AN ALGORITHM FOR MATRIX EXTENSION AND WAVELET CONSTRUCTION

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1 MATHEMATICS OF COMPUTATION Volume 65, Number 14 April 1996, Pages AN ALGORITHM FOR MATRIX EXTENSION AND WAVELET CONSTRUCTION W LAWTON, S L LEE AND ZUOWEI SHEN Abstract This paper gives a practical method of extending an n r matrix P z, r n, with Laurent polynomial entries in one complex variable z, toa square matrix also with Laurent polynomial entries If P z has orthonormal columns when z is restricted to the torus T, it can be extended to a paraunitary matrix If P z hasrankrfor each z T, it can be extended to a matrix with nonvanishing determinant on T The method is easily implemented in the computer It is applied to the construction of compactly supported wavelets and prewavelets from multiresolutions generated by several univariate scaling functions with an arbitrary dilation parameter 1 Introduction This note deals with matrix extension and a practical method for the construction of compactly supported wavelets and prewavelets from multiresolutions generated by several univariate compactly supported scaling functions with an arbitrary dilation parameter m N, m > 1 Such wavelets and prewavelets can have very small supports, a feature which may be important in numerical applications The construction of wavelets and prewavelets from a multiresolution generated by a single univariate scaling function with the dilation parameter was given in [1] and [1], respectively It is well known that wavelet and prewavelet construction from a multiresolution generated by a finite number of compactly supported scaling functions can be reduced to the problem of extending a matrix with Laurent polynomial entries This is widely studied in the case of wavelet and prewavelet construction in one and several dimensions from multiresolutions generated by one scaling function see [8, 9, 14, 15] and by several scaling functions see [5, 6, 7, 13] In practice it is necessary that the matrix extension be carried out constructively in order to obtain the wavelets explicitly However, in the existing methods such an extension requires the knowledge of some intrinsic properties of the scaling functions In this note, we shall give a constructive method which does not rely on the intrinsic properties of the scaling functions For a given finite set of compactly supported functions φ j, j =1,, r, letv denote the closed shift invariant subspace generated by φ j, j =1,, r, ie, Received by the editor February 15, 1994 and, in revised form, October 4, 1994 and January 3, Mathematics Subject Classification Primary 41A15, 41A3, 15A54, 65D7, 65F3 Key words and phrases Wavelets, prewavelets, matrix extension, splines c 1996 American Mathematical Society 73

2 74 W LAWTON, S L LEE AND ZUOWEI SHEN V := { r a j kφ j k:a j l Z}, j=1 k Z where l is the space of finitely supported sequences For ν Z, let V ν := {fm ν :f V } We say that φ j, j =1,, r, arerefinable if there are finitely supported sequences p i,j such that 11 φ i x = r p i,j kφ j mx k, j=1 k Z i =1,,r The functions φ j, j =1,, r, are called refinable or scaling functions and p i,j, i, j =1,, r, are called refinement masks If φ j, j =1,,r, are refinable and the set {φ j k:j=1,, r, k Z} forms an orthonormal or Riesz basis for V, then it is well known that V ν ν Z forms a multiresolution cf [9] Compactly supported wavelets prewavelets are compactly supported functions ψ j, j =1,, r, for which their shifts form an orthonormal Riesz basis of W := V V1, the orthogonal complement of V in V 1 Let R[z] be the univariate Laurent polynomial ring over the complex field, and let G n R be the group of all n n matrices over R[z] for which the determinants are nonvanishing on C \{} Ann n matrix P z is called paraunitary 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 Clearly a diagonal z-matrix is paraunitary Let Φ := φ 1,φ,,φ r T and write 11 in matrix form Φx = k ZP k Φmx k, where P k =p i,j k r i,j=1 Consider the r r matrices P l z := 1 P l+km z k, l =,,m 1, m k Z with Laurent polynomial entries, and form the r mr block matrix P z := P z P m 1 z, which is called a polyphase matrix It is well known that if {φ j k:j= 1,, r, k Z} forms an orthonormal basis of V,thenPzPz =I r for all z T Compactly supported wavelets corresponding to the multiresolution generated by the scaling functions φ j, j =1,, r, can be constructed by extending the matrix P to an mr mr paraunitary matrix over R[z] inwhichthefirstrrows are the matrix P In, we give a constructive method to extend an arbitrary r n matrix P, satisfying P zp z = I r for all z T, toann nparaunitary matrix over R[z] This leads to a practical method for the construction of compactly supported wavelets from a finite number of scaling functions with an arbitary scaling parameter In the construction of compactly supported wavelets, the refinement masks are usually known in most cases the scaling functions are defined by their

