Shift-Invariant Spaces and Linear Operator Equations. Rong-Qing Jia Department of Mathematics University of Alberta Edmonton, Canada T6G 2G1.

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1 Shift-Invariant Spaces and Linear Operator Equations Rong-Qing Jia Department of Mathematics University of Alberta Edmonton, Canada T6G 2G1 Abstract In this paper we investigate the structure of finitely generated shift-invariant spaces and solvability of linear operator equations. Fourier transforms and semi-convolutions are used to characterize shift-invariant spaces. Criteria are provided for solvability of linear operator equations, including linear partial difference equations and discrete convolution equations. The results are then applied to the study of local shift-invariant spaces. Moreover, the approximation order of a local shift-invariant space is characterized under some mild conditions on the generators. AMS Subject Classifications: 41 A 15, 41 A 25, 41 A 63, 42 C 99, 46 E 30, 39 A 12 Supported in part by NSERC Canada under Grant OGP

2 Shift-Invariant Spaces and Linear Operator Equations 1. Introduction The purpose of this paper is to investigate the structure of finitely generated shiftinvariant spaces and solvability of linear operator equations. Our emphasis will be placed on finitely generated local shift-invariant spaces, that is, shift-invariant spaces generated by a finite number of compactly supported functions. It will be demonstrated that linear operator equations play an important role in the study of local shift-invariant spaces. Fourier transforms and semi-convolutions will be used to characterize shift-invariant spaces. Moreover, the approximation order of a local shift-invariant space will be characterized under some mild conditions on the generators. A linear space S of functions from IR s to C is called shift-invariant if it is invariant under shifts (multi-integer translates), that is, f S = f( α) S α Z s. Let Φ be a set of functions from IR s to C. We denote by S 0 (Φ) the linear span of the shifts of the functions in Φ. Then S 0 (Φ) is the smallest shift-invariant space containing Φ. Let f be a (Lebesgue) measurable function on IR s. For 1 p <, let ( ) 1/p f p := f(x) p dx. IR s For p =, let f be the essential supremum of f on IR s. We denote by L p (IR s ) the Banach space of all measurable functions f on IR s such that f p is finite. If Φ is a subset of L p (IR s ) (1 p ), we write S p (Φ) for the closure of S 0 (Φ) in L p (IR s ). Thus, S p (Φ) is the smallest closed shift-invariant subspace of L p (IR s ) that contains Φ. The functions in Φ are called the generators of S p (Φ). If Φ is a finite subset of L p (IR s ), then S p (Φ) is said to be a finitely generated shift-invariant space. In particular, if Φ consists of a single function φ, then S p (φ) is called a principal shiftinvariant space (see [3]). There are two ways to describe the structure of a finitely generated shift-invariant space. One way is to use the Fourier transforms of the generators. The other way is to employ the semi-convolutions of the generators with sequences on Z s. 1

3 If f L 1 (IR s ), the Fourier transform ˆf is defined by ˆf(ξ) := f(x)e ix ξ dx, ξ IR s. IR s The domain of the Fourier transform can be naturally extended to include L 2 (IR s ). In [4], de Boor, DeVore, and Ron gave the following characterization of a finitely generated shift-invariant subspace of L 2 (IR s ) in terms of the Fourier transforms of the generators. For a finite subset Φ of L 2 (IR s ) and a function f L 2 (IR s ), f lies in S 2 (Φ) if and only if ˆf = τ φ ˆφ φ Φ for some 2π-periodic functions τ φ, φ Φ. The proof of this result given in [4] relies on the known characterization of doublyinvariant spaces (see [8]). On the other hand, the special case of a principal shift-invariant space was treated in [3] without recourse to the general theory of doubly-invariant spaces developed in [8]. In Section 2 we will give a simple proof of this result without appeal to the general tools used in [8] such as the range function and the pointwise projection. It is also interesting to use semi-convolution to describe the structure of a finitely generated shift-invariant space. A function from Z s to C is called a sequence. Let l( Z s ) denote the linear space of all sequences on Z s, and let l 0 ( Z s ) denote the linear space of all finitely supported sequences on Z s. a l( Z s ), the semi-convolution φ a is the sum α Z s φ( α)a(α). Given a function φ : IR s C and a sequence This sum makes sense if either φ is compactly supported or a is finitely supported. Let Φ be a finite collection of compactly supported functions from IR s to C. We use S(Φ) to denote the linear space of functions of the form φ Φ φ a φ, where a φ (φ Φ) are sequences on Z s. Following [4] we say that S(Φ) is local. For local shift-invariant spaces, see the work of Dahmen and Micchelli [5], de Boor, DeVore, and Ron [4], and Jia [9]. Let Φ be a finite collection of compactly supported functions in L p (IR s ) (1 p ). In Section 3 we will prove that S(Φ) L p (IR s ) is always closed in L p (IR s ); hence S p (Φ) is a subspace of S(Φ) L p (IR s ). In Section 4 we will show that S(Φ) L 2 (IR s ) = S 2 (Φ). Consequently, a function f L 2 (IR s ) lies in S 2 (Φ) if and only if f = φ Φ φ a φ 2

