact metric space and µ a Borel probability measure on X . Let r : X → X an N -to-1 onto Borel measurable map, i.e. | r − ( z ) | = N for µ .a.e. z ∈ X , where | · | indicates cardinality. We say that µ is strongly invariant (for r ) if for every continuous function f on X the following invariance equation is satisfied: Mathematics Subject Classification.
Key words and phrases.
Cuntz algebras, fractal, Fourier basis, Hadamard matrix, quadrature mirror filter. (1.2) Z f dµ = 1 N Z X r ( w )= z f ( w ) dµ ( z ) Assumption.
In this paper µ will be a strongly invariant measure for the N -to-1 map r : X → X as in Definition 1.1 Example 1.2.
Let T = { z ∈ C : | z | = 1 } be the unit circle. Let r ( z ) = z N , z ∈ T . Let µ be theHaar measure on T . Then µ is strongly invariant. An equivalent system can be realized on[0 , r ( x ) = N x mod 1, x ∈ [0 ,
1] with the Lebesgue measure dx on [0 , T with the unit interval [0 ,
1] by z = e πix . Example 1.3.
Let Γ be a countable discrete abelian group. Let α : Γ → Γ be an endomorphismof Γ such that α (Γ) has finite index N in Γ and(1.3) ∩ n ≥ α n (Γ) = { } Let ˆΓ be the compact dual group and let µ be the Haar measure on ˆΓ, µ (ˆΓ) = 1. Denote by α ∗ the dual endomorphism on ˆΓ, w w ◦ α ( w ∈ ˆΓ). Observe that α ∗ is surjective, | Ker α ∗ | = N so | α ∗− ( z ) | = N for all z ∈ ˆΓ, and condition (1.3) implies that ∪ n ≥ Ker α ∗ n is dense in ˆΓ. Proposition 1.4.
The Haar measure on ˆΓ is strongly invariant for α ∗ .Proof. To prove the strong invariance relation (1.2) it is enough to check it on characters on ˆΓ,which by Pontryagin duality are given by the elements of Γ and we denote them by e γ ( w ) = w ( γ ), γ ∈ Γ, w ∈ ˆΓ. Fix γ ∈ Γ, γ = 0. Pick an element g ∈ ˆΓ such that e γ ( g ) = 1. We have Z ˆΓ N X α ∗ ( w )= z e γ ( w ) dµ ( z ) = Z ˆΓ N X α ∗ ( w )= z − α ∗ ( g ) e γ ( w ) dµ ( z ) = Z ˆΓ N X α ∗ ( u )= z e γ ( u − g ) dµ ( z )= e γ ( g ) Z ˆΓ N X α ∗ ( u )= z e γ ( u ) dµ ( z ) . Since e γ ( g ) = 1 it follows that R ˆΓ 1 N P α ∗ ( w )= z e γ ( w ) dµ ( z ) = 0. Since R ˆΓ e γ ( z ) dµ ( z ) = 0 thestrong invariance of µ is obtained. (cid:3) Example 1.5.
We consider affine iterated function systems with no overlap. Let R be a d × d expansive real matrix, i.e., all the eigenvalues of R have absolute value strictly greater than 1.Let B ⊂ R d a finite set such that N = | B | . Define the affine iterated function system(1.4) τ b ( x ) = R − ( x + b ) ( x ∈ R d , b ∈ B )By [Hut81] there exists a unique compact subset X B of R d which satisfies the invariance equation(1.5) X B = ∪ b ∈ B τ b ( X B ) X B is called the attractor of the iterated function system ( τ b ) b ∈ B . Moreover X B is given by RTHONORMAL BASES GENERATED BY CUNTZ ALGEBRAS 3 (1.6) X B = ( ∞ X k =1 R − k b k : b k ∈ B for all k ≥ ) Also, from [Hut81], there is a unique probability measure µ B on R d satisfying the invarianceequation(1.7) Z f dµ B = 1 N X b ∈ B Z f ◦ τ b dµ B for all continuous compactly supported functions f on R . We call µ B the invariant measure for theIFS ( τ b ) b ∈ B . By [Hut81], µ B is supported on the attractor X B . We say that the IFS has no overlapif µ B ( τ b ( X B ) ∩ τ ′ b ( X B )) = ∅ for all b = b ′ in B .Assume that the IFS ( τ b ) b ∈ B has no overlap. Define the map r : X B → X B (1.8) r ( x ) = τ − b ( x ) , if x ∈ τ b ( X B )Then r is an N -to-1 onto map and µ B is strongly invariant for r . Note that r − ( x ) = { τ b ( x ) : b ∈ B } for µ B .a.e. x ∈ X B . Example 1.6.