3 AN ALGORITHM FOR MATRIX EXTENSION AND WAVELET CONSTRUCTION 75 refinement masks via their Fourier transforms Therefore, the above approach is quite natural If φ j, j =1,, r, form a Riesz basis of V,thenPzhasrankrfor each z T If the refinement masks, and hence the polyphase matrix P, are available, the usual way of constructing the corresponding compactly supported prewavelets is to extend the polyphase matrix P to an mr mr matrix Q over R[z] sothatqz has rank mr for all z T The standard Gram-Schmidt orthogonalization process is then applied to obtain the compactly supported prewavelets 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 implementable in the computer Based on the matrix extension, we propose another approach in the construction of prewavelets This approach gives prewavelets directly without using the Gram-Schmidt process after the matrix extension The Gram-Schmidt process requires extra computing and enlarges the supports of the prewavelets Our method usually gives prewavelets with shorter supports This approach is also applicable if the refinement masks are not available, as in the case of the multiresolution generated by cardinal Hermite splines see 4 The method is basedonthesequence 1 p i,j k :=m φ i xφ j mx kdx, R where φ j, j = 1,, r, are compactly supported scaling functions such that {φ j k:j=1,, r, k Z} forms a Riesz basis for V The sequences p i,j, i, j =1,r, are finitely supported, since φ j,j=1,,r, are compactly supported functions They are readily computed from the scaling functions If { φ j k: j=1,, r, k Z} is the dual basis of {φ j k: j= 1,,r,k Z},then 13 φ i x= r p i,j k φ j mx k, j=1 k Z i =1,,r Equivalently, Φx = P k Φmx k, k Z where Φ = φ1,, φ r T and P k = p i,j k r i,j=1 The corresponding r rm block matrix 14 P z := P z P m 1 z, where P l z := k Z P l+km z k, l =,,m 1, will be called the dual polyphase matrix of Φ It has rank r for all z T, ifφ j, j=1,, r, and their shifts form a Riesz basis for V

4 76 W LAWTON, S L LEE AND ZUOWEI SHEN In 3, we shall give an algorithm to find an m 1r mr matrix Q z over P z R[z] such that the mr mr matrix Q has rank mr for all z T and z 15 PzQ z = forz T Suppose that Q z =: Q z Q m 1 z, where each Q l,l=,,m 1,is an r r matrix with Laurent polynomial entries Q l z =: q i,j i,j l + kmz k, i,j =1,,r Then the functions 16 ψ i x := k Z r q i,j kφ j mx k, i =1,, mr r, j=1 k Z form a Riesz basis for W Indeed, ψ j k φ i for j =1,,m 1r, k Z, i =1,,r, because of 13, 15 and 16 Hence, ψ j k W, j = 1,,m 1r, k Z Further, they form a Riesz basis of W since Q z has rank m 1r for all z T This result in Hilbert space can be found in [1] If φ j, j =1,,r, and their shifts form an orthonormal basis, then P = P P and PP = I r In, we give a way to construct Q such that is a paraunitary matrix With this Q, equation 16 gives the corresponding wavelets The methods described above can also be extended to the construction of wavelets and prewavelets in Hilbert space Geronimo, Hardin and Massopust [4] have constructed two scaling functions whose shifts form an orthonormal basis of V The corresponding wavelets were constructed by Donovan, Geronimo, Hardin and Massopust [3] and also by Strang and Strela [18] Their works together with that of Goodman [5] are the main sources of motivation for this paper Paraunitary matrix extension Let Az beann 1matrixoverR[z], and suppose that the smallest degree of its jth component is k j Z, forj=1,,, n Let Then DAz is expressible in the form Dz:=diagz k1,,z kn DAz =a +a 1 z+ +a N z N, where a j C n,j=,, N, anda, a N Lemma 1 Let Az = j Z a jz j be an n 1 matrix over R[z] with Az =1 for all z T, where Az denotes the Euclidean norm Suppose that j 1 and j Z are respectively the lowest and highest degrees of Az Then a j, a j =1and a j1, a j = j Z Q