4 for some sequences a φ on Z s, φ Φ. For general p, it is a difficult question whether the two spaces S(Φ) L p (IR s ) and S p (Φ) are the same. When s = 1, it was proved in [11] that S p (Φ) = S(Φ) L p (IR) for 1 < p <. The essence of the proof given in [11] rests on the fact that S(Φ) has linearly independent generators. In Section 7 we extend this result to the case where Φ consists of a finite number of compactly supported functions in L p (IR s ) whose shifts are stable. Under such a condition we will show that S p (Φ) = S(Φ) L p (IR s ) for 1 p <. When p =, a modified result is also valid. The results of Section 7 are based on a study of discrete convolution equations. As a matter of fact, discrete convolution equations can be viewed as linear partial difference equations with constant coefficients. In turn, linear partial difference and differential equations are special forms of linear operator equations. Section 5 is devoted to an investigation of linear operator equations. The general setting is as follows. Let V be a linear space over a field K, and let Λ be a ring of commuting linear operators on V. Consider the following system of linear operator equation: λ jk u k = v j, j = 1,..., m, where λ jk Λ for j = 1,..., m and k = 1,..., n, v 1,..., v m V, and u 1,..., u n are the unknowns. We will give a criterion for solvability of such linear operator equations. The result is then applied to linear partial difference and differential equations. On the basis of Section 5, we will establish a criterion for solvability of discrete convolution equations in Section 6. Finally, the study of linear operator equations will be used to investigate approximation by shift-invariant spaces. Let Φ be a finite collection of compactly supported functions in L p (IR s ) (1 p ). Let r be a positive integer. If Φ consists of a single function φ with ˆφ(0) 0, then it is known that S(φ) provides approximation order r if and only if S(φ) contains all polynomials of degree less than r. This result was established by Ron [17] for the case p =, and by Jia [9] for the general case 1 p. In Section 8 we extend their results to finitely generated shift-invariant spaces. Let Φ = {φ 1,..., φ n }. Suppose the sequences ( ˆφ k (2πβ)) β Z s, k = 1,..., n, are linearly independent. Under this condition we will prove in Section 8 that S(Φ) provides approximation order r if and only if S(Φ) contains all polynomials of degree less than r. 3

5 2. Shift-Invariant Subspaces of L 2 (IR s ) In this section we give a new proof for the following result established by de Boor, DeVore, and Ron in [4]. Theorem 2.1. Let Φ be a finite subset of L 2 (IR s ) and f L 2 (IR s ). Then f S 2 (Φ) if and only if ˆf = τ φ ˆφ φ Φ for some 2π-periodic functions τ φ, φ Φ. Recall that L 2 (IR s ) is a Hilbert space with the inner product given by f, g := f(x)g(x) dx, f, g L 2 (IR s ), IR s where g denotes the complex conjugate of g. We say that f is orthogonal to g if f, g = 0. The orthogonal complement of a subspace V of a Hilbert space is denoted by V. It is easily seen that S 2 (f) is orthogonal to S 2 (g) if and only if f( α), g = 0 for all α Z s. The following lemma follows from basic properties of the Fourier transform. Lemma 2.2. If f, g L 2 (IR s ), then the series h(ξ) := ˆf(ξ + 2πβ) ĝ(ξ + 2πβ) β Z s converges absolutely for almost every ξ IR s. The Fourier coefficients of the 2π-periodic function h are f( α), g, α Z s. The bracket product of two functions f and g in L 2 (IR s ) is defined by [f, g](e iξ ) := ˆf(ξ + 2πβ) ĝ(ξ + 2πβ), ξ IR s. β Z s In particular, if f and g are compactly supported, then [f, g](e iξ ) = α Z s f( α), g e iα ξ ξ IR s. (2.1) The bracket product [f, g] was introduced in [14] (see Theorem 3.2 there) under some mild decay conditions on f and g. This restriction was removed in [3]. The bracket product turns out to be a convenient tool in the study of orthogonality in L 2 (IR s ). The following lemma is an easy consequence of Lemma