Let r be a rational map on the complex sphere C ∞ . Let J be its Julia set. Thenby [Bro65], [OP72] there exists a stongly invariant measure µ supported on J , which is non-atomic.The Julia set is invariant for r and the restriction r : J → J is a N -to-1 onto map where N = deg ( r ).We will show in Section 2 Proposition 2.7 how representations of the Cuntz algebra are obtainedfrom a choice of a quadrature mirror filter (QMF) basis (Definition 2.4. Then we show how QMFbases can be constructed using some unitary matrix valued functions (Theorem 2.12). This givesus a large variety of representations of the Cuntz algebras, which we use in Section 3 to constructvarious orthonormal bases.The central result of the paper is Theorem 3.1, where we present a general criterion for a Cuntzalgebra representation to generate an orthonormal basis. As a corollary (Theorem 3.5), whenapplied to some affine iterated function systems, we obtain a construction of piecewise exponentialbases on some Cantor fractal measures which extends a result of Dutkay and Jorgensen [DJ06b].In particular, we obtain piecewise exponential orthonormal bases on the middle third Cantor set(Example 3.8) which is known [JP98] not to have any orthonormal bases of exponential functions.Another corollary to our Theorem 3.1 gives us a construction of generalized Walsh bases on theunit interval starting from any unitary N × N matrix with constant first row.2. QMF bases and representations of the Cuntz algebra
Definition 2.1. A quadrature mirror filter (QMF) for r is a function m in L ∞ ( X, µ ) with theproperty that(2.1) 1 N X r ( w )= z | m ( w ) | = 1 , ( z ∈ X ) DORIN ERVIN DUTKAY, GABRIEL PICIOROAGA, AND MYUNG-SIN SONG
As shown by Dutkay and Jorgensen [DJ05, DJ07], every QMF gives rise to a wavelet theory.Various extra conditions on the filter m will produce wavelets in L ( R ) [Dau92], on Cantor sets[DJ06a, MP11], on Sierpinski gaskets [DMP08] and many others. Theorem 2.2. [DJ05, DJ07]
Let m be a QMF for r . Then there exists a Hilbert space H , arepresentation π of L ∞ ( X ) on H , a unitary operator U on H and a vector ϕ in H such that (i) (Covariance) (2.2) U π ( f ) U ∗ = π ( f ◦ r ) , ( f ∈ L ∞ ( X ))(ii) (Scaling equation) (2.3) U ϕ = π ( m ) ϕ (iii) (Orthogonality) (2.4) h π ( f ) ϕ , ϕ i = Z f dµ, ( f ∈ L ∞ ( X ))(iv) (Density) (2.5) span (cid:8) U − n π ( f ) ϕ : f ∈ L ∞ ( X ) , n ≥ (cid:9) = H Definition 2.3.
The system ( H , U, π, ϕ ) in Theorem 2.2 is called the wavelet representation asso-ciated to the QMF m .To construct a multiresolution, as in [Dau92], for a wavelet representation, one needs a QMFbasis. Definition 2.4. A QMF basis is a set of N QMF’s m , m , . . . , m N − such that(2.6) 1 N X r ( w )= z m i ( w ) m j ( w ) = δ ij , ( i, j ∈ { , . . . , N − } , z ∈ X )We can interpret these conditions in terms of a conditional expectation: Definition 2.5.
Let B be the Borel sigma-algebra on X and r − ( B ) be the sigma-algebra r − ( B ) = { r − ( B ) : B ∈ B} . Note that the r − ( B )-measurable functions are of the form f ◦ r ,where f is Borel measurable.The conditional expectation from to B to r − ( B ) is defined by(2.7) E ( f )( z ) = 1 N X r ( w )= z f ( w ) , ( z ∈ X )Alternatively E ( f ) can be defined, up to µ -measure zero as a r − ( B )-measurable function such that(2.8) Z f g ◦ rdµ = Z E ( f ) g ◦ rdµ, for all g ∈ L ∞ ( X, µ ) . Proposition 2.6.
A set of functions ( m i ) N − i =0 in L ∞ ( X, µ ) is a QMF basis if and only if (2.9) E ( m i m j ) = δ ij , ( i, j ∈ { , . . . N − } ) RTHONORMAL BASES GENERATED BY CUNTZ ALGEBRAS 5
In this case any function f ∈ L ( X, µ ) can be written in the QMF basis as (2.10) f = N − X i =0 E ( f m i ) m i Proof.
