AN ALGORITHM FOR MATRIX EXTENSION AND WAVELET CONSTRUCTION W. LAWTON, S. L. LEE AND ZUOWEI SHEN Abstract. This paper gives a practical method of exten

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1 AN ALGORITHM FOR MATRIX EXTENSION AND WAVELET ONSTRUTION W. LAWTON, S. L. LEE AND ZUOWEI SHEN Abstract. This aer gives a ractical method of extending an nr matrix P (z), r n, with Laurent olynomial entries in one comlex variable z, to a square matrix also with Laurent olynomial entries. If P (z) has orthonormal columns when z is restricted to the torus T, it can be extended to a araunitary matrix. If P (z) has rank r for each z T, it can be extended to a matrix with nonvanishing determinant on T. The method is easily imlemented in the comuter. It is alied to the construction of comactly suorted wavelets and rewavelets from multiresolutions generated by several univariate scaling functions with an arbitrary dilation arameter.. Introduction This note deals with matrix extension and a ractical method for the construction of comactly suorted wavelets and rewavelets from multiresolutions generated by several univariate comactly suorted scaling functions with an arbitrary dilation arameter m N; m >. Such wavelets and rewavelets can have very small suorts, a feature which may be imortant in numerical alications. The construction of wavelets and rewavelets from a multiresolution generated by a single univariate scaling function with the dilation arameter was given in [] and [], resectively. It is well known that wavelet and rewavelet construction from a multiresolution generated by a nite number of comactly suorted scaling functions can be reduced to the roblem of extending a matrix with Laurent olynomial entries. This is widely studied in the case of wavelet and rewavelet construction in one and several dimensions from multiresolutions generated by one scaling function (see Date: 99 Mathematics Subject lassication. Primary 4A5, 4A3, 5A54, 65D7, 65F3. Key words and hrases. Wavelets, rewavelets, matrix extension, slines.

2 W. LAWTON, S. L. LEE AND ZUOWEI SHEN [8, 9, 4, and 5]) and by several scaling functions (see [5, 6, 7, 3]). In ractice it is necessary that the matrix extension be carried out constructively in order to obtain the wavelets exlicitly. However, in the existing methods such an extension requires the knowledge of some intrinsic roerties of the scaling functions. In this note, we shall give a constructive method which does not rely on the intrinsic roerties of the scaling functions. For a given nite set of comactly suorted functions j, j = ; :::; r, let V denote the closed shift-invariant subsace generated by j, j = ; :::; r, i.e., V := f rx X j= kz a j (k) j ( k) : a j `(Z)g; where ` is the sace of nitely suorted sequences. For Z, let V := ff(m ) : f V g: We say that j, j = ; :::; r, are renable if there are nitely suorted sequences i;j such that i (x) = rx X j= kz i;j (k) j (mx k); i = ; : : : ; r: (.) The functions j, j = ; :::; r, are called renable (or scaling) functions and i;j, i; j = ; :::; r, are called renement masks. If j ; j = ; : : : ; r; are renable and the set f j ( k) : j = ; :::; r; k Zg forms an orthonormal (or Riesz) basis for V, then it is well known that (V ) Z forms a multiresolution (cf. [9]). omactly suorted wavelets (rewavelets) are comactly suorted functions j, j = ; :::; r, for which their shifts form an orthonormal (Riesz) basis of W := V? j V, the orthogonal comlement of V in V. Let R[z] be the univariate Laurent olynomial ring over the comlex eld, and let G n (R) be the grou of all n n matrices over R[z] for which the determinants are nonvanishing on n fg. An n n matrix P (z) is called araunitary if it is unitary on the unit circle T. A diagonal matrix with diagonal entries of the form z k ; k Z, is called a diagonal z-matrix. learly a diagonal z-matrix is araunitary.