6 Lemma 2.3. If φ, ψ L 2 (IR s ), then ψ is orthogonal to S 2 (φ) if and only if [φ, ψ](e iξ ) = 0 for almost every ξ IR s. Moreover, {φ( α) : α IR s } forms an orthonormal system if and only if [φ, φ](e iξ ) = 1 for almost every ξ IR s. Proof of Theorem 2.1. Denote by F (Φ) the linear space of those functions f L 2 (IR s ) for which ˆf = φ Φ τ ˆφ φ for some 2π-periodic functions τ φ (φ Φ). Suppose f F (Φ). Then for every g S 2 (Φ) and almost every ξ IR s, [f, g](e iξ ) = φ Φ τ φ (ξ)[φ, g](e iξ ) = 0. By Lemma 2.3, this shows that f is orthogonal to S 2 (Φ) ; hence f S 2 (Φ) = S 2 (Φ). In other words, F (Φ) S 2 (Φ). For the proof of F (Φ) S 2 (Φ) we shall proceed by induction on #Φ, the number of elements in Φ. Suppose #Φ = 1 and Φ = {φ}. In order to prove F (φ) = S 2 (φ), it suffices to show that F (φ) is closed. Suppose f lies in the closure of F (φ). Then there exists a sequence (f n ) n=1,2,... in F (φ) such that f n f 2 0 as n. It follows that ˆf n ˆf 2 0. By passing to a subsequence if necessary, we may assume that ˆf n converges to ˆf almost everywhere. Since f n F (φ), ˆf n = τ n ˆφ for some 2π-periodic function τn. Let E := {ξ IR s : ˆφ(ξ + 2βπ) = 0 β Z s }. Then E is 2π-periodic, i.e., ξ E implies ξ + 2βπ E for all β Z s. For each n, let λ n (ξ) := { τn (ξ) if ξ / E, 0 if ξ E. Evidently, λ n is 2π-periodic and ˆf n = λ n ˆφ. Since ˆfn converges to ˆf almost everywhere, for almost every ξ IR s, lim n ˆfn (ξ + 2βπ) = ˆf(ξ + 2βπ) for all β Z s. Let ξ be such a point. If ξ E, then λ n (ξ) = 0 for all n. If ξ / E, then ˆφ(ξ + 2βπ) 0 for some β Z s. Consequently, lim λ n(ξ) = lim n n ˆf n (ξ + 2βπ) / ˆφ(ξ + 2βπ) = ˆf(ξ + 2βπ) / ˆφ(ξ + 2βπ). This shows that λ(ξ) := lim n λ n (ξ) exists for almost every ξ IR s. Since each λ n is 2πperiodic, the limit function λ is also 2π-periodic. Taking limits of both sides of ˆf n = λ n ˆφ, we obtain ˆf = λ ˆφ. This shows that f F (φ). Therefore F (φ) is closed. 5

7 Now assume that F (Φ) = S 2 (Φ) and we wish to prove that F (Φ ψ) S 2 (Φ ψ) for any ψ L 2 (IR s ). Let P Φ denote the orthogonal projection of L 2 (IR s ) onto S 2 (Φ), and let 1 denote the identity operator on L 2 (IR s ). Let ρ := (1 P Φ )ψ. Then S 2 (Φ) is orthogonal to S 2 (ρ), and hence S 2 (Φ) + S 2 (ρ) is closed. With g := P Φ ψ S 2 (Φ) = F (Φ) we have ρ = ψ g F (Φ ψ), and so S 2 (ρ) = F (ρ) F (Φ ψ). But S 2 (Φ) + S 2 (ρ) is closed, and ψ = g + ρ S 2 (Φ) + S 2 (ρ). Therefore we have F (Φ ψ) S 2 (Φ) + S 2 (ρ) S 2 (Φ ψ). This completes the induction procedure. 3. Local Shift-Invariant Spaces Let Φ be a finite collection of compactly supported functions in L p (IR s ) (1 p ). In this section we shall show that S(Φ) L p (IR s ) is closed in L p (IR s ). A measurable function f : IR s C is called locally integrable if f 1 (K) := f(x) dx < K for every compact subset K of IR s. We denote by L loc := L loc (IR s ) the linear space of all locally integrable functions on IR s. For k = 1, 2,..., the functional p k given by p k (f) := f(x) dx [ k,k] s is a semi-norm on L loc. The family of semi-norms {p k : k = 1, 2,...} induces a topology on L loc so that L loc becomes a complete, metrizable, locally convex topological vector space. In other words, L loc is a Fréchet space (see, e.g., [7, p. 160]). Let (f n ) n=1,2,... be a sequence in L loc. Then f n converges to a function f L loc if and only if for every compact subset K of IR s, f n f 1 (K) 0 as n. The following theorem shows that a finitely generated local shift-invariant subspace of L loc (IR s ) is closed in it. Theorem 3.1. Let Φ be a finite collection of compactly supported integrable functions on IR s. Then S(Φ) is a closed subspace of L loc (IR s ). If Φ is a subset of L p (IR s ) for some p, 1 p, then S(Φ) L p (IR s ) is closed in L p (IR s ). The proof of Theorem 3.1 is based on the author s recent paper [9] on approximation order of shift-invariant spaces. Let us recall some results from [9]. 6

8 Let S = S(Φ), where Φ is a finite collection of compactly supported integrable functions on IR s. The restriction of S to the cube [0, 1) s is finite dimensional. Thus, we can find a finite collection {ψ i : i I} of integrable functions on IR s such that each ψ i vanishes outside [0, 1) s and {ψ i [0,1) s : i I} forms a basis for S [0,1) s. The shifts of the functions ψ i (i I) are linearly independent; that is, for sequences a i l( Z s ) (i I), ψ i a i = 0 = a i = 0 i I. i I Let l( Z s ) I denote the linear space of all mappings from I to l( Z s ). We define the linear mapping T from l( Z s ) I to L loc (IR s ) as follows: T (a) := ψ i a i for a = (a i ) i I l( Z s ) I. i I Let V be the range of the mapping T. Then S(Φ) is a linear subspace of V. A function f V has the following representation: f = ψ i f i, (3.1) i I where f i l( Z s ), i I. Suppose Φ = {φ j : j J}. Then f lies in S(Φ) if and only if there exist sequences u j (j J) on Z s such that f = j J φ j u j. (3.2) Since each φ j is compactly supported and belongs to V, we can find finitely supported sequences c ij on Z s (i I, j J) such that φ j = ψ i c ij, j J. (3.3) i I It follows from (3.2) and (3.3) that f = φ j ( α)u j (α) j J α Z s = ψ i ( α β)c ij (β)u j (α) j J α Z s i I β Z s = [ ] c ij (β)u j (α β) ψ i ( α). i I α Z s j J β Z s 7

