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<ul><li><p>Advances in Computational Mathematics 1(1993)109-126 109 </p><p>Wavelet bases for a set of commuting unitary operators </p><p>T.N.T . Goodman </p><p>Department of Mathematics and Computer Science, University of Dundee, Dundee DD1 4HN, Scotland, UK </p><p>S.L. Lee and W.S. Tang </p><p>Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511 </p><p>Received 9 April 1992; revised 9 February 1993 </p><p>Abst rac t </p><p>Let (U = U 1 . . . . . Ud) be an ordered d-tuple of distinct, pairwise commuting, unitary operators on a complex Hilbert space 3 , and let X := {x I . . . . . xr} c ~ such that </p><p>Z 4 n l JS U X .= {U 1 . . .U~ xj : (n a . . . . . nd ) ~ ~,d, j = 1 . . . . . r} isaRieszbasisof the dosed linear span ~0 of UZ'X. Suppose there is unitary operator D on ~ such that ~o c DT" o =: ~t and UnD = DU A" for all n E Z d, where A is a d x d matrix with integer envies and A := det(A) ~ 0. Then there is a subset F in *Ill, with r(A - 1 ) vectors, such that UZ'(F) is a Riesz basis of W 0, the orthogonal complement of *1," o in *It 1. The resulting mdtiscale and decomposition relations can be expressed in a Fourier representation by one single equation, in terms of which the duality principle follows easily. These results are a consequence of an extension, to a set of commuting unitary operators, of Robertson's Theorems on wandering subspace for a single unitary operator [24]. Conditions are given in order that UZd(F) is a Riesz basis o fW 0. They are used in the construction of a class of linear spline wavelets on a four-direction mesh. </p><p>Keywords: Hilbert space, commuting unitary operators, Riesz basis, wandering subspaces, multiresolution approximation, duality principle, box splines. </p><p>Subject classification: 41A15, 42C15, 47B37. </p><p>1. Introduction </p><p>Cons ider the H i lber t space ~ = L2(R d) o f square - in tegrab le complex -va lued funct ions on the d -d imens iona l Euc l idean space R d. For any vector v ~ R a, let T,, : L2(R a) ---> L2(R a) be def ined by </p><p> LC. Baltzer AG, Science Publishers </p></li><li><p>110 T~I.T. Goodman et al., Wavelet bases </p><p>T , f (x ) := f (x -u ), f E L2(Rd), x E R d. (1.1) </p><p>Let A be a d x d matrix with entries in 7, and det(A) ~ 0, and define DA : L2(R d) L2(R d) by </p><p>OAf(X ) := ( IdetA l ) l l2 f (Ax) , f E L2(Rd), x E R d. (1.2) </p><p>Clearly, DA and T,, are unitary operators on L2(Rd), and T,,DA = DATA,. In particular, if we let Ty = Tel, j = 1 . . . . . d, where e j := (t~y,t)f=l are the standard unit vectors in R d, then T : = (7"1 . . . . . Td) is a d-tuple of commuting unitary operators such that </p><p>TnDA = DA TAn, n E Z a, (1.3) </p><p>where we have used the multi-index notation T n : = TI n~ . . . T~ d, n = (n I . . . . . rid) eZ d. </p><p>Let r be a positive integer. A sequence (Tin)raG z of closed linear subspaces of L2(R d) is called a multiresolution approximation of Lz(R d) of multiplicity r with dilation matrix A if the following conditions hold: </p><p>T, n c Tm+1, m EZ, (1.4) </p><p>U Tm is dense in L2(R a) and A Tm= [0}, (1.5) meZ mel </p><p>fET