OF THE SHIFTS OF FINITELY. 27 February underlying renable function. Unfortunately, several desirable properties are not available with

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1 STBILITY ND INDEPENDENCE OF THE SHIFTS OF FINITELY MNY REFINBLE FUNCTIONS Thomas. Hogan 7 February 1997 BSTRCT. Typical constructions of wavelets depend on the stability of the shifts of an underlying renable function. Unfortunately, several desirable properties are not available with compactly supported orthogonal wavelets, e.g. symmetry and piecewise polynomial structure. Presently, multiwavelets seem to oer a satisfactory alternative. The study of multiwavelets involves the consideration of the properties of several (simultaneously) renable functions. In Section of this paper, we characterize stability and linear independence of the shifts of a nite renable function set in terms of the renement mask. Several illustrative examples are provided. The characterizations given in Section actually require that the renable functions be minimal in some sense. This notion of minimality is made clear in Section 3, where we provide sucient conditions on the mask to ensure minimality. The conditions are shown to be also necessary under further assumptions on the renement mask. n example is provided illustrating how the software package MPLE can be used to investigate at least the case of two simultaneously renable functions. 1. Introduction In this paper, we provide a characterization of the stability and linear independence of the shifts of nitely many compactly supported renable functions L (IR). The characterization is in terms of the renement mask. The nature of these results leads us to a question involving the \length" of the space generated by these functions. Some notes on this topic are also provided. We begin with some notation used throughout this paper. The symbols IN and ZZ + denote the set of natural numbers and non-negative integers respectively. We also use TT :=[::) frequently, but with the understanding that any reference to a function f dened on the half-open interval TT also refers to the -periodization of f, f(x +j):=f(x) for every x TT; j ZZ MS (MOS) Subject Classication. 4C15. Key words and phrases. Multiresolution, (global) linear independence, stability, Riesz basis, matrix renement mask, shift-invariant spaces, multiwavelets. 1

2 THOMS HOGN and vice versa. For any set S, we denote by `(S), the set of all a C S satisfying kak` := ( X ss ja(s)j ) 1= < 1: nd we denote by L (IR) the set of all measurable functions f :IR! C satisfying kfk L := Z and equipped with the usual inner product hf;gi := IR Z jf(x)j dx 1= < 1; IR f(x)g(x) dx: nite set L (IR) is said to have stable shifts if there exist positive constants c 1 and c such that c 1 kak` k X X jzz a (j)(,j)k L c kak` for any sequence a `( ZZ) (it is often said that provides a Riesz basis (for its closed linear span) in this case); a compactly supported nite set L (IR) is said to have linearly independent shifts if the map a 7! X X jzz a (j)(,j) is one-to-one on C ZZ. In [JM1], Jia and Micchelli gave necessary and sucient conditions for the linear independence of the shifts of a nite compactly supported set L (IR). They showed that has linearly independent shifts if and only if the sequences b( +j) jzz ( ) (1:1) are linearly independent for every C. In [JM], moreover, they showed that a nite compactly supported set L (IR) has stable shifts if and only if the sequences in (1.1) are linearly independent for every IR. In this paper, we are concerned with the stability and independence of the shifts of renable functions. compactly supported nite set L (IR) is said to be renable if ( is not identically zero and) there exists a nitely supported sequence a :ZZ! C

3 satisfying STBILITY ND INDEPENDENCE 3 = X jzz a(j)(,j) = Equivalently, is renable X X jzz a ; (j) (,j) b(!) =(e,i! )b (!) for all! C; 1 : where := ( ; ) ; and ; (z) := 1 X jzz a ; (j)z j (z Cn): Unless stated otherwise, whenever we refer to a renable with (renement) mask, we mean that =(a ; ) ; is a matrix of trigonometric polynomials; that 1 is an eigenvalue of () and a right 1-eigenvector v is specied, i.e., ()v = v 6= ; and that L (IR) is a compactly supported solution to the renement equation b() =b ; b () = v: (1:) The notions of stability and linear independence of the shifts of renable functions are important in many areas including subdivision, nite element analyses, approximation from shift-invariant spaces, and wavelets. nd most recently, they have proved to be important in the study of what are now being referred to as multiwavelets. Multiwavelets have been studied by Donovan, Geronimo, Hardin, Kessler, and Massopust beginning with [HKM], in which continuous scaling functions were provided. Then in [GHM], arbitrarily smooth scaling functions were constructed including a pair with orthonormal shifts, and these scaling functions were then used to construct compactly supported symmetric/anti-symmetric orthonormal multiwavelets in [DGHM]. Multiwavelets have also been constructed independently by Goodman and Lee. In [GL], they dealt with Riesz bases and dual bases for a set S of spline functions on IR with multiple knots on ZZ. They further provided a Riesz basis for the orthogonal complement ofs in the set f s() : s S g (see also [GLT]). [SS] provides an alternative pair of piecewise linear multiwavelets. s with wavelets, it seems important to understand the properties of multiwavelets in terms of the renement mask of a renable function set. lready much work has been done in this area. pproximation orders of renable function sets have been characterized in [HSS]. This characterization appeared independently in [P], which also featured a factorization of the mask. This factorization has already been put to use in [CDP] to study the regularity or smoothness of renable function sets. ll of these results were arrived at under the assumption that the shifts of the renable set be stable or independent.