9 Comparing this with (3.1), we conclude that f lies in S(Φ) if and only if j J β Z s c ij (β)u j (α β) = f i (α) α Z s and i I. (3.4) Given α Z s, we denote by τ α the difference operator on l( Z s ) given by τ α a := a( + α), a l( Z s ). If p = α Z s c αz α is a Laurent polynomial, where c α = 0 except for finitely many α, then p induces the difference operator p(τ) := α Z s c α τ α. For i I and j J, let g ij denote the Laurent polynomial given by g ij (z) = Then (3.4) can be rewritten as β Z s c ij (β)z β, z (C \ {0}) s. g ij (τ)u j = f i, i I. (3.5) j J We observe that (3.5) is a system of linear partial difference equations with constant coefficients. This system of partial difference equations is said to be consistent if it can be solved for (u j ) j J. It is said to be compatible if the following compatibility conditions are satisfied: For Laurent polynomials q i (i I), q i g ij = 0 j J = q i (τ)f i = 0. i I i I It was proved in [9, Theorem 3] that the system of partial difference equations in (3.5) is consistent if and only if it is compatible. This result is an extension of the well-known Ehrenpreis principle for solvability of linear partial differential equations with constant coefficients (see [6]). Now let P denote the set of all Laurent polynomials in s variables. Let Q be the subset of P I given by Q := { (q i ) i I : q i g ij = 0 i I j J }. Thus, we arrive at the following conclusion. 8

10 Lemma 3.2. There exists a subset Q of P I such that a function f = i I ψ i f i lies in S(Φ) if and only if q i (τ)f i = 0 (q i ) i I Q. (3.6) i I We are in a position to establish Theorem 3.1. Proof of Theorem 3.1. First, we show that V is a closed linear subspace of L loc (IR s ). Since {ψ i [0,1) s : i I} is linearly independent, there exist two positive constants C 1 and C 2 such that for f = T (a) V and for all β Z s, C 1 a i (β) f 1 (β + [0, 1) s ) C 2 a i (β). (3.7) i I Let (f (n) ) n=1,2,... be a sequence in V converging to a function f in L loc (IR s ). Suppose f (n) = T (a (n) ). Then by (3.7) we have (m) a i (β) a (n) i (β) C 1 1 f (m) f (n) 1 (β + [0, 1) s ) for all i I and β Z s. This shows that (a (n) i (β)) n=1,2,... is a Cauchy sequence of complex numbers. Let a i (β) := lim n a (n) i (β) and a := (a i ) i I. Using (3.7) again, we see that for all β Z s T (a) T (a (n) ) 1 (β + [0, 1) s ) C 2 ai (β) a (n) i (β). Hence T (a (n) ) converges to T (a) in L loc (IR s ). In other words, f = T (a), thereby proving that V is closed in L loc (IR s ). Next, we show that S(Φ) is closed in V. Let ( f (n)) be a sequence in S(Φ) n=1,2,... converging to f V. Suppose f (n) = T (a (n) ) for each n and f = T (a). Then the preceding paragraph tells us that for each i I and each β Z s, a (n) i (β) converges to a i (β) as n. In other words, a (n) i converges to a i pointwise. Since each f (n) lies in S(Φ), by Lemma 3.2 we have i I i I i I q i (τ)a (n) i = 0 (q i ) i I Q. (3.8) For a fixed element (q i ) i I Q and a fixed β Z s, q i (τ)a i (β) only involves finitely many a i (α), α Z s. Letting n in (3.8) we conclude that q i (τ)a i = 0 (q i ) i I Q. i I 9

11 This shows that f = T (a) lies in S(Φ), by Lemma 3.2. Therefore, S(Φ) is a closed subspace of L loc (IR s ). Finally, suppose that Φ is a subset of L p (IR s ) for some p, 1 p. If (f (n) ) n=1,2,... is a sequence in S(Φ) L p (IR s ) converging to f in L p (IR s ), then f (n) converges to f in the topology of L loc (IR s ). Hence f lies in S(Φ) by what has been proved. This shows that S(Φ) L p (IR s ) is closed in L p (IR s ). 4. Local Shift-Invariant Subspaces of L 2 (IR s ) Let Φ be a finite collection of compactly supported functions in L 2 (IR s ). In [3] de Boor, DeVore, and Ron demonstrated that the two spaces S(Φ) L 2 (IR s ) and S 2 (Φ) provide the same approximation order. However, they left the question open whether these two spaces are the same. In this section we show that these two spaces are indeed the same. Consequently, we give a characterization for S 2 (Φ) in terms of the semi-convolutions of the generators with sequences on Z s. Theorem 4.1. Let Φ be a finite collection of compactly supported functions in L 2 (IR s ). Then S(Φ) L 2 (IR s ) = S 2 (Φ). Consequently, a function f L 2 (IR s ) lies in S 2 (Φ) if and only if f = φ a φ φ Φ for some sequences a φ on Z s, φ Φ. In our proof we use the following two basic facts. First, if f L 2 (IR s ) and a l 0 ( Z s ), then f a(ξ) = ˆf(ξ)ã(e iξ ), ξ IR s, (4.1) where ã(z) := α Z s a(α)zα is the symbol of a. Second, if f L 2 (IR s ) and g = f a for some nontrivial sequence a l 0 ( Z s ), then S 2 (f) = S 2 (g) (see [4, Corollary 2.5]). Indeed, g = f a implies ĝ(ξ) = ˆf(ξ)ã(e iξ ) and ˆf(ξ) = ĝ(ξ)/ã(e iξ ), for a.e. ξ IR s, where ã(e iξ ) is a 2π-periodic trigonometric polynomial. So f S 2 (g) and g S 2 (f) by Theorem 2.1. We also need the following lemma (cf. [14, Theorem 4.4] and [4, Theorem 3.38]). 10