5 AN ALGORITHM FOR MATRIX EXTENSION AND WAVELET CONSTRUCTION 77 Proof As is observed before the lemma, there exists a diagonal z-matrix D such that each component of DA is a polynomial Hence, we need only to prove the case for which Az =a +a 1 z+ +a N z N, with Az =1forz Tand a,a N WenotethatonT, the constant term of Az is N j= a j, a j and the coefficient of the highest degree is a, a N The results then follow from the fact that Az =1forz T Suppose that D 1 is a diagonal z-matrix such that 1 D 1 Az =a 1 + a 1 1 z + +a1 N zn, where a 1 j C n, j =,,N, and a 1, a1 N LetU 1 be a unitary matrix over C such that U 1 a 1 =α,,, T, α By Lemma 1, U 1 a 1,U 1a 1 N = Hence, the first entry U 1a 1 N 1 of the vector U 1 a 1 N is zero Multiplying 1 by U 1 followed by an appropriate diagonal z-matrix D, we can express D U 1 D 1 Az =a + a 1 z + +a M zm, where a, a M, andm<n Furthermore, D U 1 D 1 Az =1, z T, and a, a M =, by Lemma 1 Repeating the above procedure gives a sequence of unitary matrices U j, j =1,, k, overcand a sequence of diagonal z-matrices D j, j =1,, k, such that U k D k U k 1 D k 1 D U 1 D 1 Az=1,,, T, z T Let 3 P z :=U k D k U 1 D 1 z, z C\{} Then P is paraunitary, 4 and for z T, P zaz =1,,, T, z T, 5 Az =Pz 1,,, T, where P z is the conjugate transpose of P z The relation 5 shows that the first column of the matrix P z coincides with Az Thus, P z is a paraunitary extension of Az The above algorithm can also be applied to extend an n r matrix Az with orthonormal columns and with entries in R[z], to a paraunitary matrix Theorem 1 Suppose that A j z, j =1,,r,aren 1column vectors over R[z], r<n, which are orthonormal on T, and that Az :=A 1 z A r z is an n r matrix over R[z] Then there exists an n n paraunitary matrix Q such that 6 QAz = Ir

6 78 W LAWTON, S L LEE AND ZUOWEI SHEN Furthermore, 7 where 8 Q = Q r Q r 1 Q 1, Ik 1 Q k = P k, k =1,, r, is an n n paraunitary matrix and P k is an n k +1 n k+1 paraunitary matrix of the form 3 Proof The observation before the theorem shows that there is a paraunitary matrix Q 1 of the form 3 such that Q 1 A 1 z =1,,, T, z T Since Q 1 is a paraunitary matrix, it follows that for i, j =1,,r, Q 1 A i z,q 1 A j z =δ i,j, z T Hence, the first component Q 1 A j z 1 of Q 1 A j z is zero for z T, j =,, r, and 1 Q 1 Az = A, z T, z A r z where A j z, j =,, r, are n 1 1 matrices, with A i z,a j z = δ i,j for all z T Suppose that there are paraunitary matrices Q 1,, Q k 1,k<r, such that Ik 1 Q k 1 Q Q 1 Az= A k, z T, k z Ak r z where A k j z, j=k,, r, are n k+1 1 matrices, with A k i z,a k j z =δ i,j, z T LetP k be an n k +1 n k+ 1 paraunitary matrix of the form 3 such that P k A k k z =1,,,T, z T By a similar argument as above, we have for i, j = k,,r, and P k A k i z,p k A k j z =δ i,j, z T, P k A k l z =, l = k+1,, r, z T 1 Ik 1 Let Q k = P k Then clearly Q k is paraunitary, and Ik Q k Q k 1 Q 1 Az= P k A k k z P ka k r z Finally, letting Q := Q r Q r 1 Q 1, we have Ir QAz =, z T This completes the proof

7 AN ALGORITHM FOR MATRIX EXTENSION AND WAVELET CONSTRUCTION 79 Remark 1 If Qz satisfies 6, then Qz is a paraunitary extension of Az Further, the proof gives an algorithm for the construction of Qz This can be applied in the construction of wavelets if the scaling functions and refinement masks are known Example 1 Consider the scaling functions φ 1 and φ constructed by Geronimo, Hardin and Massopust [4] They satisfy the matrix dilation equation where P = 1 1 φ1 x φ x = P = k= P k φ1 x k φ x k,, P 1 = , P 3 = 1 1 The corresponding polyphase matrix P is given by P z = 1+9z 9 z 3 3z 1 Let Az = 6 1+9z 8 3 3z 9 z 6 1 Symbolic computation using the above algorithm produces a 4 4 paraunitary matrix z z z Q = 9 z z z satisfying the relation 9 z QAz = 3 z I z The matrix Q is a productof Householdermatrices and a diagonalz-matrix Indeed, where Q 1 = Q = Q 3 DQ Q 1, ,,