4 4 THOMS HOGN Yet none of these papers provided a means for testing the mask to determine stability or independence. In Section, we provide a characterisation of stability and linear independence in terms of the mask. This however brings up another issue which does not appear when dealing with a single renable function. The issue concerns the notion which de Boor, DeVore, and Ron refer to in [BDR] as length. necessary condition for a set to have stable shifts is that be a minimally generating set. This unfortunately is not a condition which is easily characterised in terms of the mask. For this reason, Section 3 is devoted to notes on the length of a renable space and how it depends on the mask.. Stability and independence.1. General This section deals with L (IR) which satisfy len S() = #, where len S() := min f # : S( ) = S() g; S() := the L -closure of S (); and S () := the set of all nite linear combinations of integer translates of all the : The results of [BDR] imply that for any compactly supported nite set L (IR) with len S() = #, the sequences in (1.1) are linearly independent for almost every C. If len S() < #, on the other hand, the sequences were shown to be linearly dependent for almost every C. Evidently, if len < #, then does not have stable shifts. The main results of this section characterize stability and linear independence of a renable set in terms of the renement mask under the assumption that len S() = #. Then, we complete the analysis by providing, in Section 3, necessary and sucient conditions for len S() = # in terms of the mask. Before stating our main results, we provide some additional denitions. For a given integer m, we say that a point! IR ism-cyclic in TT if m! =! 6= in TT; i.e., if!= ZZ, while m!,! ZZ. Equivalently,! IR ism-cyclic in TT if and only if! = m, 1 for some ZZn( m, 1)ZZ. It follows that e,i! C is a primitive odd root of unity if and only if! is cyclic, i.e. is m-cyclic in TT for some m. We point out that if! is m-cyclic in TT, then so is `! for any ` ZZ +. lso, if! is cyclic, then! + is acyclic, i.e. is not m-cyclic in TT for any integer m. Finally, itwas shown in [H, pp. 1{11] that, if! is m-cyclic in TT for some integer m, then for any j ZZ,! +j = mn! + mn,k (:1) for some n IN, k f; 1; :::; m, 1g, and ZZnZZ (see also [JW, p. 113]).

5 STBILITY ND INDEPENDENCE 5 s in [H], we say that a -periodic function f :C! C has a -periodic zero in IR (resp. C) if there exists z IR (resp. C) such that f(z) =f(z + ) = ; and a contaminating zero if there exists an integer m and! which ism-cyclic in TT, such that f( k! + ) = for all k f; 1; :::; m, 1g: We also dene, for any k; n ZZ, n;k := Y n>`>k (`) :=( n,1 )( n, ) ( k+1 ); with the understanding that the (empty) product represents the identity matrix when n, 1 <k+1. Lastly, for C,we dene := X and X 1 a ; :.. Statement of main results Theorem.1. Suppose is a solution to the renement equation (1.) and len S() = #. Then has stable (resp. linearly independent) shifts if and only if for every C n, (i) if b () =, then there exists n ZZ+ so that n ()() 6= ; (ii) if (!) =for some! IR (resp.! C), then (! + ) 6= ; and (iii) for any integer m and any! IR which ism-cyclic in TT, there exists n IN and k f; 1; :::; m, 1g so that mn;k (!)( k! + ) 6= : s stated in Section.1, if has stable shifts then len S() = #. This provides the following necessary conditions. Corollary.. Suppose is a solution to the renement equation (1.). If has stable (resp. linearly independent) shifts, then for every C n, (i) if b () =, then there exists n ZZ+ so that n ()() 6= ; (ii) if (!) =for some! IR (resp.! C), then (! + ) 6= ; and (iii) for any integer m and any! IR which ism-cyclic in TT, there exists n IN and k f; 1; :::; m, 1g so that mn;k (!)( k! + ) 6= : It should be noted that, in both of the conditions (i) and (iii) of the above theorem and corollary, n can be bounded. Specically, one need only verify condition (i) for n = ; 1; :::; #,1; and one need only verify condition (iii) for n = 1; ; :::; #. s regards