The first statement is clear. For the second, define for f ∈ L ( X, µ ) the vector-valuedfunction F ( f )( z ) = ( f ( w )) r ( w )= r ( z ) ∈ C N . Note that the QMF basis property implies that( F ( √ N m i )( z )) N − i =0 is an orthonormal basis in C n . Then for z ∈ XF ( f )( z ) = N − X i =0 (cid:28) F ( f )( z ) , F ( 1 √ N m i )( z ) (cid:29) C N F ( 1 √ N m i )( z ) = N − X i =0 E ( f m i )( z ) F ( m i )( z )Then looking at the first component (since r ( z ) = r ( z ) one can take w = z ) we get (2.10). (cid:3) Next, we show how a QMF basis induces a representation of the Cuntz algebra.
Proposition 2.7.
Let ( m i ) N − i =0 be a QMF basis. Define the operators on L ( X, µ )(2.11) S i ( f ) = m i f ◦ r, i = 0 , . . . , N − Then the operators S i are isometries and they form a representation of the Cuntz algebra O N , i.e. (2.12) S ∗ i S j = δ ij , i, j = 0 , . . . , N − , N − X i =0 S i S ∗ i = I The adjoint of S i is given by the formula (2.13) S ∗ i ( f )( z ) = 1 N X r ( w )= z m i ( w ) f ( w ) Proof.
We compute the adjoint: take f , g in L ( X, µ ). We use the strong invariance of µ . h S ∗ i f , g i = Z f m i g ◦ r dµ = Z N X r ( w )= z m i ( w ) f ( w ) g ( z ) dµ ( z )Then (2.13) follows. The Cuntz relations in (2.12) are then easily checked with Proposition 2.6. (cid:3) Every QMF basis generates a multiresolution for the wavelet representation associated to m .Since the ideas are simple and are the same as in the classical wavelet theory presented in [Dau92],we omit the proof. Note though, that the intersection of the resolution spaces might be non-trivial(for example, if m = 1 then 1 is contained in this intersection). Proposition 2.8.
Let ( m i ) N − i =0 be a QMF basis. Let ( H , U, π, ϕ ) be the wavelet representationassociated to m . Define (2.14) V := span { π ( f ) ϕ : f ∈ L ∞ ( X ) } , V n = U − n V , n ∈ Z (2.15) ψ i = U − π ( m i ) ϕ, i = 1 , . . . , N − DORIN ERVIN DUTKAY, GABRIEL PICIOROAGA, AND MYUNG-SIN SONG (2.16) W i := span { π ( f ) ψ i : f ∈ L ∞ ( X ) } Then (i) ∪ n ∈ Z V n = H (ii) V = V ⊕ W ⊕ · · · ⊕ W N − (iii) If ∩ n ∈ Z V n = { } then M n ∈ Z U n ( W ⊕ · · · ⊕ W N − ) = H A particular case which we will use in Section 3, is that of QMF bases generated by Hadamardmatrices which are defined from a finite set B and its spectrum Λ. Definition 2.9.
Denote by e λ ( x ) := e πiλ · x for λ, x ∈ R d . Let B be a finite subset of R d , | B | =: N .We say that a finite set Λ in R d is a spectrum for B if | Λ | = N and the matrix1 √ N [ e πib · λ ] λ ∈ Λ b ∈ B is unitary. Let B and L be finite subsets of Z d , | B | =: N = | L | and let R be an expansive d × d integer matrix. We say that ( B, L ) is a Hadamard pair with scaling factor R if L is a spectrum for R − B ; equivalently, the matrix 1 √ N [ e πiR − b · l ] l ∈ Lb ∈ B is unitary. Example 2.10.
Consider the setting in Example 1.5. We have the following equivalence:
Proposition 2.11.