m :~DAfETm+ 1, mEZ, (1.6) </p><p>f ET m :~ TA- , ,n f ET m, mEZ, nEZ a, (1.7) </p><p>3X = {I . . . . . r] c L2(R d) such that TZ ' (x ) is a Riesz basis of T o, (1.8) </p><p>where T za (X) : = {Tnj : n E 7I d, j = 1 . . . . . r}. If r = 1, the corresponding multi- resolution approximation of L2(R d) will be called simple. Lemari6 and Meyer [17] have constructed the tensor product scaling function o from a one-dimensional multiresolution approximation such that 0 generates a simple multiresolution approximation (~m)m~l of L2(R d) with A = diag(2 . . . . . 2), and TZ~({o}) is an orthonormal basis of To. Furthermore, there are 2 d - 1 functions ~, j ~ {0, 1 }d\{0}, such that </p><p>{llf j(X -- n) : n E 7/d, j E {0, 1}d\{0} } </p><p>is an orthonormal basis for the orthogonal complement ~W'o of To in T1, and </p><p>{2ma/211/j(2mx- n) : m ~ Z ,n E gd, j ~ {0, 1}d\{0}} </p><p>is an orthonormal basis of L2(Rd). </p></li><li><p>T.N.T. Goodman et al., Wavelet bases 111 </p><p>Riemenschneider and Shen [22] have constructed an orthonormal box-spline wavelet basis of L2(R a) for d = 1, 2, 3, using the orthonormalization technique of Mallat [18]. The orthonormal box-spline wavelet basis is a multivariate analogue of the Lemari6 basis [16]. Recently, in an attempt to extend the earlier works of Chui and Wang [3,4], Chui et al. [5], Jia and Micchelli [12], and Riemenschneider and Shen [23] have studied compactly supported non-orthonormal box-spline wavelets. </p><p>Our objective is to study the general setting of wavelet bases for a set of commuting unitary operators in a Hilbert space, and to apply the general theory in the construction of new wavelet bases in L2(Ra). </p><p>Let U = (U1 . . . . . Ud) be an ordered d-tuple of distinct unitary operators on a complex Hilbert space ~ such that Uk,,~ = UiUk, k, j = 1 . . . . . d. We shall use the multi-index notation U== U~I. . .U,~ ~ for m = (ml , ma)~Z d, with the convention that Uj is the identity operator on ~, j = 1 . . . . . d and assume that U m is the identity only if m = 0. </p><p>A closed linear subspace ~ of ~ is called a wandering subspace for U if Um(~g) 1 Un(~g) for all m, n ~ Z d, m ;~ n. Further, if ~ = Y.n~ldUn(~,), then we say that ~ is a complete wandering subspace of ~ for U. We observe that the results of Robertson (theorem 1, corollary and theorem 2 in [24]) on wandering subspaces for a single unitary operator can be easily extended to a set of commuting unitary operators. An analogue of Robertson's results involving non-orthonormal Riesz bases for a single unitary operator is more suitable for application to B-spline wavelets (see [15,3,4]). In section 2, we state results on Riesz bases generated by a set of commuting unitary operators on a finite set of vectors. These are applied in section 3 to the study of wavelet bases in Hilbert space. The general theory is applied, in section 4, in the construction of a linear wavelet on a four-directional mesh in R 2. This example has many desirable features which suggest that it may be more useful than the existing multivariate wavelets. </p><p>Throughout this paper, we shall denote the space of all square-integrable