6 6 THOMS HOGN condition (i), this is obvious since is a matrix. To see that n can also be bounded in condition (iii), we introduce, for given integer m and! which ism-cyclic in TT, the matrix M(!) := nm+m;nm,1 (!): Since nm+`!, m+`! =`nm, m m, 1 (m!,!) ZZ for n IN and ` =; 1; ;m, 1, M(!) is independent ofn IN. Moreover, mn;k (!) =M n,1 (!) m;k (!); which are spanned already by the cases n =1; ; ; # for any particular k. Note also that if det is not identically zero, then m could also be bounded in terms of the number (and locations) of zeros of det. We also provide the following sucient conditions. Corollary.3. Suppose is a solution to the renement equation (1.) and len S() = #. If satises the following three conditions, then the shifts of are stable (resp. linearly independent): (i) the matrix () b () has full rank; (ii) det has no -periodic zeros in IR (resp. C); and (iii) det has no contaminating zeros. t rst, the condition (iii) appearing in Theorem.1 and Corollary. may seem to be nearly inaccessible. However, if one rst tries to apply Corollary.3, and nds that the mask does not satisfy the easily checked condition (iii) of Corollary.3, then one should begin to analyze the (left-)nullspaces of the matrix at the contaminating zero of det. Verifying condition (iii) of Theorem.1 at these points, is then straight-forward..3. Lemmata For a single function L (IR), we refer to S() :=S(fg) as the principal shiftinvariant, or PSI space, generated by. This terminology is used in the rst of the following results both of which can be found in [BDR]. Result.4. For any compactly supported nite set L (IR), S() is the orthogonal sum of nitely many PSI spaces, each generated by a compactly supported function having linearly independent shifts. Result.5. If the compactly supported nite set L (IR) has linearly independent shifts and S() is compactly supported, then = X for some nitely supported a C ZZ. X jzz (,j)a (j) Result.4 guarantees, for any compactly supported nite set L (IR), the existence of a compactly supported set L (IR) having linearly independent shifts for which

7 STBILITY ND INDEPENDENCE 7 S() = S( ). Then Result.5 implies that each is a nite linear combination of the shifts of the elements of. In the Fourier domain, this implies the existence of a matrix U of trigonometric polynomials satisfying b =U b : If len S() = #, then U is a square matrix. Moreover, since has linearly independent shifts, the sequences (1.1) are linearly dependent at if and only if U() is singular. More specically, for C, b ( +j) jzz =() U() =: (:) ssuming that len S() = #, the sequences (1.1) are linearly independent for almost every C. This implies that det U is a non-trivial trigonometric polynomial, which in turn implies that the matrix B, of rational trigonometric polynomials, is well-dened by the relation U()B := U: (:3) This denition leads to the relation U()b () = b () = b =U b =U()B b ; which implies that b () =B b (:4) almost everywhere. The characterization of S( ) given in [BDR] then implies that (=) S( ). Since the shifts of are linearly independent, Result.5 implies that the entries of the matrix B are trigonometric polynomials. Now, an immediate consequence of (.3) is that n;k (!)U( k+1!)=u( n!)b n;k (!) (:5) for any! C and integers n k Proof of main results Proof of Theorem.1. To see that condition (i) is necessary for to have independent shifts, note that every element ofzzn has the form n+1 (k +1) for some n ZZ + and k ZZ, and that b( n+1 (k +1)) = n ()()b ((k +1)): It is then clear that if does not satisfy condition (i), then does not satisfy condition (ii), then b (j) jzz =. lso, if b (! +4j)=(!) b (! +j) = and b (! + +4j)=(! + ) b (! + +j) = for all j ZZ:

8 8 THOMS HOGN So the sequences (1.1) are linearly dependent at =!. Of course! IR =) IR. Now, if does not satisfy condition (iii), we show that b (! +j) is zero for all j ZZ: Suppose! is m-cyclic in TT for some integer m and suppose j ZZ is given. Then it follows from equation (.1) that b (! +j)= b, mn! + mn,k = mn;k (!)b ( k+1! +) = mn;k (!)( k! + )b ( k! + ); for some n IN, k f1; ; :::; m, 1g, ZZnZZ. So b (! +j) is zero if condition (iii) is not satised. This completes the proof of necessity. To prove suciency, suppose that the shifts of are dependent. Suppose, moreover, that satises conditions (i) and (ii). We show that then (iii) is violated. ssuming that the shifts of are dependent, (.) implies the existence of C n and # TT+iIR so that U(#) =. We will show that this implies that # is cyclic. Since U(#) =, equation (.3) implies that # U # = U(#)B and # # + U + = U(#)B # = # + =: Condition (ii) then implies that U is singular at either # or # + (if the shifts of are in fact not stable, then # IR). Since det U is a trigonometric polynomial, it has only nitely many zeros in TT+iIR, and the arguments used in the proof of [JW, Lemma 1] imply that m #, # ZZ for some m. We must still show that #= ZZ. Suppose U() =, then b () = and (i) implies that n ()() is not zero for some n ZZ +. Evidently, n ()()U() =U()B n ()B() =: But this is a contradiction, since is acyclic, so U() isinvertible. ccepting the existence of the integer m and the m-cyclic # for which U(#) is zero, we proceed to prove that this violates condition (iii). Since mn!,! = mn, 1 m, 1 (m!,!) ZZ;