8 73 W LAWTON, S L LEE AND ZUOWEI SHEN Q = , Q 3 = and D = diag1, 1/z, 1, 1 The conjugate transpose Qz of Qz is a paraunitary extension of Az Hence, the corresponding wavelets can be constructed using the last two columns of Q and 16 The resulting wavelets are a symmetric and antisymmetric pair 3 Matrix extension In Theorem 1 it was assumed that the columns of the matrix Az areorthonormal on T, and we obtain a paraunitary matrix Q such that Ir QAz = The conjugate transpose Qz of Qz is a paraunitary extension of Az In this section we shall impose no conditions on Az and consider the extension of Az overr[z] Theextensioncanbeachievedinthesamewayasinthecase of paraunitary extension However, the transformations are accomplished by a sequence of elementary matrices and diagonal z-matrices instead of by unitary matrices Theorem 31 Let Az be an n 1 matrix over R[z] Then there is a matrix P G n R such that 31 PAz=pz,,, T, where p R[z] Furthermore, P is a product of elementary matrices and diagonal z-matrices Proof Suppose D 1 z is a diagonal z-matrix such that D 1 Az =a 1 + +a 1 N zn, where a 1 j C n and a 1, a1 N In practice, D 1z is chosen so that the vector a 1 has as many nonzero entries as possible, in order to speed up the process Let E 1 be a product of elementary matrices which reduce the n matrixa 1 a1 N to its echelon form with E 1 a 1 =α,,, T,whereα Further, if a 1 and a 1 N are linearly independent, we can choose E 1 so that the first entry of the vector E 1 a 1 N In the case is zero E 1 a 1 j = α j E 1 a 1 =α j,,, T, j =1,, N,

9 AN ALGORITHM FOR MATRIX EXTENSION AND WAVELET CONSTRUCTION 731 then 31 holds with P z :=E 1 D 1 zand pz:=1+α 1 z+ +α N z N, z C\{} Otherwise, E 1 a 1 j is not a multiple of E 1 a 1 for some j Multiplying E 1 D 1 Az by an appropriate diagonal z-matrix D z gives D E 1 D 1 Az=a + +a N 1 z N1, z C\{}, where N 1 N, anda, a N 1 Recall that D is chosen so that the vector a has as many nonzero entries as possible The case N 1 = N occurs if and only if E 1 a 1 N = α NE 1 a 1 for some α N, ie, a 1 and a 1 N are linearly dependent But in this case the vectors a and a N 1 are linearly independent Applying the above procedure, we can constructively find an invertible constant matrix E which reduces a a N 1 to its echelon form, with E a = α,,, T, α Further, if N 1 = N, thena and a N 1 are linearly independent Hence, E canbechosensuchthate a N 1 1 = Again, either 31 holds with P := E D E 1 D 1, or we can choose an appropriate diagonal z-matrix D 3 z such that 3 D 3 E D E 1 D 1 Az =a 3 + +a 3 N z N, where N <N,anda 3, a3 N Since each entry of D 1 Az is a polynomial, and since the procedure reduces the degree of the entries, the process will stop after a finite number of steps Hence, there is a sequence of invertible matrices E j,j=1,, k, and a sequence of diagonal z-matrices D j,j=1,, k, such that E k D k E k 1 D k 1 E 1 D 1 Az=pz,,, T for some p R[z] Therefore, 31 holds with 33 as desired P z :=E k D k E k 1 D k 1 E 1 D 1 z G n R, Direct application of Theorem 31 gives the following results z C\{}, Theorem 3 Suppose that A j z, j =1,,r,aren 1column vectors over R[z], r<n,andaz:=a 1 z A r z is an n r matrix over R[z] Then there exists Q G n R such that 34 QAz = Br z where B r z is an upper triangular r r matrix over R[z] Furthermore, 35 Q = Q r Q r 1 Q 1, where Ik 1 36 Q k =, k =1,,r, P k and P k is an n k +1 n k+1matrix in G n k+1 R of the form 33,