A finite set Λ in R d is a spectrum for R − B if and only if ( e λ ) λ ∈ Λ is a QMFbasis. Let L be a finite subset of Z d . Then ( B, L ) is a Hadamard pair with scaling factor R if andonly if ( e l ) l ∈ L is a QMF basis.Proof. We have1 N X r ( w )= z e λ ( w ) e λ ′ ( w ) = 1 N X b ∈ B e πiτ b ( z ) · ( λ − λ ′ ) = 1 N X b ∈ B e πiR − ( z + b ) · ( λ − λ ′ ) = e πiR − ( z ) · ( λ − λ ′ ) N X b ∈ B e πiR − b · ( λ − λ ′ ) Thus, the QMF basis condition is equivalent to1 N X b ∈ B e πiR − b · ( λ − λ ′ ) = δ λλ ′ which is exactly the orthogonality of the columns of the matrix1 √ N [ e πiR − b · λ ] λ ∈ Λ b ∈ B The equivalence for Hadamard pairs follows as a particular case. (cid:3)
RTHONORMAL BASES GENERATED BY CUNTZ ALGEBRAS 7 If B is a finite set and R − B has spectrum Λ, then the set { e λ : λ ∈ Λ } is a QMF basis, byProposition 2.11. Then, with Proposition 2.7, the operators S λ f = e λ f ◦ r form a representationof the Cuntz algebra. Such representations were studied in [DJ12].The next theorem shows how QMF bases can be constructed from unitary matrix valued functionsas in the work of Bratteli and Jorgensen [BJ02a, BJ02b, BEJ00, BJ97], now in a more generalcontext. Theorem 2.12.
Fix ( m i ) N − i =0 a QMF basis. There is a one-to-one correspondence between thefollowing two sets: (i) QMF bases ( m ′ i ) N − i =0 (ii) Unitary valued maps A : X → U N ( C ) Given a QMF basis ( m ′ i ) N − i =0 the matrix A with entries (1) A ij ( z ) = 1 N X r ( w )= z m ′ i ( w ) m j ( w ) , ( z ∈ X, i, j = 0 , . . . , N − is unitary.Given a unitary-valued map A : X → U N ( C ) , the functions form a QMF basis (2) m ′ i ( z ) = N − X j =0 A ij ( r ( z )) m j ( z ) , ( z ∈ X, i = 0 , . . . N − These correspondences are inverse to each other.Proof.
The result requires some simple computations N − X j =0 A ij ( z ) A i ′ j ( z ) = 1 N X j X r ( w )= z m ′ i ( w ) m j ( z ) · X r ( w ′ )= z m ′ i ′ ( w ′ ) m j ( w ′ ) =1 N X w,w ′ m ′ i ( w ) m ′ i ′ ( w ′ ) · X j m j ( w ) m j ( w ′ ) = 1 N X w,w ′ m ′ i ( w ) m ′ i ′ ( w ′ ) δ w,w ′ = δ ii ′ Note that we used the equality X j m j ( w ) m j ( w ′ ) = δ ww ′ which follows from the fact that the matrix1 √ N [ m i ( w )] i =0 ,...N − w ∈ r − ( z ) is unitary, which, in turn, is a consequence of the QMF property. Hence A is unitary.If A is unitary, we check the QMF relations:1 N X r ( w )= z m ′ i ( w ) m ′ j ( w ) = 1 N X w X k A ik ( r ( w )) m k ( w ) X l A jl ( r ( w )) m l ( w ) =1 N X k,l A ik ( z ) A jl ( z ) X w m k ( w ) m l ( w ) = X k,l A jk ( z ) A jl ( z ) δ kl = δ ij DORIN ERVIN DUTKAY, GABRIEL PICIOROAGA, AND MYUNG-SIN SONG
Hence ( m ′ i ) N − i =0 is a QMF basis.The fact that the two correspondences are inverse to each other follows from the next computation: X j A ij ( r ( z )) m j ( z ) = X j N X r ( w )= r ( z ) m ′ i ( w ) m j ( w ) m j ( z ) = X r ( w )= r ( z ) m ′ i ( w ) · N X j m j ( w ) m j ( z )= X r ( w )= r ( z ) m ′ i ( w ) δ wz = m ′ i ( z ) (cid:3) Remark 2.13.
Note that the equation (1) can be reformulated as A ij ( r ( z )) = E ( m ′ i m j ). The con-ditional expectation E can be regarded as a L ∞ ( X, µ )-valued inner product h f , g i L ∞ ( X,µ ) = E ( f g )for f , g ∈ L ∞ ( X, µ ). The QMF basis condition is equivalent to the orthogonality of ( m i ) N − i =0 withrespect to this inner product. Since the dimension of L ∞ ( X, µ ) as a module over E ( L ∞ ( X, µ )) = L ∞ ( X, r − ( B ) , µ ) is N , the completeness is automatic, so ( m i ) N − i =0 is an orthonormal basis for thisinner product. Thus A ◦ r is the change of base matrix from ( m i ) to ( m ′ i ). Equation (2) can beunderstood in the sense that a unitary matrix maps orthonormal bases into orthonormal bases.3. Orthonormal bases generated by Cuntz algebras
Next, we present the central result of our paper. It gives a general criterion for a family generatedby the Cuntz isometries to be an orthonormal basis.