3 AN ALGORITHM FOR MATRIX EXTENSION AND WAVELET ONSTRUTION 3 Let := ( ; ; : : : ; r ) T and write (.) in matrix form (x) = X kz P k (mx k); where P k = ( i;j (k)) r i;j= : onsider the r r matrices P `(z) := m X kz P`+km z k ; ` = ; : : : ; m ; with Laurent olynomial entries, and form the r mr block matrix P (z) := P (z)j jp m (z) ; which is called a olyhase matrix. It is well known that if f j ( k) : j = ; :::; r; k Zg forms an orthonormal basis of V, then P (z)p (z) = I r for all z T. omactly suorted wavelets corresonding to the multiresolution generated by the scaling functions j, j = ; :::; r, can be constructed by extending the matrix P to an mr mr araunitary matrix over R[z] in which the rst r rows are the matrix P. In x, we give a constructive method to extend an arbitrary r n matrix P, satisfying P (z)p (z) = I r for all z T, to an nn araunitary matrix over R[z]. This leads to a ractical method for the construction of comactly suorted wavelets from a nite number of scaling functions with an arbitary scaling arameter. In the construction of comactly suorted wavelets, the renement masks are usually known (in most cases the scaling functions are dened by their renement masks via their Fourier transforms). Therefore, the above aroach is quite natural. If j, j = ; :::; r, form a Riesz basis of V, then P (z) has rank r for each z T. If the renement masks, and hence the olyhase matrix P, are available, the usual way of constructing the corresonding comactly suorted rewavelets is to extend the olyhase matrix P to an mr mr matrix Q over R[z] so that Q(z) has rank mr for all z T. The standard Gram-Schmidt orthogonalization rocess is then alied to obtain the comactly suorted rewavelets. Section 3 gives an algorithmic method to extend a general matrix. The method uses only elementary transformations and transformations by z-matrices and is easily imlementable in

4 4 W. LAWTON, S. L. LEE AND ZUOWEI SHEN the comuter. ased on the matrix extension, we roose another aroach in the construction of rewavelets. This aroach gives rewavelets directly without using the Gram-Schmidt rocess after the matrix extension. The Gram-Schmidt rocess requires extra comuting and enlarges the suorts of the rewavelets. Our method usually gives rewavelets with shorter suorts. This aroach is also alicable if the renement masks are not available, as in the case of the multiresolution generated by cardinal Hermite slines (see x4). The method is based on the sequence ~ i;j (k) := m Z R i(x) j (mx k)dx; (.) where j, j = ; :::; r, are comactly suorted scaling functions such that f j ( k) : j = ; :::; r; k Zg forms a Riesz basis for V. The sequences ~ i;j ; i; j = ; : : : r, are nitely suorted, since j ; j = ; : : : ; r, are comactly suorted functions. They are readily comuted from the scaling functions. If f ~ j ( k) : j = ; :::r; k Zg is the dual basis of f j ( k) : j = ; : : : ; r; k Zg, then i (x) = Equivalently, rx X j= kz ~ i;j (k) ~ j (mx k); i = ; : : : ; r: (.3) (x) = X kz ep k e (mx k); where e = ~ ; : : : ; ~ T r and r epk = (~i;j(k)) : i;j= The corresonding r rm block matrix ep (z) := e P (z)j j e P m (z) ; (.4) where ep `(z) := X kz ep`+km z k ; ` = ; : : : ; m ; will be called the dual olyhase matrix of : It has rank r for all z T, if j, j = ; :::; r, and their shifts form a Riesz basis for V.

5 AN ALGORITHM FOR MATRIX EXTENSION AND WAVELET ONSTRUTION 5 In x3, we shall give an algorithm to nd an (m )r mr matrix Q (z) over R[z] such that the mr mr e P (z) Q (z) A has rank mr for all z T and ep (z)q (z) = for z T: (.5) Suose that Q (z) =: Q (z)j jq m (z) ; where each Q`; ` = ; : : : ; m ; is an r r matrix with Laurent olynomial entries Then the functions Q` i(x) := X (z) =: q i;j (` + km)z k ; i; j = ; : : : ; r : i;j kz rx X j= kz q i;j (k) j (mx k); i = ; :::; mr r; (.6) form a Riesz basis for W. Indeed, j ( k)? i for j = ; : : : ; (m )r; k Z; i = ; : : : ; r, because of (.3), (.5) and (.6). Hence, j ( k) W ; j = ; : : : ; (m )r; k Z. Further, they form a Riesz basis of W since Q (z) has rank (m )r for all z T. This result in Hilbert sace can be found in []. If j ; j = ; : : : ; r, and their shifts form an orthonormal basis, then e P = P and P P = I r. In x, we give a way to construct Q such P Q A is a araunitary matrix. With this Q, equation (.6) gives the corresonding wavelets. The methods described above can also be extended to the construction of wavelets and rewavelets in Hilbert sace. Geronimo, Hardin and Massoust [4] have constructed two scaling functions whose shifts form an orthonormal basis of V. The corresonding wavelets were constructed by Donovan, Geronimo, Hardin and Massoust [3] and also by Strang and Strela [8]. Their works together with that of Goodman [5] are the main sources of motivation for this aer.