12 Lemma 4.2. Let Φ be a finite collection of compactly supported functions in L 2 (IR s ), and let P Φ denote the orthogonal projection of L 2 (IR s ) onto S 2 (Φ). Then there exists a nontrivial sequence b l 0 ( Z s ) such that for every compactly supported function g L 2 (IR s ), P Φ (g b) is compactly supported. Proof. The proof proceeds by induction on #Φ. Suppose Φ consists of a single function φ 0. For a compactly supported function g L 2 (IR s ), let h and u be the functions determined by ĥ(ξ) = [φ, φ](e iξ )ĝ(ξ) and û(ξ) = [g, φ](e iξ ) ˆφ(ξ), ξ IR s. Let b and c be the sequences such that b(e iξ ) = [φ, φ](e iξ ) and c(e iξ ) = [g, φ](e iξ ), ξ IR s. Note that the sequence b is independent of g. Since both g and φ are compactly supported, (2.1) tells us that both b and c are finitely supported. Moreover, by (4.1) we have h = g b and u = φ c. We find that [h u, φ] = [h, φ] [u, φ] = [φ, φ][g, φ] [g, φ][φ, φ] = 0, so u = P φ h = P φ (g b). But u = φ c is compactly supported. Now assume that the lemma is valid for a finite set Φ of compactly supported functions in L 2 (IR s ). We wish to prove that it is also true for Φ ψ, where ψ is a compactly supported function in L 2 (IR s ). By the induction hypothesis, there exists a nontrivial sequence b l 0 ( Z s ) such that for every compactly supported function g L 2 (IR s ), P Φ (g b) is compactly supported. Let ρ := ψ b P Φ (ψ b). Then ρ is compactly supported. Moreover, since S 2 (ψ) = S 2 (ψ b), the space S 2 (Φ ψ) = S 2 ( Φ (ψ b) ) = S 2 (Φ ρ) is the orthogonal sum of S 2 (Φ) and S 2 (ρ). By what has been proved, there exists a nontrivial sequence c such that for every compactly supported function g L 2 (IR s ), P ρ (g c) is compactly supported. Note that g (b c) = (g b) c = (g c) b. Therefore, for every compactly supported function g L 2 (IR s ), P Φ ψ (g (b c)) = P Φ (g (b c)) + P ρ (g (b c)) is compactly supported. 11

13 Proof of Theorem 4.1. Theorem 3.1 shows S(Φ) L 2 (IR s ) S 2 (Φ), so we only have to show S 2 (Φ) S(Φ) L 2 (IR s ). The latter was proved in [3, Theorem 2.16] for the case #Φ = 1. For the general case we argue as follows. Let f = φ Φ φ a φ S(Φ) L 2 (IR s ). We wish to prove f S 2 (Φ). supported function g S 2 (Φ), For this purpose, we observe that for every compactly f, g = φ( α), g a φ (α) = 0. φ Φ α Z s By Lemma 4.2 we can find a nontrivial sequence b l 0 ( Z s ) such that for every function h L 2 (IR s ) with compact support, P Φ (h b) is compactly supported. Let g S 2 (Φ). Then P Φ (g b) = 0. There exists a sequence (g n ) n=1,2,... of compactly supported functions in L 2 (IR s ) such that g n g 2 0 as n. Let h n := g n b P Φ (g n b). Then each h n is compactly supported and h n S 2 (Φ). Hence f, h n = 0 for n = 1, 2,.... Furthermore, lim f, h n = f, g b P Φ (g b) = f, g b. n This shows f, g b = 0. Let c be the sequence given by c(α) = b( α) for all α Z s. Then f, g b = 0 implies f c, g = α Z s f( α), g c(α) = α Z s f, g( + α)c(α) = f, g b = 0. This is true for all g S 2 (Φ) ; hence f c S 2 (Φ) = S 2 (Φ). Therefore we have f S 2 (f c) S 2 (Φ), as desired. 12