Theorem 3.1.
Let H be a Hilbert space and ( S i ) N − i =0 be a representation of the Cuntz algebra O N .Let E be an orthonormal set in H and f : X → H a norm continuous function on a topologicalspace X with the following properties: (i) E = ∪ N − i =0 S i E . (ii) span { f ( t ) : t ∈ X } = H and || f ( t ) || = 1 , for all t ∈ X . (iii) There exist functions m i : X → C , g i : X → X , i = 0 , . . . , N − such that (3.1) S ∗ i f ( t ) = m i ( t ) f ( g i ( t )) , t ∈ X. (iv) There exist c ∈ X such that f ( c ) ∈ span E . (v) The only function h ∈ C ( X ) with h ≥ , h ( c ) = 1 , ∀ c ∈ { x ∈ X : f ( x ) ∈ span E} , and (3.2) h ( t ) = N − X i =0 | m i ( t ) | h ( g i ( t )) , t ∈ X are the constant functions.Then E is an orthonormal basis for H .Proof. Define h ( t ) := X e ∈E |h f ( t ) , e i| = || P f ( t ) || , t ∈ X where P is the orthogonal projection onto the closed linear span of E .Since t f ( t ) is norm continuous we get that h is continuous. Clearly h ≥
0. Also, if f ( c ) ∈ span E , then || P f ( c ) || = || f ( c ) || = 1 so h ( c ) = 1. In particular, from (ii) and (iv), h ( c ) = 1. We RTHONORMAL BASES GENERATED BY CUNTZ ALGEBRAS 9 check (3.2). Since the sets S i E , i = 0 , . . . N − t ∈ X : h ( t ) = N − X i =0 X e ∈E |h f ( t ) , S i e i| = N − X i =0 X e ∈E |h S ∗ i f ( t ) , e i| = N − X i =0 | m i ( t ) | X e ∈E |h f ( g i ( t )) , e i| == N − X i =0 | m i ( t ) | h ( g i ( t ))By (v), h is constant and, since h ( c ) = 1, h ( t ) = 1 for all t ∈ X . Then || P f ( t ) || = 1 for all t ∈ X . Since || f ( t ) || = 1 it follows that f ( t ) ∈ span E for all t ∈ X . But the vectors f ( t ) span H sospan E = H and E is an orthonormal basis. (cid:3) Remark 3.2.
The operators of the form Rh ( t ) = N − X i =0 | m i ( t ) | h ( g i ( t )) , t ∈ X, h ∈ C ( X ) , that appear in (3.2), are sometimes called Ruelle operators or transfer operators, see e.g. [Bal00].3.1. Piecewise exponential bases on fractals.
We apply Theorem 3.1 to the setting of Example2.10, in dimension d = 1 for affine iterated function systems, when the set R B has a spectrum L . Definition 3.3.
Let L in R , | L | = N , R > L is a spectrum for the set R B . We saythat c ∈ R is an extreme cycle point for ( B, L ) if there exists l , l , . . . , l p − in L such that, if c = c , c = c + l R , c = c + l R . . . c p − = c p − + l p − R then c p − + l p − R = c , and | m B ( c i ) | = 1 for i = 0 , . . . , p − m B ( x ) = 1 N X b ∈ B e πibx x ∈ R . Definition 3.4.
We denote by L ∗ the set of all finite words with digits in L , including the emptyword. For l ∈ L let S l be given as in (2.11) where m l is replaced by the exponential e l . If w = l l . . . l n ∈ L ∗ then by S w we denote the composition S l S l . . . S l n . Theorem 3.5.