9 STBILITY ND INDEPENDENCE 9 U(!) = implies that U ( mn!) =. By (.5) and the periodicity ofu, mn;k (!)( k! + )U( k! + ) = mn;k (!)U( k+1!)b( k! + ) = U( mn!)b mn;k (!)B( k! + ) =: Since, k! + is acyclic, U( k! + ) isinvertible, so for every n IN, k f; 1; :::; m, 1g. mn;k (!)( k! + ) = Proof of Corollary.3. We rst point out that conditions (i) and (ii) of Corollary.3 clearly imply conditions (i) and (ii) of Theorem.1. Note, then, that it is sucient to prove the corollary as it relates to stability. For, suppose that det has no - periodic zeros in C, then it certainly has no -periodic zeros in IR. If, moreover, condition (i) is satised then Theorem.1 implies that the sequences (1.1) are independent for all CnIR. If condition (iii) is also satised, and we accept the stability statement of the corollary, then the shifts of must in fact be independent. So, suppose that (i) and (ii) are satised, and that the shifts of are not stable. We will show that condition (iii) is violated. The proof of Theorem.1 implies that the matrix U of (.) has at least one singularity, and that all such singularities are at points which are cyclic. Now, we show that for any! which is cyclic, det U (!) =and det U (!) ==) (:6) det (! + ) =: First, equation (.3) implies that det( (! + ) U (! + )) = det(u (!) B (! + )) = : nd, since! + is acyclic, U (! + ) isinvertible. This implies that det (! + ) =. Condition (ii) then implies that det (!) 6= : (:7) Next, the denition of B also implies that det( (!) U (!)) = det(u (!) B (!))= which together with (.7) implies that det U (!) = ; proving (.6). Now, if U has a singularity at a point! which ism-cyclic in TT, then det U( m!)= det U(!) =. pplying (.6) repeatedly, shows that det has a contaminating zero.

10 1 THOMS HOGN.5. Examples Example.6. This example demonstrates the necessity of the hypothesis len S()=#. Then Let v := ( 1 ) T and (!) := e (e,i! ) where e(z) := z,z+, z+1 4 z+1,z + z () = e (1) = 1, 1 1 satises ()v = v. Now, det (z) e =(z +1)=, which means that is invertible except at. Therefore, satises the conditions (ii) and (iii) of Corollary.3. Moreover, b() () = v e (,1) = 1, has full rank. The mask satises all three conditions of Corollary.3. However, there is a solution L (IR) which satises the renement equation (1.) for this, which does not have independent (or even stable) shifts. Let! 1 := [::) : [::1), [1::) Then b(!) = e,i!,1,i!,e,i! +e,i!,1,i! satises the renement equation (1.). Moreover, b((j +1)) = 4 (j+1)i!!! : for every j ZZ; demonstrating that the sequences (1.1) are linearly dependent for =. This implies that does not have stable shifts. In this case, S() = S( [::1) ). So len S() < #. Example.7. For this example, let v := ( 1 e(z) := 1 1 z z z : 1) T and (!) := e (e,i! ) where If, for some! C, (!) = then is a scalar multiple of ( e,i!,1). So if (!) = (! + ) =, then =. This implies that satises condition (ii) of Corollary.. Moreover, b() 1 1=,1= () = 1,1= 1=

11 STBILITY ND INDEPENDENCE 11 has full rank, so satises condition (i) of Corollary.. However, the shifts of are not stable, since does not satisfy condition (iii) of Corollary.. To see this, note that is -cyclic in TT. Dening z 3 := e,i=3,wehave 1 z where ( )= 1 3 z z 1 z, ( 4 )= 1 3 z, z We also point out that n;k k = M := Now, setting := ( z,1), Moreover, M =. So n;k 3 ; ( 3 )= 1 1 z ; ( 5 3 )= 1 z z, 1 z,1 z,1 z,,, M n, M n,1, 3 3 = 1 1 z z, 1 5 = =: 3 3 k 3 + = : ; : if k =, and if k =1; for k =; 1 and any n IN. This violates condition (iii) of Corollary., so the shifts of are not stable. For completeness, we point out that = 3 [:: 3 ) [ 1 ::) is a solution to the renement equation (1.) and indeed its shifts are not stable. The above example, in which det is identically zero, should not lead one to believe that this property implies the instability of the shifts of. counter-example to such a conjecture is provided by for which e(z) := := 1+z 1,z 1 ; v := ; [::1), [:: 1 ) [ 1 ::1) satisfying the renement equation (1.), has linearly independent shifts. ;

12 1 THOMS HOGN Example.8. In this example, we use the results of this section to investigate the renable functions constructed in [DGHM]. These were actually shown, in [DGHM], to have orthonormal shifts. Let v := ( p e(z) := 1) T and (!) := e (e,i! ) where p 6 p +6 p z 16,1+9z +9z, z 3,3 p +1 p z, 3 p : z Then is dened as the solution to the renement equation (1.). It follows that det e (z) =, 1 4 (z +1)3 : So conditions (ii) and (iii) of Corollary.3 are satised. Moreover, the matrix b() () = p 4 p =5 1,8=5 has full rank. So Corollary.3 implies that the shifts are linearly independent (as long as len S() =, which we will show in Example 3.16). 3. Notes on length 3.1. Background shift-invariant subspace of L (IR) is a closed linear subspace S of L (IR) which is also closed under shifts, i.e., which, for each S, also contains (,j) for every integer j ZZ. nitely generated shift-invariant, or FSI space, is then a space of the form S() := span f (,j) : ; j ZZ g for some nite set L (IR). The cardinality of a smallest generating set for an FSI space S is called the length of S. We write len S := min f # : S = S() g: This terminology and notation is from [BDR]. In Section, stability and linear independence of the shifts of a nite set of mutually renable functions were characterized in terms of the mask under the assumption that len S() = #, i.e., that the FSI space generated by could not be generated by any set consisting of fewer elements. However, no means for verifying this assumption by investigation of the mask were provided. In this section we give acharacterization of this assumption in terms of the mask.