6 6 W. LAWTON, S. L. LEE AND ZUOWEI SHEN. Paraunitary matrix extension Let A(z) be an n matrix over R[z], and suose that the smallest degree of its j th comonent is k j Z, for j = ; ; :::; n. Let Then DA(z) is exressible in the form D(z) := diag(z k ; : : : ; z kn ): DA(z) = a + a z + + a N z N ; where a j n ; j = ; :::; N, and a ; a N 6= : Lemma.. Let A(z) = P jz a j z j be an n matrix over R[z] with ka(z)k = for all z T, where ka(z)k denotes the Euclidean norm. Suose that j and j Z are resectively the lowest and highest degrees of A(z). Then X ha j ; a j i = and ha j ; a j i = : jz Proof. As is observed before the lemma, there exists a diagonal z-matrix D such that each comonent of DA is a olynomial. Hence, we need only to rove the case for which A(z) = a + a z + + a N z N ; with ka(z)k = for z T and a 6=, a N 6=. We note that on T, the constant term of ka(z)k is P N j= ha j; a j i and the coecient of the highest degree is ha ; a N i. The results then follow from the fact that ka(z)k = for z T. Suose that D is a diagonal z-matrix such that D A(z) = a () + a () z + + a () N z N ; (.) where a j () n ; j = ; : : : ; N; and a () ; a () 6=. Let U N be a unitary matrix over such that U a () = (; ; : : : ; ) T ; 6= : y Lemma., hu a () ; U a () N i =. Hence, the rst entry (U a () N ) of the vector U a () N is zero. Multilying (.) by U followed by an aroriate diagonal z-matrix

7 AN ALGORITHM FOR MATRIX EXTENSION AND WAVELET ONSTRUTION 7 D, we can exress D U D A(z) = a () + a () z + + a () M z M ; (.) where a () ; a M () 6=, and M < N. Furthermore, kd U D A(z)k = ; z T; and ha () ; a () i =, by Lemma.. Reeating the above rocedure gives a sequence M of unitary matrices U j ; j = ; :::; k, over and a sequence of diagonal z-matrices D j ; j = ; :::; k; such that Let Then P is araunitary, and for z T, U k D k U k D k D U D A(z) = (; ; : : : ; ) T ; z T: P (z) := U k D k U D (z); z n fg: (.3) where P (z) is the conjugate transose of P (z). P (z)a(z) = (; ; :::; ) T ; z T; (.4) A(z) = P (z) (; ; : : : ; ) T ; (.5) The relation (.5) shows that the rst column of the matrix P (z) coincides with A(z): Thus, P (z) is a araunitary extension of A(z). The above algorithm can also be alied to extend an nr matrix A(z) with orthonormal columns and with entries in R[z], to a araunitary matrix. Theorem.. Suose that A j (z); j = ; : : : ; r, are n column vectors over R[z], r < n, which are orthonormal on T, and that A(z) := (A (z)j ja r (z)) is an n r matrix over R[z]. Then there exists an n n araunitary matrix Q such that QA(z) I r A : (.6)