functions on the d-dimensional torus Ra/(2rcZ a) by L2((0, 2~)a), and let Ls2((0, 2n:) a) (respectively, -2 Lms((0, 2a)a)) be the set of all row vectors with s components (respectively, the set of all m x s matrices with entries) in L2((0, 2a)a). </p><p>2. Riesz bases for a finite set of commuting unitary operators </p><p>Let U := (U1 . . . . . Ud) be an ordered d-tuple of commuting distinct unitary operators on a complex Hilbert space ~. For V c ~, we shall write </p><p>O zd (V) ","" {Un"O ". n ~ Z d, v ~ V}, </p><p>and let </p></li><li><p>112 T.N.T. Goodman et al., Wavelet bases </p><p>Let Y= {Yl . . . . . Ys} c ~ and suppose </p><p>((yk, Unyt))neZa ~. /2(Za), k , l = 1 . . . . . s, (2.1) </p><p>where ( , ) denotes the inner product in ~. Then </p><p>( / s Or(0) := ~, (Yk, Unyt) ein o (2.2) </p><p>held kl I </p><p>is a Hermitian matrix with entries in L2((0, 2~)d). By Bochner's Theorem on positive definite functions ([25], p. 19), the matrix eDr(O) is positive semi-definite for almost all 0 ~ R a. </p><p>Clearly, U za (Y) is an orthonormal basis of (U zd (Y)) if and only if CDr(0) is the identity matrix for almost all 0 ~ R e. In the general case, let ,q,j(0), j = 1 . . . . . s, be the eigenvalues of ~r(O). Then we have the following characterization of U zd (Y) as a Riesz basis of its closed linear span (U z~ (Y)), which is a multivariate analogue of theorem 2.1 in [15]. </p><p>THEOREM 2.1 </p><p>The set U zd(Y) is a Riesz basis of (UZd(Y)) if and only if there exist positive constants CI and 6"2 such that </p><p>cl - __. c2 (2.3) </p><p>for almost every 0 ~ R a. </p><p>We shall omit the proof, which is essentially the same as that of theorem 2.1 in [15]. </p><p>The process of orthonormalization of a Riesz basis for a set of distinct commuting unitary operators, to produce an orthonormal basis for the same set of operators, can be accomplished by the following variant of the Fuglede-Putnam Theorem. </p><p>THEOREM 2.2 </p><p>Let ~f and ~ be complex Hilbert spaces, and for j = 1 . . . . . d, let Aj : ~ ---) and Bj : ~ ---) ~ be normal operators. If J : ~K --+ ~ is a bounded invertible operator such that Ai J = JBj , j = 1 . . . . . d, then there exists a unitary operator U : ~ ---) such that AjU = UBj, j = 1 . . . . . d. </p><p>The proof is a minor modification of that for the original Fuglede-Putnam Theorem (see [9], solution 192). The operator U is the unitary factor in the polar decomposition of J (see [9], problem 134), and is independent of Aj and Bj , j = 1 . . . . . d. </p></li><li><p>T.N.T. Goodman et al., Wavelet bases 113 </p><p>THEOREM 2.3 </p><p>Suppose that U zd (Y) is a Riesz basis of (U z~ (Y)). Then there exists a set I~ c (U z ~(Y)) of cardinality s, such that U zd (I?) is an orthonormal basis of (U za (Y)). </p><p>Proof Let e j, j = 1 . . . . . d, be the standard unit vectors in R a, and let Ej" 12(zd) s </p><p>--~ 12(zd) ~ be the shift operator in the direction of e y defined by </p><p>Eja(n) = a(n - e J), a 12(zd) s, n Z d. </p><p>Then {El . . . . . Ed} is a set of commuting unitary operators on 12(Za) s, and with E := (El . . . . . Ed), we have Ema(n) = a(n - m) for a ~ 12(Zd) ~, and