13 STBILITY ND INDEPENDENCE 13 In [BDR], nitely generated shift-invariant subspaces of L (IR) were characterized as follows. For any nite L (IR) and any f L (IR), f S() if and only if bf = X b (3:1) for some -periodic functions. They say that is a basis for S() if the functions in (3.1) are uniquely determined by f. In this case, the functions were shown to be rational trigonometric polynomials. They also prove that every nite set L (IR) of compactly supported functions contains a basis for S(). It follows that for any nite set L (IR) of compactly supported functions, there exists and a matrix U of rational polynomials, so that rank U = # and b(!) =U(e,i! )b (!) for almost every! IR: (3:) If is a basis for S(), then # = len S(). It follows that rank U =# =lens() in (3.). In fact, for compactly supported, is a basis for S() if and only if len S() = #. The necessity of the conditions provided in this section will not require that L (IR). The background we provide here dealing with this more general setting is primarily from [HC] and [CDP]. s with L (IR), we say that the compactly supported nite set D (IR) is renable with mask if is a matrix of trigonometric polynomials, v is a given right 1-eigenvector of (), and D (IR) is a compactly supported solution to b() =b ; b () = v: (3:3) Evidently, b = ny j=1 b j n for any n IN. Now, suppose that b is the only entire function satisfying (3.3) and that the limit, := lim n!1 ny j=1 j v (3:4) converges uniformly on compact sets. Clearly,,() =, and,() = b (). Therefore,, must be the Fourier-Laplace transform of. When proving the necessity of the conditions provided in this section, I will assume that b is the only entire function satisfying (3.3). Sucient conditions for this property are provided in [HC] and [CDP]. The following theorem from [CDP] provides sucient conditions for the convergence of (3.4). Result 3.9. Suppose is a square matrix of trigonometric polynomials. Suppose that v is a right 1-eigenvector of (), i.e., ()v = v 6=. Suppose <, where := (()) := max fjj : ()u = u for some u 6= g:

14 14 THOMS HOGN Then the limit (3.4) converges uniformly on compact sets. Throughout this section, we use to denote the set of all polynomials of one variable with complex coecients and we use F to denote the eld of (univariate) rational polynomials with complex coecients, i.e., Then, for any set S, the set F := f f=g : f; g ; g6= g: F S := fp : S!F: s 7! p s g is a vector space (over the eld F). We interpret any nite subset P of F S also as a matrix P F SP and We identify both of these with the linear map and vice versa; P : F P!F S : u 7! Pu:= X pp pu p We make no distinction between these three interpretations of P. For such P and any z C, P (z) C S, equivalently P (z) C SP, equivalently P (z) :C P! C S denotes P evaluated at z. When #S <1, the subspace P? F S is dened for P F S by P? := ( v F S : hp; vi := X ss p s v s =8p P ) : That is, P? =ker P T, where P T : F S!F P : v 7! P T v := (hp; vi) pp : Evidently, rank P + dim P? = dim F S =#S whenever #P <1; and P?? = P whenever P is a subspace of F S. We denote by, the map which takes any function p dened on C and maps it to the function z 7! p(z ). That is, for any function p with domain C, p also has domain C and is dened by the rule (p)(z) :=p(z ) for all z C: Moreover, if P is any set of functions on C, then P is dened by P := f p : p P g: Evidently, P whenever P and (B) =()(B) whenever B is well-dened. Lastly, for any matrix of rational trigonometric polynomials, I denote by e the matrix of rational polynomials dened by e(e,i! )=(!): With these denitions in place, we begin our analysis with a