14 5. Linear Operator Equations In this section we establish a criterion for solvability of linear operator equations and then apply the result to linear partial difference and differential equations with constant coefficients. The study of linear operator equations is important for our investigation of local shift-invariant spaces. Let K be a field, and let V be a linear space over K. Given a linear mapping λ on V, we use ker λ to denote its kernel {u V : λ u = 0}. Thus, λ is one-to-one if ker λ = {0}. If λ is both one-to-one and onto, then we say that λ is invertible. Let L(V ) be the set of all linear mappings on V. Then L(V ) is a ring under addition and composition. The identity mapping on V is the identity element of L(V ). In general, L(V ) is noncommutative. subring. We are interested in commutative subrings of L(V ) with identity. Let Λ be such a The ideal generated by finitely many elements λ 1,..., λ m in Λ is denoted by (λ 1,..., λ m ). An ideal I of Λ is said to be invertible if I contains an invertible linear mapping. Note that the inverse of an invertible linear mapping λ Λ is not required to lie in Λ. The kernel of I, denoted ker I, is the intersection of all ker λ, λ I. Consider the following system of linear operator equations: where λ jk λ jk u k = v j, j = 1,..., m, (5.1) Λ for j = 1,..., m and k = 1,..., n, v 1,..., v m V, and u 1,..., u n are the unknowns. Our purpose is to give a criterion for solvability of (5.1). Linear operator equations with one unknown (n = 1) were investigated by Jia, Riemenschneider, and Shen in [15]. We say that the system (5.1) is consistent if there exist u 1,..., u n V that satisfy the equations in (5.1). Two systems of linear operator equations are said to be equivalent if they have the same solutions. We say that (5.1) is compatible if for any µ 1,..., µ m Λ with m j=1 µ jλ jk = 0, k = 1,..., n, one must have m j=1 µ jv j = 0. Evidently, if (5.1) is consistent, then it is compatible. If we replace every vector v j (j = 1,..., m) in (5.1) by the zero vector, then the resulting system is called the associated homogeneous system. Thus, the solutions of (5.1) are unique if and only if the associated homogeneous system only has the trivial solution. 13

15 Theorem 5.1. Let Λ be a commutative subring of L(V ) with identity. Suppose that every finitely generated ideal I of Λ with ker I = {0} is invertible. Then the system (5.1) of linear operator equations is uniquely solvable for u 1,..., u n in V if and only if it is compatible and the associated homogeneous system only has the trivial solution. Proof. It is obvious that the two conditions are necessary for the system (5.1) to be uniquely solvable. The proof of sufficiency proceeds by inductions on n. Suppose n = 1 and consider the system of linear operator equations λ j u = v j, j = 1,..., m. (5.2) By the assumption, the associated homogeneous system λ j u = 0, j = 1,..., m, only has the trivial solution. In other words, ker (λ 1,..., λ m ) = {0}; hence (λ 1,..., λ m ) is invertible. Thus, there exist µ 1,..., µ m Λ such that ν := µ 1 λ 1 + +µ m λ m is invertible. Let u := ν 1 (µ 1 v µ m v m ). We claim that u satisfies the equations in (5.2). Indeed, since (5.2) is compatible, we have λ j v k = λ k v j for j, k {1,..., m}. Therefore m m m νλ j u = λ j (νu) = λ j µ k v k = µ k (λ j v k ) = µ k λ k v j = νv j. But ν is invertible, so it follows that λ j u = v j for j = 1,..., m. Let n > 1 and assume that the theorem has been verified for n 1. We shall prove that (5.1) is uniquely solvable under the conditions stated in the theorem. Note that the kernel of the ideal (λ 11,..., λ m1 ) is trivial, for otherwise the associated homogeneous system would have nontrivial solutions. Thus, there exist µ 1,..., µ m Λ such that the linear mapping ν := µ 1 λ µ m λ m1 is invertible. Apply ν to both sides of each equation in (5.1): νλ jk u k = νv j, j = 1,..., m. (5.3) Since ν is invertible, two systems (5.1) and (5.3) are equivalent. Let m m v 0 := µ j v j and λ 0k := µ j λ jk, k = 1,..., n. j=1 j=1 14

16 Then λ 01 = ν, and the equation λ 0k u k = v 0 (5.4) is a consequence of (5.1). For each j = 1,..., m, apply λ j1 to both sides of (5.4) and subtract the resulting equation from (5.3). In this way we obtain ( ) λ01 λ jk λ j1 λ 0k uk = λ 01 v j λ j1 v 0, j = 1,..., m. (5.5) k=2 The system consisting of the equations in (5.5) and the equation in (5.4) is equivalent to the original system of equations in (5.1). Now let us show that (5.5) is uniquely solvable for (u 2,..., u n ). By the induction hypothesis, it suffices to verify that (5.5) satisfies the two conditions stated in the theorem. First, since the original system (5.1) is compatible, so is (5.5). Second, the homogeneous system associated to (5.5) only has the trivial solution. Indeed, if (u 2,..., u n ) is a nontrivial solution of the homogeneous system, then we can find u 1 V such that λ 01 u 1 = (λ 02 u λ 0n u n ), because λ 01 = ν is invertible. Thus, (u 1, u 2,..., u n ) would be a nontrivial solution to the homogeneous system associated with (5.1), which is a contradiction. We have proved that (5.5) is uniquely solvable. Let (u 2,..., u n ) be the solution. Since λ 01 = ν is invertible, we can find u 1 V such that νu 1 = v 0 λ 0k u k. k=2 Consequently, (u 1, u 2,..., u n ) is the unique solution of (5.1). Next, we discuss two special linear operator equations: linear partial difference equations and linear partial differential equations. Theorem 5.1 will be used to give criteria for solvability of those equations. Let Π(C s ) denote the linear space of all polynomials of s variables with coefficients in C. For a nonnegative integer d, we denote by Π d (C s ) the subspace of all polynomials of (total) degree less than or equal to d. If no ambiguity arises, we write Π for Π(C s ) and Π d for Π d (C s ), respectively. 15