8 8 W. LAWTON, S. L. LEE AND ZUOWEI SHEN Furthermore, Q = Q r Q r Q ; (.7) where Q k I k P k A ; k = ; :::; r; (.8) is an n n araunitary matrix and P k is an (n k + ) (n k + ) araunitary matrix of the form (.3). Proof. The observation before the theorem shows that there is a araunitary matrix Q of the form (.3) such that Q A (z) = (; ; : : : ; ) T ; z T: Since Q is a araunitary matrix, it follows that for i; j = ; : : : ; r, hq A i (z); Q A j (z)i = i;j ; z T: Hence, the rst comonent (Q A j (z)) of Q A j (z) is zero for z T, j = ; :::; r, and Q A(z) A () (z) A () (z) r A ; z T; where A () j (z); j = ; :::; r; are (n ) matrices, with ha () i (z); A () j (z)i = i;j for all z T. Suose that there are araunitary matrices Q ; :::; Q k ; k < r, such that where A (k) j Q k Q Q A(z) I k A (k) k (z) A (k) r (z) A ; z T; (z); j = k; :::; r; are (n k + ) matrices, with ha (k) i (z); A (k) j (z)i = i;j, z T. Let P k be an (n k + ) (n k + ) araunitary matrix of the form (.3) such that P k A (k) k (z) = (; ; : : : ; ) T ; z T: y a similar argument as above, we have for i; j = k; : : : ; r; hp k A (k) i (z); P k A (k) j (z)i = i;j ; z T;

9 AN ALGORITHM FOR MATRIX EXTENSION AND WAVELET ONSTRUTION 9 and Let Pk A (k) ` (z) = ; ` = k + ; :::; r; z T: Q k = Then clearly Q k is araunitary, and Finally, letting Q k Q k Q A(z) I k P k A I k P k A (k) k (z) P k A (k) r (z) Q := Q r Q r Q ; A : we have This comletes the roof. QA(z) I r A ; z T: Remark. If Q(z) satises (.6), then Q(z) is a araunitary extension of A(z). Further, the roof gives an algorithm for the construction of Q(z). This can be alied in the construction of wavelets if the scaling functions and renement masks are known. Examle. onsider the scaling functions and constructed by Geronimo, Hardin and Massoust [4]. They satisfy the matrix dilation (x) (x) A = 3X k= P (x k) (x k) A ; where P = P A ; P = A ; P3 6 A ; A :

10 W. LAWTON, S. L. LEE AND ZUOWEI SHEN The corresonding olyhase matrix P is given by Let P (z) = A(z) z 9 z +9z z 9 z 6 Symbolic comutation using the above algorithm roduces a 4 4 araunitary matrix satisfying the relation Q z 4 5 3(z +) 9 z + 3(z +) 9 z + 3( z +) QA(z) = 3 5 z I A : z 9 A : z 9 The matrix Q is a roduct of Householder matrices and a diagonal z-matrix. Indeed, Q = Q 3 D Q Q ; A : A where Q A ; Q ) 3 6( (3 ) +3 A ;

11 AN ALGORITHM FOR MATRIX EXTENSION AND WAVELET ONSTRUTION Q 3 A and D = diag(; =z; ; ) : The conjugate transose Q(z) of Q(z) is a araunitary extension of A(z). Hence, the corresonding wavelets can be constructed using the last two columns of Q and (.6). The resulting wavelets are a symmetric and antisymmetric air. 3. Matrix extension In Theorem. it was assumed that the columns of the matrix A(z) are orthonormal on T, and we obtain a araunitary matrix Q such that QA(z) I r The conjugate transose Q(z) of Q(z) is a araunitary extension of A(z). In this section we shall imose no conditions on A(z) and consider the extension of A(z) over R[z]. The extension can be achieved in the same way as in the case of araunitary extension. However, the transformations are accomlished by a sequence of elementary matrices and diagonal z-matrices instead of by unitary matrices. A : Theorem 3.. Let A(z) be an n matrix over R[z]. Then there is a matrix P G n (R) such that P A(z) = ((z); ; : : : ; ) T ; (3.) where R[z]. Furthermore, P is a roduct of elementary matrices and diagonal z-matrices. Proof. Suose D (z) is a diagonal z-matrix such that D A(z) = a () + + a () N z N ;