15 STBILITY ND INDEPENDENCE 15 Lemma 3.1. Suppose #S <1. Suppose X is a subspace of F S. Then X has a basis U S. Moreover, such a basis exists with the property that rank U(z) =#U for every z C; equivalently, det(u T U) is a non-zero constant. Proof. ny basis of X consists of nitely many vectors from F S. Since #S <1, a basis from S may be obtained simply bymultiplying by a polynomial which is a common multiple of all denominators of a given basis. This proves the rst assertion. Next we show that such a basis U S,iseverywhere of full rank if and only if the polynomial p := det(u T U) is a non-zero constant. Clearly, p(z) 6= if and only if U(z) is of full rank. Moreover, the polynomial p is a non-zero constant polynomial if and only if it never vanishes, hence if and only if U is everywhere of full rank. Now, suppose U is a basis for X. Then we can complete U to a basis V =: [ U U ] for F S. Clearly, ifv is everywhere of full rank, then U is as well. Suppose V is not everywhere of full rank. Specically, suppose 9r C such that det V (r) =. Then, det V (,r). Moreover, since V (r) is not of full rank, there exists a C V F V such that V (r)a =. Since V S is a basis, Va6= (i.e., Vais not the zero element off S ). Therefore, Va (,r) S n. If a v 6= for some v U, then we can replace v in U with the polynomial vector Va=(,r). This obviously has no eect on the basis U for X, while it reduces the degree of det V by one (indeed, det V has been multiplied by a v =(,r)), since det V 6=. If on the other hand, a U =, then there is some v U for which a v 6=. gain, replacing v in U with Va=(,r), reduces the degree of det V by one. Moreover, although we have altered the basis U, the resulting set is still a basis for X, since a v 6=, while a U =. 3.. Length of an FSI space In this section we provide conditions on the mask which are sucient to conclude that len S() = # under the assumption that L (IR). These conditions are shown to be also necessary if the mask satises the hypotheses of Result 3.9. The assumption L (IR) is not required to conclude the necessity of the conditions. In the satement of the following theorem, we assume that is a square matrix of trigonometric polynomials; that 1 is an eigenvalue of (); and that v 6= is a right 1- eigenvector of (). We assume moreover, that L (IR) is a (compactly supported) solution to the renement equation (3.3). Theorem If ` := len S() < #, then there exists P with #P = rank P = #, `, and such that (i) P T (1)b ()=, while rank P (1)=#P ; and (ii) (P T ) e P? =. Proof. Suppose ` := len S() < #. Then there exists, as in (3.), with # = `, and such that b =U(e,i )b for some U F. Moreover, rank U = `.

16 16 THOMS HOGN Let P be a basis for U?. Then #P = rank P =#, `. nd since P T U =, P T (1)b () = P T (1)U(1)b () = : Moreover, by Lemma 3.1, we may assume that P T (1) is of full rank. lso, P T (e,i )U(e,i )b =P T (e,i )b =P T (e,i )b () =(P T U)(e,i )b () =; again since P T U =. Now, is compactly supported and len S( ) = ` = #, so is a basis for S( ). Therefore P T (e,i )U(e,i ) =. Equivalently, (P) T e U =. Since U is of full rank, it is a basis for U?? = P?,so (P) T e P? =: To deal with necessity, we drop the assumption that L (IR). The theorem below is valid for general compactly supported distributions. We will not specify precisely what is meant by S() in this case. However, the proof of the theorem below provides, under certain assumptions on the mask, a set satisfying (i) # < #; (ii) is a nite linear combination of the shifts of ; and (iii) is a nite linear combination of the shifts of. Regardless of the denitions of S() and len S, it is reasonable to say that points (i), (ii), and (iii) imply that len S() # < #. In the statement of the following theorem, we assume that satises the hypotheses of Result 3.9. That is, we assume that 1 is an eigenvalue of (); that v 6= is a right 1-eigenvector; and that the spectral radius (()) <. We assume moreover that is b the only entire function satisfying (3.3). Theorem 3.1. If there exists P with rank P =#P and satisfying conditions (i) and (ii) of Theorem 3.11, then len S() #, #P. Proof. We will provide with # = #, #P, so that the elements of are nite linear combinations of the shifts of the elements of and vice versa. By Lemma 3.1, there exists a basis U for P? with the properties that U(z) has full rank everywhere and det(u T U) is a constant. This second property implies (say by Cramer's rule) that (U T U),1, and hence the left inverse U,L := (U T U),1 U T ; of U, also consist of polynomials. Since rank P =#P,#U =#, #P. Now, condition (ii) implies that e P? P?. nd since U is a basis for P?, this implies the existence of e B F UU for which eu =(U) e B:

17 STBILITY ND INDEPENDENCE 17 We claim that B also satises the hypotheses of Result 3.9. Evidently, B has trigonometric polynomial entries, since e B = (U,L ) e U. b() P? (1) by condition (i), so b () = U(1)v for some v C U.For this v, lso, U(1) e B(1)v = e (1)U(1)v = e (1) b () = b () = U(1)v: nd since U(1) is of full rank, it follows that B()v = v: Now suppose B()u = u for some u 6=. Then U(1)u 6=, while e(1)u(1)u = U(1) e B(1)u = U(1)u: That is, every eigenvalue of B() is also an eigenvalue of (). Therefore, (B()) (()) < : By Result 3.9,, := lim n!1 ny j=1 B j v converges uniformly on compact sets. Moreover,,isanentire function satisfying b () = U(1),(), as well as U(e,i ),() =U(e,i )B, =U(e,i ),: Since we assumed that b was the only entire function satisfying (3.3), we must have b =U(e,i ),: (3:5) Consequently,,=U,L (e,i )b. Since U,L consists of polynomials, this implies that, is the Fourier-Laplace transform of some compactly supported set D (IR) which is a nite linear combination of the shifts of. Then, (3.5) implies that the elements of are also nite linear combinations of the shifts of the elements of. Since # = #U =#, #P, the proof is complete.