10 73 W LAWTON, S L LEE AND ZUOWEI SHEN Note that the last n r rows of Q, whichwedenotebyq r+1,,q n,forman n r nmatrix Q := Q r+1 of rank n r for all z C \{} Further, by 34, 37 Az Q z =, z C\{} If Az =A 1 z A r z is an n r matrix over R[z] ofrankrfor each z T or z C \{}, then the n n matrix Az Q z is an extension of A satisfying 37 and of rank n for all z T or z C \{} That Az Q z isofrank nfollows from 37 and the fact that Az andq z are of ranks r and n r, respectively, for all z T or z C \{} This observation leads to Corollary 31 Suppose that Az =A 1 z A r z is an n r matrix over R[z] of rank r for each z T or z C \{}and that Q is a product of elementary matrices and diagonal z-matrices satisfying 34 IfQ r+1,, Q n are the last n r rows of Q and Q r+1 z Q z :=, Q n z then the matrix Az Q z is an extension of A of rank n for all z T or z C \{} satisfying 37 We note that this corollary can be applied directly in the construction of univariate prewavelets from a multiresolution generated by several scaling functions with an arbitrary dilation parameter m Z Since our proof of Theorem 31 is constructive and can be implemented in the computer step by step, this leads to a constructive method for the construction of prewavelets We further remark that the Quillen-Suslin Theorem shows the existence of such an extension when the n r matrix Az overr[z]hasrankrfor all z C \{} However, in the prewavelet construction, we usually assume that the scaling functions and their shifts form a Riesz basis of V, hence the corresponding n r matrix A is of rank r only on T Further, our proof here for the univariate case is elementary and constructive We remark that further row operations may be performed on the last n r rows of the matrix Q in Theorem 3 to obtain wavelets with desirable properties, like smallest support and symmetry These operations preserve the relations 37 This is illustrated in the example in the following section 4 Cardinal Hermite spline wavelets For nonnegative integers n r, lets n 1,r S be the space of spline functions of degree n 1 defined on R with knots of multiplicity r on the set S The space S n 1,r Z has a basis comprising functions φ l, l =1,,r, with minimal supports The functions are called cardinal Hermite B-splines and are uniquely determined by the condition that they vanish outside [, n r + ] and that they satisfy the Hermite interpolating conditions 41 φ k 1 l ν =c ν δ l,k, ν =1,,m r+1,k,l=1,,r, Q n

11 AN ALGORITHM FOR MATRIX EXTENSION AND WAVELET CONSTRUCTION 733 where c ν, ν =1,,n r + 1, are the coefficients of the generalized Euler- Frobenius polynomial 4 Π n 1,r λ = n r ν= c ν+1 λ ν, which may be defined as the minor of order n r n r obtained by removing the first r rows and last r columns of the matrix n 1 k F n 1 λ := λδ k,l l k,l= see [17, 11] Let V be the shift invariant subspace generated by φ l, l =1,,r In [6] it was shown that the shifts φ l x k, l=1,,r, k Z, form a Riesz basis of V Let V ν := {f ν x:f V }, ν Z Then V ν V ν+1,ν Z,and{V ν } ν Z is a multiresolution of L R of multiplicity r Since the scaling functions are explicitly known, their dual polyphase matrix P z can be computed using 1, and hence the method of 3 is applicable for the construction of the corresponding prewavelets The following example shows the construction of cubic cardinal Hermite spline wavelets from two interpolatory cubic Hermite scaling functions This corresponds to the case m = r = Example The scaling functions φ 1, φ are supported on [, ], and are given by { 3x φ 1 x = x 3, x 1, 4+1x 9x +x 3, 1 x, { x φ x= +x 3, x 1, 4+8x 5x +x 3, 1 x They are C 1 on R and satisfy the interpolation conditions φ 1 ν +1=δ ν, φ ν=, ν Z, φ ν +1=δ ν, φ ν=, ν Z The dual polyphase matrix for φ 1, φ is 67 Pz = z 15 15z z z 138z z 34z 1 34z 41z 1 +41z 1z z Let Az := Pz,z TThemethodof 3givesa4 4matrixQsatisfying the relation QA = B, where z /15 B= 1358/19