17 A mapping a from Z s to C is called a polynomial sequence, if there is a polynomial q of s variables with coefficients in C such that a(α) = q(α) for all α Z s. The degree of a is the same as the degree of q. Let IP( Z s ) denote the linear space of all polynomial sequences on Z s. Suppose p(z) = α Z c αz α is a Laurent polynomial of s variables with coefficients s in C, where c α = 0 except for finitely many α. Let e denote the s-tuple (1,..., 1). Then p(e) = α Z c α. The polynomial p induces the difference operator p(τ) = s α Z c ατ α. s It is easily seen that p(τ) maps IP( Z s ) to itself. For a sequence a on Z s we have (τ α 1)a = a( + α) a. Hence (τ α 1)a = 0 if a is a constant sequence. Moreover, if a is a polynomial sequence, then (τ α 1)a is also a polynomial sequence of degree less than the degree of a. Thus, if p(e) = 0, then the difference operator p(τ) is degree-reducing; that is, for any polynomial sequence a, p(τ)a is a polynomial sequence of degree less than the degree of a. Consequently, p(τ) is invertible on IP( Z s ) if and only if p(e) 0. Indeed, if p(e) = 0, then p(τ)a = 0 for any constant sequence a. If p(e) 0, then we can write p = c p 0, where c = p(e) and p 0 (e) = 0. Thus, p 0 (τ) is degree-reducing, and so p n+1 0 (τ)a = 0 for all polynomial sequences a of degree n. Given a polynomial sequence a of degree n, the equation p(τ)r = a has a unique solution r = [ 1/c + p 0 (τ)/c p n 0 (τ)/c n+1 ]a. This shows that p(τ) is invertible. Let Λ be the ring of all partial difference operators of the form p(τ), where p is a Laurent polynomial of s variables with coefficients in C. Then Λ is a commutative ring with identity. If I is a finitely generated ideal of Λ with ker I = {0}, then I is invertible. To see this, let I be the ideal generated by p 1 (τ),..., p m (τ). If ker I = {0}, then for at least one j, p j (e) 0, for otherwise the constant sequences would lie in the kernel of I. But p j (e) 0 implies that p j (τ) is invertible. This shows that I is invertible. Theorem 5.2. Let p jk (j = 1,..., m; k = 1,..., n) be Laurent polynomials of s variables with coefficients in C. The homogeneous system of linear partial difference equations p jk (τ)u k = 0, j = 1,..., m, (5.6) 16

18 only has the trivial solution if and only if the matrix P := ( p jk (e) ) 1 j m,1 k n has rank n. Consequently, for given polynomial sequences v 1,..., v m, the system of equations p jk (τ)u k = v j, j = 1,..., m, is uniquely solvable for (u 1,..., u n ) IP( Z s ) n if and only if the matrix P has rank n and the system is compatible. Proof. If the rank of P is less than n, then there exists a nonzero vector (a 1,..., a n ) in C n \ {0} such that p jk (e)a k = 0 for j = 1,..., m. (5.7) For each k, let u k be the constant sequence α a k, α Z s. Then (u 1,..., u n ) is a nontrivial solution to the homogeneous system (5.6). Conversely, suppose that the homogeneous system (5.6) has a nontrivial solution (u 1,..., u n ). We observe that for any polynomial q, (q(τ)u 1,..., q(τ)u n ) is also a solution of (5.6). We can find a polynomial q such that q(τ)u 1,..., q(τ)u n are constant sequences but q(τ)u k 0 for at least one k. Let a k = q(τ)u k (0) for k = 1,..., n. Then the complex vector (a 1,..., a n ) satisfies (5.7). Hence the rank of the matrix P is less than n. This proves the first part of the theorem. The second part of the theorem follows immediately from the first part of the theorem and Theorem 5.1. The rest of this section is devoted to a study of linear partial differential equations. For this purpose we need the multi-index notation. Let IN be the set of positive integers, and let IN 0 := IN {0}. An element in IN s 0 is called a multi-index. If α = (α 1,..., α s ) is a multi-index, then its length α is defined by α := α α s, and its factorial is defined by α! := α 1! α s!. For two multi-indices α = (α 1,..., α s ) and β = (β 1,..., β s ), by α β, or β α, we mean α j β j for j = 1,..., s. Let α IN s 0 be a multi-index. The differential operator D α on Π(C s ) is defined by D α ( β IN s 0 ) b β z β := β! b β (β α)! zβ α. β α 17