18 18 THOMS HOGN 3.3. Two and three functions Suppose # =, then the results of Section 3. take the following simpler form, in which we also have a bound for deg P.swe are assuming that b () is not zero, len S() is either one or two. For L (IR), the sucient conditions then simplify to Corollary Suppose e =: (aij ) is a two-by-two matrix of polynomials. Suppose v 6= is a right 1-eigenvector of (). Suppose L (IR) is a solution to the renement equation (3.3). If len S() = 1, then there exist polynomials p 1 and p satisfying deg p 1 maxfdeg a ; deg a 1 + deg p g deg p maxfdeg a 11 ; deg a 1 + deg p 1 g; and such that (i) p 1 (1)v 1 + p (1)v =, while one of p 1 (1) or p (1) is non-zero; and (ii) ( p 1 (z ) p (z a11 (z) a )) 1 (z) p (z) =: a 1 (z) a (z),p 1 (z) We point out that the bounds on the degrees of p 1 and p could be replaced with the simpler bounds deg p 1 max deg a ; deg a 11 + deg a 1 ; deg a 1 + deg a 1 3 deg p max deg a 11 ; deg a + deg a 1 ; deg a deg a 1 3 or the even simpler bounds deg p k max i;j fdeg a ij g which, though weaker than the bounds in the statement of the theorem, more clearly display that the degrees of p 1 and p can be bounded a priori. Proof. It is enough to provide the bounds for deg p 1 and deg p (the rest follows immediately from Theorem 3.11). We may assume that p 1 and p are relatively prime. Then condition (ii) implies that The bounds on deg p 1 and deg p follow. p 1 divides a p 1, a 1 p ; and p divides a 11 p, a 1 p 1 : The results of Section 3. take a simple form in the case # = 3 as well. Now there are two possibilities, len S() = 1 or len S() =. Note that the three equations in condition (ii) of the next two corollaries are inherently redundant. However, since we don't know which ifanyofthep k might be zero, we have nochoice but to consider all three equations. To handle the possibility that len S() = 1, we oer the

19 STBILITY ND INDEPENDENCE 19 Corollary Suppose e =: (aij ) is a three-by-three matrix of polynomials. Suppose v 6= is a right 1-eigenvector of (). Suppose L (IR) is a solution to the renement equation (3.3). If len S() = 1, then there exist polynomials p 1, p and p 3 satisfying deg p 1 max deg a 11 ; deg a 1 + deg p deg p max deg a1 + deg p 1 deg p 3 max deg a31 + deg p 1 ; deg a 13 + deg p 3 ; deg a ; deg a 3 + deg p 3 ; deg a 3 + deg p ; deg a 33 and such that (i) p (1)v 3, p 3 (1)v = p 3 (1)v 1, p 1 (1)v 3 = p 1 (1)v, p (1)v 1 =, while not all of p 1 (1), p (1), and p 3 (1) are zero; and p 3 (z ),p 3(z ) p (z ),p 1 (z ),p (z ) p 1 (z ) a 11(z) a 1 (z) a 13 (z) a 1 (z) a (z) a 3 (z) a 31 (z) a 3 (z) a 33 (z) 1 p 1(z) p (z) =: p 3 (z) gain we point out that the bounds above imply that deg p k max i;j fdeg a ij g. Proof. ssuming, as we may, that the polynomials p 1, p, and p 3 are relatively prime, condition (ii) implies that p 1 divides a 11 p 1 + a 1 p + a 13 p 3 ; p divides a 1 p 1 + a p + a 3 p 3 ; and p 3 divides a 31 p 1 + a 3 p p 3 : The bounds given now follow easily. The rest follows from Theorem To handle the possibility that len S()=,we make the additional assumption that det is not identically zero. For this case, we oer the Corollary Suppose e =: (aij ) is a three-by-three matrix of polynomials, for which det e is not identically zero. Suppose v 6= is a right 1-eigenvector of (). Suppose L (IR) is a solution to the renement equation (3.3). If len S()=, then there exist polynomials p 1, p, and p 3 satisfying deg p k max i;j deg a ij for k =1;, and 3, and such that (i) p 1 (1)v 1 + p (1)v + p 3 (1)v 3 =, while one of p 1 (1), p (1), orp 3 (1) is non-zero; and (ii) ( p 1 (z ) p (z ) p 3 (z ))@ a 11(z) a 1 (z) a 13 (z) a 1 (z) a (z) a 3 (z) a 31 (z) a 3 (z) a 33 (z) 1 3(z) p (z) p 3 (z),p 1 (z) =:,p (z) p 1 (z)

20 THOMS HOGN Proof. Let M := (m ij ) be the matrix of cofactors of e. That is, m ji := (,1) i+j det (a i j ) i 6=i;j 6=j (i; j =1; ; 3): Since det e is not identically zero, e is almost everywhere invertible, and M T = (det e ) e,1 : Moreover, deg m ij max i ;j fdeg a i jg for i; j =1; ; 3. By Theorem 3.11, there exists P T =: ( p 1 p p 3 ) such that P T e = up T for some u =: f=g F. This in turn implies that g det e P T = P T M : f s we may assume that p 1, p, and p 3 are relatively prime and as the elements of M are polynomials, f must divide g det e and, in turn, P T divides P T M. This then implies that every element ofp is of degree less than max i;j fdeg m ij g max i;j fdeg a ij g Example Example We complete the analysis of the renable functions from [DGHM] begun in Example.8. Recall that p e(z) = 6 p +6 p z 16,1+9z +9z, z 3,3 p +1 p z, 3 p ; v = z p : 1 By Corollary 3.13, if len S() = 1, then there would be polynomials p 1 and p satisfying ( p 1 (z ) p (z )) e (z) p (z),p 1 (z) = (3:6) as well as p p 1 (1) + p (1) =. The bounds provided there for the degrees of these polynomials implies that deg p 1 and deg p 1. We use Maple to try to nd such p 1 and p.we let p 1 (z) =:a + a 1 z + a z and p (z) =:b + b 1 z: The following Maple code then sets up these two conditions and attempts to solve for the coecients a ;a 1 ;a ;b ;b 1.