12 W. LAWTON, S. L. LEE AND ZUOWEI SHEN where a j () n and a () ; a N () 6= : In ractice, D (z) is choosen so that the vector a () has as many nonzero entries as ossible, in order to seed u the rocess. Let E be a roduct of elementary matrices which reduce the n matrix (a () ja() N ) to its \echelon form" with E a () = (; ; : : : ; ) T, where 6=. Further, if a () and a () N E a () N are linearly indeendent, we can choose E so that the rst entry of the vector is zero. In the case E a () j = j E a () = ( j ; ; : : : ; ) T ; j = ; :::; N; then (3.) holds with P (z) := E D (z) and (z) := + z + + N z N ; z n fg: Otherwise, E a () j is not a multile of E a () for some j. Multilying E D A(z) by an aroriate diagonal z-matrix D (z) gives D E D A(z) = a () + + a () N z N ; z n fg; where N N, and a () ; a N () 6=. Recall that D is choosen so that the vector a () has as many nonzero entries as ossible. The case N = N occurs if and only if E a () N = N E a () for some N 6=, i.e., a () and a () N in this case the vectors a () and a () N are linearly indeendent. are linearly deendent. ut Alying the above rocedure, we can constructively nd an invertible constant matrix E which reduces (a () ja() N ) to its \echelon form", with E a () = ( ; ; : : : ; ) T, 6=. Further, if N = N, then a () and a N () are linearly indeendent. Hence, E can be choosen such that (E a N () ) =. Again, either (3.) holds with P := E D E D, or we can choose an aroriate diagonal z-matrix D 3 (z) such that where N < N, and a (3) ; a (3) N 6=. D 3 E D E D A(z) = a (3) + + a (3) N z N ; (3.) Since each entry of D A(z) is a olynomial, and since the rocedure reduces the degree of the entries, the rocess will sto after a nite number of stes. Hence, there is a sequence of invertible matrices E j ; j = ; :::; k; and a sequence of diagonal

13 AN ALGORITHM FOR MATRIX EXTENSION AND WAVELET ONSTRUTION 3 z-matrices D j ; j = ; :::; k; such that E k D k E k D k E D A(z) = ((z); ; : : : ; ) T for some R[z]. Therefore, (3.) holds with as desired. P (z) := E k D k E k D k E D (z) G n (R); z n fg; (3.3) Direct alication of Theorem 3. gives the following results. Theorem 3.. Suose that A j (z); j = ; : : : ; r, are n column vectors over R[z], r < n, and A(z) := (A (z)j ja r (z)) is an n r matrix over R[z]. there exists Q G n (R) such that QA(z) r(z) Then A ; (3.4) where r (z) is an uer triangular r r matrix over R[z]. Furthermore, Q = Q r Q r Q ; (3.5) where Q k I k P k A ; k = ; : : : ; r; (3.6) and P k is an (n k + ) (n k + ) matrix in G n k+ (R) of the form (3.3). Note that the last n r rows of Q; which we denote by Q r+; : : : ; Q n, form an (n r) n matrix Q Q r+ of rank n r for all z n fg. Further, by (3.4), A(z) Q (z) = ; z n fg: (3.7) If A(z) = (A (z)j ja r (z)) is an n r matrix over R[z] of rank r for each z T (or z n fg), then the n n matrix (A(z)jQ (z) ) is an extension of A satisfying. Q n A