m, n Z d. </p><p>Let J"/2(Ze)" --, (U zd (Y)) be defined by </p><p>$ </p><p>J (a 1 . . . . . a s ) := ~ ~ aJ(n)Un(yj) j=l n~Z d </p><p>Then UnJ = JE n, for all n 7/a. By the Fuglede-Putnam Theorem, there exists a unitary operator H : 12(7/a) s --> (U zd (Y)) such that UnH = HE", for all n ~ 7/a. </p><p>Let e be the unit sequence in/2(7/d) such that e(n)= So ,,, n 7/a, and for j = 1 . . . . . s, l e te j := (t~i, ke)~= 1 ~12(7/d) ". Then {Ene j : n 7/a, ) = 1 . . . . . s} isan orthonormal basis of 12(Zd) s. Let 1~ := {He . . . . . HeS}. Then Uzd(Y) is an orthonormal basis of (uZd(Y)). [] </p><p>Now suppose {Yl . . . . . Ys} c ~ such that U Zd (Y) is a Riesz basis of (U zd (Y)), and let </p><p>F" (U ze (Y)) --o (U zd (Y)) </p><p>be defined by $ </p><p>F(v) = ~ ~ (v, Unyj)Unyj, u (U"(Y)). (2.4) j=l nEZ d </p><p>PROPOSITION 2.1 </p><p>Let F be defined as in (2.4). Then F is a positive, bounded invertible operator which commutes with U i, j = 1 . . . . . d. </p><p>Proof The positivity and invertibility of F is well known ([27], p. 185). The </p><p>commutativity o fF and Uj follows by applying F to Ujv in (2.4), and using the fact that U~ . . . . . Ua are commuting unitary operators. [] </p></li><li><p>114 T.N.T. Goodman et al., Wavelet bases </p><p>Let ~" := F -~ (y), </p><p>and define </p><p>Yk := F-I(yk), k = 1 . . . . . s. (2.5) </p><p>Then uZd(Y) and u zd (~ ") are biorthogonal Riesz bases of (Uza(Y)). Any v ~ (u Zd (Y)} Can be written as $ </p><p>v = ~ ~, (v, V'yy)Unyj (2.6) j= l n~Z a </p><p>or $ Z Z (2.7) j= l neZ a </p><p>For any subset V = {1) 1 . . . . . Din} of (uzd(Y)}, let </p><p>and I </p><p>lm,$ ev(O) := ~_, (vk,U"yDe i ' ' </p><p>n Z d : k=l,l=l </p><p>(2.8) </p><p>Pv(O) := ~ (v,,U"yt)e i"' (2.9) n ~ I a k=l,/=l </p><p>for 0 ~ R d. Then Pv, -.Pv ~ L2ms(( 0, 2z0a) We shall call Pv and t' v the Fourier representations of U zd (V) with respect to U zd zd - (Y) and U (Y), respectively. By (2.6) with v = vk, we have </p><p>$ (Vk, UVyt) = ~ ~ (VK, U"Yj)(U"yj,U"Yt) (2.10) </p><p>j= l n~Z a </p><p>for k = 1 . . . . . m, l = 1 . . . . . s, and v E Z a. Equation (2.10) is a relation between Fourier coefficients whose Fourier series satisfy the corresponding relation </p><p>Pv = ~v,t,r. (2.11) </p><p>In particular, P~ = P@Y- By (2.8) and (2.9), ~ = I, the identity matrix, and P~ = @~. It follows that </p><p>~ = ~1. (2.12) </p><p>For a given finite set of vectors Y which generates a Riesz basis U zd Y, the set Y which generates the dual basis may be obtained by computing @~ using (2.12). The coefficients (Yk,u"Y/), k, l= 1 . . . . . s, n ~7/a, in the expansion </p></li><li><p>T~.T. Goodman et al., Wavelet bases 115 </p><p>$ </p><p>Yl~ = ~_, ~ (Yk,Unyt)UnYl, k = 1, . . . . s, i=i n~Z d </p><p>are then obtained from the entries of ~ . If in addition ((vk, U%t))vGZ d ~ f(Zd), k, l= 1, . . . . m, then a similar </p><p>argument leads to PvP = ~v. (2.13) </p><p>If uZd(v) i s a Riesz basis of its closed linear span, we shall write = {Vl . . . . . ~=}, where U z~ (V) and U zd (V) are biorthogonal bases of (U zd (V)). </p><p>The following results on duality and extension of Riesz bases for a finite set of commuting