21 STBILITY ND INDEPENDENCE 1 > :=sqrt()/*matrix([[6*sqrt()*(1+z),16], > [-1+9*z+9*z^-z^3, sqrt()*(-3+1*z-3*z^)]]); > v:=matrix([[sqrt()],[1]]); > p1:=a+a1*z+a*z^; p:=b+b1*z; > spt:=matrix([[subs(z=z^,p1),subs(z=z^,p)]]); > Pperp:=matrix([[p],[-p1]]); > condi:=subs(z=1,evalm(spt&*v))[1,1]; > exprii:=evalm(spt&*&*pperp)[1,1]; > condii:=coeffs(collect(exprii,z),z); > solve({condi,condii},{a,a1,a,b,b1}); Maple's response is fb =;a =;b 1 =;a 1 =;a =g fb 1 = b 1 ;a = 1 p b1 ;a =, 1 p ;a1 =;b =,b 1 g Corresponding to p 1 = p = ; and p 1 = p (z, 1), p = z, 1, respectively. However, neither of these choices for p 1, p satises the condition that ( p 1 (1) p (1) ) be of full rank. Therefore, len S() 6= 1. It is worth noting that the polynomials p 1 (z) = p (1 + z) and p (z) = 1 satisfy condition (3.6). Moreover P T (1) = ( p 1 (1) p (1) ) = ( p 1 ) has full rank. However, P T (1)v =36=. Multiplying p 1 and p by z, 1 does not eect (3.6) and clearly makes P T (1)v =. But in this case, P T (1) = ( ) is not of full rank. So this example also illustrates that all three conditions are necessary to conclude that len S() < #. References [BDR] Boor, C. de, R. DeVore, and. Ron (1994), \The structure of nitely generated shift-invariant spaces in L (IR d )", J. Funct. nal. 119, 37{78. [CDP] Cohen,., I. Daubechies, and G. Plonka (1995), \Regularity of renable function vectors", preprint. [DGHM] Donovan, G. C., J. S. Geronimo, D. P. Hardin, and P. R. Massopust (1996), \Construction of orthogonal wavelets using fractal interpolation functions", SIM J. Math. nal. 7, 1791{1815. [GHM] Geronimo, J. S., D. P. Hardin, and P. R. Massopust (1994), \Fractal functions and wavelet expansions based on several scaling functions", J. pprox. Theory 78, 373{ 41.

22 THOMS HOGN [GL] Goodman, T. N. T. and S. L. Lee (1994), \Wavelets of multiplicity r", Trans. mer. Math. Soc. 34, 37{34. [GLT] Goodman, T. N. T., S. L. Lee, and W. S. Tang (1993), \Wavelets in wandering subspaces", Trans. mer. Math. Soc. 338, 639{654. [H] Hogan, T. (1995), \Stability and independence of the shifts of a multivariate renable function", CMS. Techn. Rep [HKM] Hardin, D. P., B. Kessler, and P. R. Massopust (199), \Multiresolution analyses based on fractal functions", J. pprox. Theory 71, 14{1. [HC] Heil, C. and D. Colella (1996), \Matrix renement equations: existence and uniqueness", J. Fourier nal. ppl., 363{377. [HSS] Heil, C., G. Strang, and V. Strela (1996), \pproximation by translates of renable functions", Numer. Math. 73, 75{94. [JM1] Jia, Rong-Qing and C.. Micchelli (199), \On linear independence for integer translates of a nite number of functions", Proc. Edinburgh Math. Soc. 36, 69{85. [JM] Jia, Rong-Qing and C.. Micchelli (1991), \Using the renement equations for the construction of pre-wavelets II: powers of two", in Curves and Surfaces (P.-J. Laurent,. Le Mehaute, and L. L. Schumaker, eds), cademic Press (New York), 9{46. [JW] Jia, Rong-Qing and Jianzhong Wang (1993), \Stability and linear independence associated with wavelet decompositions", Proc. mer. Math. Soc. 117, 1115{114. [P] Plonka, G. (to appear), \pproximation order of shift-invariant subspaces of L (IR) generated by renable function vectors", Constr. pprox. xxx, xx{xx. [SS] Strang, G. and V. Strela (1995), \Finite element multiwavelets", in pproximation Theory, Wavelets and pplications (S. P. Singh, ntonio Carbone, and B. Watson, eds), Kluwer (Dordrecht, Netherlands), 485{496.