14 4 W. LAWTON, S. L. LEE AND ZUOWEI SHEN (3.7) and of rank n for all z T (or z n fg). That (A(z)jQ (z) ) is of rank n follows from (3.7) and the fact that A(z) and Q (z) are of ranks r and n r, resectively, for all z T (or z n fg). This observation leads to orollary 3.. Suose that A(z) = (A (z)j ja r (z)) is an n r matrix over R[z] of rank r for each z T (or z n fg) and that Q is a roduct of elementary matrices and diagonal z-matrices satisfying (3.4). If Q r+; :::; Q n rows of Q and Q (z) Q r+(z). Q n(z) A ; are the last n r then the matrix (A(z)jQ (z) ) is an extension of A of rank n for all z T (or z n fg) satisfying (3.7). We note that this corollary can be alied directly in the construction of univariate rewavelets from a multiresolution generated by several scaling functions with an arbitrary dilation arameter m Z. Since our roof of Theorem 3. is constructive and can be imlemented in the comuter ste by ste, this leads to a constructive method for the construction of rewavelets. We further remark that the Quillen- Suslin Theorem shows the existence of such an extension when the n r matrix A(z) over R[z] has rank r for all z n fg. However, in the rewavelet construction, we usually assume that the scaling functions and their shifts form a Riesz basis of V, hence the corresonding n r matrix A is of rank r only on T. Further, our roof here for the univariate case is elementary and constructive. We remark that further row oerations may be erformed on the last n r rows of the matrix Q in Theorem 3. to obtain wavelets with desirable roerties, like smallest suort and symmetry. These oerations reserves the relations (3.7). This is illustrated in the examle in the following section. 4. ardinal Hermite sline wavelets For nonnegative integers n r, let S n ;r (S) be the sace of sline functions of degree n dened on R with knots of multilicity r on the set S. The sace

15 AN ALGORITHM FOR MATRIX EXTENSION AND WAVELET ONSTRUTION 5 S n ;r (Z) has a basis comrising functions `; ` = ; : : : ; r; with minimal suorts. The functions are called cardinal Hermite -slines and are uniquely determined by the condition that they vanish outside [; n r + ] and that they satisfy the Hermite interolating conditions (k ) ` () = c `;k ; = ; : : : ; m r + ; k; ` = ; : : : ; r; (4.) where c ; = ; : : : ; n r+, are the coecients of the generalized Euler-Frobenius olynomial n ;r () = n r X = c + ; (4.) which may be dened as the minor of order (n r) (n r) obtained by removing the rst r rows and last r columns of the matrix! n k F n () := ` k;`! (see [7, ]). k;`= Let V be the shift invariant subsace generated by `; ` = ; : : : ; r. In [6] it was shown that the shifts `(x k); ` = ; : : : ; r; k Z, form a Riesz basis of V. Let V := ff( x) : f V g; Z: Then V V + ; Z, and fv g Z is a multiresolution of L (R) of multilicity r. Since the scaling functions are exlicitly known, their dual olyhase matrix e P (z) can be comuted using (.), and hence the method of x3 is alicable for the construction of the corresonding rewavelets. The following examle shows the construction of cubic cardinal Hermite sline wavelets from two interolatory cubic Hermite scaling functions. This corresonds to the case m = r =. Examle. The scaling functions ; are suorted on [; ], and are given by 8 < 3x x 3 ; x ; (x) = : 4 + x 9x + x 3 ; x ; (x) = 8 < : x + x 3 ; x ; 4 + 8x 5x + x 3 ; x :

16 6 W. LAWTON, S. L. LEE AND ZUOWEI SHEN They are on R and satisfy the interolation conditions ( + ) = ; () = ; Z; ( + ) = ; () = ; Z: The dual olyhase matrix for ; is 67 e P(z) z z 38z z 4z + 4z 5 5z z 34z 34z z + 64 z A : Let A(z) := e P (z) ; z T. The method of x3 gives a 4 4 matrix Q satisfying the relation QA =, where 34 (94 z 794) =5 358=9 A :

17 AN ALGORITHM FOR MATRIX EXTENSION AND WAVELET ONSTRUTION 7 Further row oerations on the last two rows of Q roduce another matrix, which we shall still denote by Q = (Q ij ) and which still saties QA = : The entries of Q are Q(z) = 9 ; Q (z) = ; Q 3 (z) = ; Q 4 (z) = ; Q (z) = z z 966 ; Q (z) = z z ; Q 3 (z) = 679 z 48 ; Q 4(z) = z ; Q 3 (z) = z 56 z ; Q 3 (z) = z 538 ; Q 33 (z) = + z; Q 34 (z) = z 769 ; Q 4 (z) = z ; Q 4 (z) = z z ; Q 43 (z) = z; Q 44 (z) = z : Let Q be the 4 matrix comrising the last two rows of Q. Then (A(z)jQ (z) ) is an extension of A(z) of rank 4 satisfying (3.7). If we write Q = Q jq Q Q Q Q A ; Q Q Q Q and dene q i;j (k); i; j = ; ; k = ; : : : ; 4; by Q` i;j(z) =: X k= q i;j (` + k)z k ; ` = ; ; the corresonding wavelets f ; g which generate a Riesz basis for W by (x) = 4X k= q ; (k) (x k) + q ; (k) (x k) are given