Let B ⊂ R , ∈ B , | B | = N , R > and let µ B be the invariant measure associatedto the IFS τ b ( x ) = R − ( x + b ) , b ∈ B . Assume that the IFS has no overlap and that the set R B has a spectrum L ⊂ R , ∈ L . Then the set E ( L ) = { S w e − c : c is an extreme cycle point for ( B, L ) , w ∈ L ∗ } is an orthonormal basis in L ( µ B ) . Some of the vectors in E ( L ) are repeated but we count themonly once.Proof. Let c be an extreme cycle point. Then | m B ( c ) | = 1. Using the fact that we have equality inthe triangle inequality (1 = | m B ( c ) | ≤ N P b ∈ B | e πibc | = 1) , and since 0 ∈ B , we get that e πibc = 1so bc ∈ Z for all b ∈ B . Also there exists another extreme cycle point d and l ∈ L such that d + lR = c .Then we have: S l e − c ( x ) = e πilx e πi ( Rx − b )( − c ) , if x ∈ τ b ( X B ). Since bc ∈ Z and R ( − c ) + l = − d ,we obtain(3.3) S l e − c = e − d We use this property to show that the vectors S w e − c , S w ′ e − c ′ are either equal or orthogonal for w, w ′ in L ∗ and c, c ′ extreme cycle points for ( B, L ). Using (3.3), we can append some letters atthe end of w and w ′ suh that the new words have the same length: S w e − c = S wα e − d , S w ′ e − c ′ = S w ′ β e − d ′ , | wα | = | w ′ β | where d, d ′ are cycle points.Moreover, repeating the letters for the cycle points d and d ′ as many times as we want, we canassume that α ends in a repetition of the letters associated to d and similarly for β and d ′ . But,since | wα | = | w ′ β | , the Cuntz relations imply that S wα e − d ⊥ S w ′ β e − d ′ or wα = w ′ β . Assume | w | ≤ | w ′ | . Then α = w ′′ β for some word w ′′ . Then S wα e − d ⊥ S w ′ β e − d iff S α e − d ⊥ S w ′′ β e − d ′ .Also, α consists of repetitions of the digits of the cycle associated to d and similarly for d ′ . So S α e − d = e − f , S w ′′ β e − d ′ = e − f ′ , and all points d, d ′ , f, f ′ , c, c ′ all belong to the same cycle. So theonly case when S w e − c is not orthogonal to S w ′ e − c ′ is when they are equal.Next we check that the hypotheses of Theorem 3.1 are satisfied. We let f ( t ) = e − t ∈ L ( µ B ). Tocheck (i) we just to have to see that e − c ∈ ∪ l ∈ L S l E ( L ). But this follows from (3.3). Requirement(ii) is clear. For (iii) we compute S ∗ l e − t ( x ) = 1 N X b ∈ B e − πil · R ( x + b ) e − πit · R ( x + b ) = e − πx · R ( t + l ) N X b ∈ B e − πib ( t + lR ) == m B (cid:18) t + lR (cid:19) e − t + lR ( x )So (iii) is satisfied with m l ( t ) = m B ( t + lR ), g l ( t ) = t + lR .For (iv) take c = − c for any extreme cycle point ( 0 is always one). For (v), take h continuouson R , 0 ≤ h ≤ h ( c ) = 1 for all c with e − c ∈ span E ( L ), and h ( t ) = X l ∈ L (cid:12)(cid:12)(cid:12)(cid:12) m B (cid:18) t + lR (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) h (cid:18) t + LR (cid:19) := Rh ( t )In particular, we have h ( c ) = 1 for every extreme cycle point c . Assume h
1. First we will restrictour attention to t ∈ I := [ a, b ] with a ≤ min LR − , b ≥ max LR − , and note that g l ( I ) ⊂ I for all l ∈ L . Let m = min t ∈ I h ( t ). Then let h ′ = h − m , assume m <
1. Then Rh ′ ( t ) = h ′ ( t ) for all t ∈ R , h ′ has azero in I and h ≥ I , h ′ ( z ) = 0. But this implies that | m B ( g l ( z )) | h ′ ( g l ( z )) = 0 for all l ∈ L .Since P l ∈ L | m B ( g l ( z )) | = 1, it follows that for one of the l ∈ L we have h ′ ( g l ( z )) = 0. Byinduction, we can find z n = g l n − · · · g l z such that h ′ ( z n ) = 0. We prove that z is a cycle point.Suppose not. Since m B has finitely many zeros, for n large enough g α k · · · g α z n is not a zero for m B , for any choice of digits α , . . . , α k in L . But then, by using the same argument as above we getthat h ′ ( g α k · · · g α z n ) = 0 for any α , . . . , α k ∈ L . The points { g α k · · · g α z n : α , ...α k ∈ L, k ∈ N } are dense in the attractor X L of the IFS { g l } l ∈ L , thus h ′ is constant 0 on X L . But the extremecycle points c are in X L and since h ( c ) = 1 we have 0 = h ′ ( c ) = 1 − m , so m = 1. Thus h = 1 on I . Since we can let a → −∞ and b → ∞ we obtain that h ≡ (cid:3) Remark 3.6.