12 734 W LAWTON, S L LEE AND ZUOWEI SHEN Further row operations on the last two rows of Q produce another matrix, which we shall still denote by Q =Q ij and which still satifies QA = B The entries of Q are Q11z = 19 1, Q 1z =1,Q 13 z =,Q 14 z =, Q 1 z = z z , Q z = z z , Q 3 z = 679 z 48, Q 4z = z 58444, Q 31 z = z 561 z , Q 3 z = z 1538, Q 33 z = 1+z, Q 34 z = z 769, Q 41 z = z , Q 4 z = z z , Q 43 z = 1 z, Q 44 z = z Let Q be the 4 matrix comprising the last two rows of Q ThenAz Q z is an extension of Az of rank 4 satisfying 37 If we write Q = Q Q 1 = Q 11 Q 1 Q 1 11 Q 1 1 Q 1 Q Q 1 1 Q 1 and define q i,j k, i,j =1,,k=1,,4, by Q l i,jz =: q i,j l +kz k, l =,1, k= the corresponding wavelets {ψ 1,ψ } which generate a Riesz basis for W are given by and ψ 1 x = ψ x =, 4 q 1,1 kφ 1 x k+q 1, kφ x k k= 4 q,1 kφ 1 x k+q, kφ x k, k= where the coefficients q 1,j k andq,j k, up to 1 significant figures, are given in Tables 1 and

13 AN ALGORITHM FOR MATRIX EXTENSION AND WAVELET CONSTRUCTION 735 Table 1 Coefficients q 1,j k k\j Table Coefficients q,j k k\j Figures 1 and show the graphs of ψ 1 and ψ, respectively Note that ψ 1 is symmetric and ψ is antisymmetric about 3, and both have support on [, 3] Figure 1 Graph of ψ 1

14 736 W LAWTON, S L LEE AND ZUOWEI SHEN Figure Graph of ψ References 1 Chui, K Charles and Wang, Jian-zhong, On compactly supported spline wavelets and a duality principle, Trans Amer Math Soc , MR 9f:41 Daubechies, L, Orthonormal bases of compactly supported wavelet, Comm Pure and Appl Math , MR 9m:439 3 Donovan, G, Geronimo, J S, Hardin, D P and Massopust, P R, Construction of orthogonal wavelets using fractal interpolation functions, SIAM J Math Anal, to appear 4 Geronimo, J S, Hardin, D P and Massopust, P R, Fractal functions and wavelet expansions based on several scaling functions, J Approx Theory , CMP 94:17 5 Goodman, T N T, Interpolatory Hermite spline wavelets, J Approx Theory , CMP 94:15 6 Goodman, T N T, Lee S L and Tang W S, Wavelets in wandering subspaces, Trans Amer Math Soc , MR 93j:417 7 Goodman, T N T and Lee S L, Wavelets of multiplicity r, Trans Amer Math Soc , MR 94k: Jia, R Q and C A Micchelli, Using the refinement equation for the construction of prewavelets II: Powers of two, Curves and Surfaces P J Laurent, A Le Méhauté and L L Schumaker, eds, Academic Press, New York, 1991, pp 9 46 MR 93e:654 9 Jia, R Q and Z Shen, Multiresolution and Wavelets, Proc Edinburgh Math Soc , 71 3 CMP 94:14 1 Lee, S L, W S Tang, Tan H H, Wavelet bases for a unitary operator, Proc Edinburgh Math Soc , 33 6 CMP 95:13 11 Lee, S L, B-splines for cardinal Hermite interpolation, Linear Algebra Appl , 69 8 MR 5: Mallat, S, Multiresolution approximations and wavelet orthonormal bases of L R, Trans Amer Math Soc , MR 9e: Micchelli, C A Using the refinement equation for the construction of prewavelets VI: Shift invariant subspaces, Approximation Theory, Spline Functions and Applications, S P Singh ed, Kluwer Academic Publishers, 199, 13 MR 94f: Riemenschneider, S D and Z Shen, Box splines, cardinal series and wavelets, in Approximation Theory and Functional Analysis, C K Chui, ed, Academic Press, New York, 1991, pp CMP 91:7 15 Riemenschneider, S D and Z Shen, Wavelets and prewavelets in low dimensions, J Approx Theory , MR 94d:446

15 AN ALGORITHM FOR MATRIX EXTENSION AND WAVELET CONSTRUCTION Schoenberg, IJ, Cardinal spline interpolation, CBMS-NSF Series in Appl Math, Vol 1, SIAM Publ, Philadelphia, 1973 MR 54: Schoenberg, I J and Sharma, A, Cardinal interpolation and spline functions V The B- splines for cardinal Hermite interpolation, Linear Algebra Appl , 1 4 MR 57: Strang, G and Strela V, Short Wavelets and Matrix Dilation Equations, preprint Institute of Systems Science, National University of Singapore, Heng Mui Keng Terrace, Kent Ridge, Singapore 511 address, WLawton:wlawton@issnussg Department of Mathematics, National University of Singapore, 1 Kent Ridge Crescent, Singapore 511 address, S L Lee: matleesl@mathnussg address, Z Shen: matzuows@mathnussg