18 8 W. LAWTON, S. L. LEE AND ZUOWEI SHEN and (x) = 4X k= q ; (k) (x k) + q ; (k) (x k); where the coecients q ;j (k) and q ;j (k), u to signicant gures, are given in the following tables: Table. oecients q ;j (k) knj Table. oecients q ;j (k) knj Figures and show the grahs of and, resectively. Note that is symmetric and is antisymmetric about 3 ; and both have suort on [; 3]: ibliograhy [] hui, K. harles and Wang, Jian-zhong, On comactly suorted sline wavelets and a duality rincile, Trans. Amer. Math. Soc. 33 (99), [] Daubechies, L., Orthonormal bases of comactly suorted wavelet, omm. Pure and Al. Math. 4 (988), [3] Donovan, G., Geronimo, J. S., Hardin, D. P. and Massoust, P. R., onstruction of orthogonal wavelets using fractal interolation functions, SIAM J. Math. Anal., to aear.

19 AN ALGORITHM FOR MATRIX EXTENSION AND WAVELET ONSTRUTION 9 [4] Geronimo, J. S., Hardin, D. P. and Massoust, P. R., Fractal functions and wavelet exansions based on several scaling functions, J. Arox. Theory, 78 (994), [5] Goodman, T. N. T., Interolatory Hermite sline wavelets, J. Arox. Theory, 78 (994), [6] Goodman, T. N. T., Lee S. L. and Tang W. S., Wavelets in wandering subsaces, Trans. Amer. Math. Soc. 338 (993), [7] Goodman, T. N. T. and Lee S. L., Wavelets of multilicity r, Trans. Amer. Math. Soc. 34 (994), [8] Jia, R. Q. and. A. Micchelli, Using the renement equation for the construction of rewavelets II: Powers of two, urves and Surfaces (P. J. Laurent, A. Le Mehaute and L. L. Schumaker, eds.), Academic Press, New York, 99, [9] Jia, R. Q. and Z. Shen, Multiresolution and Wavelets, Proc. Edinburgh Math. Soc. 37 (994), 7-3. [] Lee, S. L., W. S. Tang, Tan H. H., Wavelet bases for a unitary oerator, Proc. Edinburgh Math. Soc. (To aear). [] Lee, S. L., -slines for cardinal Hermite interolation, Linear Algebra Al. (975), [] Mallat, S., Multiresolution aroximations and wavelet orthonormal bases of L (R), Trans. Amer. Math. Soc. 35 (989), [3] Micchelli,. A. Using the renement equation for the construction of rewavelets VI: Shift invariant subsaces, Aroximation Theory, Sline Functions and Alications, S. P. Singh (ed.), Kluwer Academic Publishers, 99, 3 -. [4] Riemenschneider, S. D. and Z. Shen, ox slines, cardinal series and wavelets, in \Aroximation Theory and Functional Analysis",. K. hui, ed., Academic Press, New York, 99, [5] Riemenschneider, S. D. and Z. Shen, Wavelets and rewavelets in low dimensions, J. Arox. Theory 7 (99), [6] Schoenberg, I.J., ardinal sline interolation, MS-NSF Series in Al. Math., Vol., SIAM Publ., Philadelhia, 973. [7] Schoenberg, I. J. and Sharma, A., ardinal interolation and sline functions V. The -slines for cardinal Hermite interolation, Linear Algebra Al. 7 (973), -4. [8] Strang, G. and Strela V., Short Wavelets and Matrix Dilation Equations, rerint. Institute of Systems Science, National University of Singaore, Heng Mui Keng Terrace, Kent Ridge, Singaore 5

20 W. LAWTON, S. L. LEE AND ZUOWEI SHEN Deartment of Mathematics, National University of Singaore, Kent Ridge rescent, Singaore 5