The functions in E ( L ) are piecewise exponential. The formula for S l ...l n e − c is(3.4) S l ...l n e − c ( x ) = e α ( b,l,c ) · e l + Rl + ... + R n − l n − + R n ( − c ) ( x ) RTHONORMAL BASES GENERATED BY CUNTZ ALGEBRAS 11 where α ( b, l, c ) = − [ b l + ( Rb + b ) l + ... + ( R n − b + ... + b n − ) l n ] + ( R n − b + ... + b n ) · c if x ∈ τ b ...τ b n X B . We have S l ...S l n e − c ( x ) = e l ( x ) e l ( rx ) ...e l n ( r n − x ) e c ( r n x )If x ∈ τ b ...τ b n X B then rx ∈ τ b ...τ b n X B , r n − x ∈ τ b n X B . So rx = Rx − b r x = Rrx − b = R x − Rb − b ... r n − x = R n − x − R n − b − ... − Rb n − − b n − r n x = R n x − R n − b − R n − b − ... − Rb n − − b n . The rest follows from a direct computation.
Corollary 3.7.
In the hypothesis of Theorem 3.1, if in addition
B, L ⊂ Z and R ∈ Z , then thereexists a set Λ such that { e λ : λ ∈ Λ } is an orthonormal basis for L ( µ B ) .Proof. If everything is an integer then, it follows from Remark 3.6 that S w e − c is an exponentialfunction for all w and extreme cycle points c . Note that, as in the proof of Theorem 3.1, bc ∈ Z forall b ∈ B . (cid:3) Example 3.8.
We consider the IFS that generates the middle third Cantor set: R = 3, B = { , } .The set { , } has spectrum L = { , / } . We look for the extreme cycle points for ( B, L ).We need | m B ( − c ) | = 1 so | e πi c | = 1, therefore c ∈ Z . Also c has to be a cycle for the IFS g ( x ) = x/ g / ( x ) = x +3 / so 0 ≤ c ≤ / − = 3 /
8. Thus, the only extreme cycle is { } . ByTheorem 3.1 E = { S w w ∈ { , / } ∗ } is an orthonormal basis for L ( µ B ). Note also that thenumbers e πiα ( b,l,c ) in formula (3.4) are ± πiB · L ⊂ πi Z .3.2. Walsh bases.
In the following, we will focus on the unit interval, which can be regarded asthe attractor of a simple IFS and we use step functions for the QMF basis to generate Walsh-typebases for L [0 , Example 3.9.
The interval [0 ,
1] is the attractor of the IFS τ x = x , τ x = x +12 , and the invariantmeasure is the Lebesgue measure on [0 , r defined in Example 1.5 is rx = 2 x mod1. Let m = 1, m = χ [0 , / − χ [1 / , . It is easy to see that { m , m } is a QMF basis. Therefore S , S defined as in Proposition 2.7 form a representation of the Cuntz algebra O . Proposition 3.10.
The set E := { S w w ∈ { , } ∗ } is an orthonormal basis for L [0 , , theWalsh basis.Proof. We check the conditions in Theorem 3.1. To see that (i) holds note that S f ( t ) = e t , t ∈ R . (ii) is clear. For (iii) we compute S ∗ e t ( x ) = 12 ( e πit · x/ + e πit · ( x +1) / ) = e πit · x/
12 (1 + e πit/ ) S ∗ e t ( x ) = 12 ( e πit · x/ − e πit · ( x +1) / ) = e πit · x/
12 (1 − e πit/ ) Thus (iii) holds with m ( t ) = (1 + e πit/ ), m ( t ) = (1 − e πit/ ), g ( t ) = g ( t ) = t . Since e = 1it follows that (iv) holds.For (v) take h continuous on R , 0 ≤ h ≤ h ( c ) = 1 for all c ∈ R with e t ∈ span E , in particular h (0) = 1 and h ( t ) = (cid:12)(cid:12)(cid:12)(cid:12)
12 (1 + e πit/ ) (cid:12)(cid:12)(cid:12)(cid:12) h ( t/
2) + (cid:12)(cid:12)(cid:12)(cid:12)
12 (1 − e πit/ ) (cid:12)(cid:12)(cid:12)(cid:12) h ( t/
2) = h ( t/ h ( t ) = h ( t/ n ) for all t ∈ R , n ∈ N . Letting n → ∞ and using the continuity of h , we get h ( t ) = h (0) = 1 for all t ∈ R . Since all conditions hold, we get that E is an orthonormal basis.That E is actually the Walsh basis follows from the following calculations: for | w | = n in { , } ∗ let n = P i x i i be the base 2 expansion of n . Because S f = f ◦ r , S f = m f ◦ r and m ≡ S w x ) = m ( r i x ) · m ( r i x ) · · · m ( r i k x ) , where i , i , . . . , i k correspond to those i with x i = 1 . Also m ( r i x ) = m (2 i x mod i ) are the Rademacher functions and thus we obtain the Walsh basis(see e.g. [SWS90]). (cid:3) The Walsh bases can be easily generalized by replacing the matrix1 √ (cid:18) − (cid:19) which appears in the definition of the filters m , m , with an arbitrary unitary matrix A withconstant first row and by changing the scale from 2 to N . Theorem 3.11.