unitary operators are extensions of similar results for a single operator. They may be derived using similar arguments as in the proofs of theorems 3.1 and 3.2, and propositions 3.1 and 3.2 of [15]. </p><p>THEOREM 2.4 </p><p>Let X = {xl . . . . . x,} and Y = {Yl . . . . . y,} be finite subsets of ~ and suppose Z d Zd that U zd (X) and U zd (Y) are Riesz bases of (U (X)) and (U (r)), respectwely. </p><p>If (U zd (X)) c (U zd (Y)} and r = s, then </p><p>(1) Px, P~:, Px and P2 are invertible and </p><p>px 1 = ~, p~l = ~; (2.14) </p><p>(2) the following relations hold: </p><p>Px = PxdPr, PX = Px~ ; (2.15) </p><p>(3) (u = (v z'(Y)). </p><p>THEOREM 2.5 </p><p>LetX = {xl . . . . . x,} and Y = {Yl . . . . . Ys} be finite subsets of ~ and suppose that uZd(x) and U zd (Y) are Riesz bases of (U zd (X)) and (U zd (Y)) respectively. If (U zd (X)) c (U zd (Y)) and r < s, then 3z 1, . . Zd , Zs_ , e (U (Y)), such that F := {z I . . . . . zs_r} .k (U zd (X)), and U zd (X u F) is a Riesz basis of (U zd (Y)) The set F is not unique. Such a set can be constructed by either of the following formulas: </p><p>where </p><p>$ </p><p>zk = Z Z ?t)k(n)UnY) ' (2.16) j=l neZ d </p><p>Z ni(")e'""/ =: Pc(O) E L~s_r)xs((O, 2n:) a) n ~ Z d )k=l , j= l </p></li><li><p>116 T.N.T. Goodman et al., Wavelet bases </p><p>such that ~'r(O)Px(O)* = 0 (2.17) </p><p>and the eigenvalues of/3r(0)Pr(0)* satisfy (2.3), </p><p>or </p><p>where </p><p>$ </p><p>zk = Y-~ Z 4(n)Unyj ' (2.18) j= l ncZ d </p><p>y-r.s ~'2 er0, 2~)a) Z ai(n)ei"'| =: Pr(0 n e Z d Jk=l, j= l </p><p>such that Pr(O)Px(O)* = 0 (2.19) </p><p>and the eigenvalues of r'r(O)Pr(O)* satisfy (2.3). </p><p>3. Wavelet bases for a finite set of commuting unitary operators </p><p>As in section 2, let U := (U1 . . . . . Ua) be an ordered d-tuple of distinct commuting unitary operators on a complex Hilbert s~ace ~, and X := {xl . . . . . x,} c ~ such that uZd(x) is a Riesz basis of~'o = (U z (X)). Suppose there is a unitary operator D on ~ such that </p><p>To c Ol/'o =: l/" l (3.1) and </p><p>UnD = DU An, n ~ Z a, (3.2) </p><p>where A is a d d matrix with integer entries and det(A) ~ 0. The map A Z a --~ Z a, taking n ---) An, is a homomorphism which is injective. The image AZ a of Z a under this map is a subgroup of Z a. Let Za/AZ a denote the quotient group of Z a by AZ a. Then Za/AZ a is a finite group. The cardinality of Za/AZ a will be called the index of AZ d in Z d. We have </p><p>PROPOSrHON 3.1 </p><p>The index of AZ a in Z a is Idet(A)l. </p><p>Proof Let Ma denote the set of all d d matrices with integer entries. Since A ~ Ma, </p><p>there exists invertible matrices P, Q ~Ma with P-~, Q-1 EMd such that PQA = A = diag(~q . . . . . ~,a) ([10], p. 176). Clearly, PZ d = Z a and QZ a = Z a. Hence, Za/AZ a is isomorphic to Zall~7_ a. However, the index of AZ a in Z a is I 1-Ija___l ~,jl = Idet(A)l, and since Idet(P)l and Idet(Q)l are 1, we have Idet(A)l = Idet(A)l. [] </p></li><li><p>T.N.T. Goodman et al., Wavelet bases 117 </p><p>For simplicity, we shall write A := tdet(A)l, and denote the coset 7'+ A7:a by [7']. Then 7/a is partitioned into disjoint cosets [~], j = 0 . . . . . A -...</p></li></ul>