Let N ∈ N , N ≥ . Let A = [ a ij ] be an N × N unitary matrix whose first row isconstant √ N . Consider the IFS τ j x = x + jN , x ∈ R , j = 0 , . . . , N − with the attractor [0 , andinvariant measure the Lebesgue measure on [0 , . Define m i ( x ) = √ N N − X j =0 a ij χ [ j/N, ( j +1) /N ] ( x ) Then { m i } N − i =0 is a QMF basis. Consider the associated representation of the Cuntz algebra O N .Then the set E := { S w w ∈ { , ...N − } ∗ } is an orthonormal basis for L [0 , .Proof. We check the conditions in Theorem 3.1. Let f ( t ) = e t , t ∈ R .To check (i) note that S ≡
1. (ii) is clear. For (iii) we compute: S ∗ k e t = 1 N N − X j =0 m k ( τ j x ) e t ( τ j x ) = 1 √ N N − X j =0 a kj e πit · ( x + j ) /N = e πit · x/N √ N N − X j =0 a kj e πit · j/N So (iii) is true with m k ( t ) = √ N P N − j =0 a kj e πit · j/N and g k ( t ) = tN .(iv) is true with c = 0. For (v) take h ∈ C ( R ), 0 ≤ h ≤ h ( c ) = 1 for all c ∈ R with e c ∈ span E ( in particular h (0) = 1), and h ( t ) = N − X k =0 | m k ( t ) | h ( t/N ) = h ( t/N ) N − X k =0 N | N − X j =0 a kj e − πit · j/N | = h ( t/N ) · N || Av || where v = ( e − πit · j/N ) N − j =0 . Since A is unitary, || Av || = || v || = N . Then h ( t ) = h ( t/N n ). Letting n → ∞ and using the continuity of h we obtain that h ( t ) = 1 for all t ∈ R . Thus, Theorem 3.1implies that E is an orthonormal basis. (cid:3) Remark 3.12.
We can read the constants that appear in the step function S w A with itself n times, where n is the length of the word w .Let A be an N × N matrix, B an M × M matrix. Then A ⊗ B has entries :( A ⊗ B ) i + Mi ,j + Mj = a i j b i j , i , j = 0 , . . . , N − i , j = 0 , . . . , M − A ⊗ B = Ab , Ab , · · · Ab ,M − Ab , Ab , · · · Ab ,M − ... ... . . . ... Ab M − , Ab M − , · · · Ab M − ,M − The matrix A ⊗ n is obtained by induction, tensoring to the left: A ⊗ n = A ⊗ A ⊗ ( n − .Thus A ⊗ A ⊗ A ⊗ · · · ⊗ A , n times, has entries A ⊗ ni + Ni + N i + ··· + N n − i n − ,j + Nj + ··· + N n − j n − = a i j a i j . . . a i n − j n − Now compute for i , . . . i n − ∈ { , . . . , N − } : S i ...i n − x ) = m i ( x ) m i ( rx ) . . . m i n − ( r n − x )Suppose x ∈ [ kN n , k +1 N n ), 0 ≤ k < N n and k = N n − j + N n − j + · · · + N j n − + j n − , where0 ≤ j , . . . , j n − < N .Then x ∈ [ j N , j +1 N ), rx = ( N x )mod1 ∈ [ j N , j +1 N ) , . . . , r n − x = ( N n − x )mod1 ∈ [ j n − N , j n − +1 N ), so m i ( x ) = √ N a i j , m i ( rx ) = √ N a i j , . . . , m i n − ( r n − x ) = √ N a i n − j n − hence S i ...i n − x ) = √ N n a i j . . . a i n − j n − = √ N n A ⊗ ni + Ni + N i + ··· + N n − i n − ,j Nj + ··· + N n − j n − Figure 1.
Walsh functions S w w of length 2. Example 3.13.
The pictures in Figure 1 show the Walsh functions that correspond to the scale N = 4 and the matrix A =
12 12 12 12 √ − √ √ − √ − − for the words of length 2, indicated at the top. RTHONORMAL BASES GENERATED BY CUNTZ ALGEBRAS 15
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