19 A polynomial p = α IN a s α z α induces the differential operator p(d) := 0 α IN a s α D α. 0 The differential operator p(d) is invertible on Π if and only if p(0) 0. Indeed, if p(0) = 0, then p(d) 1 = 0. Conversely, if p(0) 0, then we may write p = c p 0, where c = p(0) and p 0 is a polynomial with p 0 (0) = 0. Then for any polynomial q of degree n, the equation p(d)r = q has a unique solution r = [ 1/c + p 0 (D)/c p n 0 (D)/c n+1] q. This shows that p(d) is invertible. Let Λ be the ring of all linear partial differential equations of the form p(d), where p is a polynomial of s variables with coefficients in C. Then Λ is a commutative ring with identity. If I is a finitely generated ideal of Λ with ker I = {0}, then I is invertible. To see this, let I be the ideal generated by p 1 (D),..., p m (D). If ker I = {0}, then p j (0) 0 for at least one j, for otherwise the constants would lie in ker I. But p j (0) 0 implies that p j (D) is invertible on Π. This shows that I is invertible. The following theorem can be proved in the same way as Theorem 5.2 was done. Theorem 5.3. Let p jk (j = 1,..., m; k = 1,..., n) be polynomials of s variables with coefficients in C. The homogeneous system of linear partial differential equations p jk (D)u k = 0, j = 1,..., m, only has the trivial solution if and only if the matrix P := ( p jk (0) ) 1 j m,1 k n has rank n. Consequently, for given polynomials v 1,..., v m, the system of equations p jk (D)u k = v j, j = 1,..., m, is uniquely solvable for (u 1,..., u n ) Π n if and only if the matrix P has rank n and the system is compatible. 18

20 6. Discrete Convolution Equations In this section we shall give a criterion for solvability of discrete convolution equations. Recall that l( Z s ) is the linear space of complex-valued sequences on Z s, and l 0 ( Z s ) is the linear space of all finitely supported sequences on Z s. Moreover, we use c 0 ( Z s ) to denote the linear space of all sequences a on Z s such that lim α a(α) = 0, where α := α α s for α = (α 1,..., α s ) Z s. Given a sequence a on Z s, we define ( ) 1/p a p := a(α) p, 1 p <. α Z s For p =, we define a to be the supremum of { a(α) : α Z s }. For 1 p we denote by l p ( Z s ) the Banach space of all sequences a on Z s such that a p <. Given a l( Z s ), the formal Laurent series α Z s a(α)zα is called the symbol of a, and denoted by ã(z). If a l 1 ( Z s ), then the symbol ã is a continuous function on the torus T s := {(z 1,..., z s ) C s : z 1 =... = z s = 1}. If a l 0 ( Z s ), then ã is a Laurent polynomial. For a, b l( Z s ), we define the convolution of a and b by a b(α) := β Z s a(α β)b(β), α Z s, whenever the above series is absolutely convergent. For example, if δ is the sequence given by δ(α) = 1 for α = 0 and δ(α) = 0 for α Z s \ {0}, then a δ = a for all a l( Z s ). Evidently, for a l 0 ( Z s ) and b l( Z s ), the convolution a b is well defined. Let a be an element in l 0 ( Z s ) such that ã(z) 0 for all z T s. For given v l ( Z s ), the discrete convolution equation a u = v has a unique solution for u l ( Z s ). To see this, let c(α) := 1 1 e iα ξ (2π) s [0,2π) ã(e iξ dξ, α Z s. ) s Then the sequence c decays exponentially fast, and c(z)ã(z) = 1 for all z T s. Hence c a = δ. If a u = v, then it follows that u = δ u = (c a) u = c (a u) = c v. 19

21 This proves uniqueness of the solution. Moreover, if v lies in l p ( Z s ) for some p, 1 p, then the solution u lies in l p ( Z s ); if v c 0 ( Z s ), then the solution u also lies in c 0 ( Z s ). Consider the system of discrete convolution equations a jk u k = v j, j = 1,..., m, (6.1) where a jk l 0 ( Z s ) (j = 1,..., m; k = 1,..., n) and v j l( Z s ) (j = 1,..., m). We say that this system of equations is compatible if for any c 1,..., c m m j=1 c j a jk = 0, k = 1,..., n, one must have m j=1 c j v j = 0. l 0 ( Z s ) with Theorem 6.1. Let v 1,..., v m l ( Z s ). Suppose that the system of discrete convolution equations in (6.1) is compatible. If the matrix A(z) := ( ã jk (z) ) 1 j m,1 k n has rank n for every z T s, then the system of equations in (6.1) is uniquely solvable for (u 1,..., u n ) (l ( Z s )) n. Furthermore, if v 1,..., v m lie in l p ( Z s ) for some p, 1 p <, then the solutions u 1,..., u n also lie in l p ( Z s ); if v 1,..., v m lie in c 0 ( Z s ), then the solutions u 1,..., u n also lie in c 0 ( Z s ). Proof. For j = 1,..., m, let c j be the sequence given by c j (α) = a j1 ( α), α Z s. Then c j (z) = ã j1 (z) for z T s. Let a 0k := m j=1 c j a jk, k = 1,..., n. Since A(z) has rank n, the Laurent polynomials ã 11 (z),..., ã m1 (z) do not have common zeros in T s. Hence ã 01 (z) = m c j (z)ã j1 (z) = j=1 m ãj1 (z) 2 > 0 z T s. j=1 Let us consider the case n = 1 first. In this case, (6.1) implies a 01 u 1 = m c j v j =: v 0. j=1 Since ã 01 (z) > 0 for all z T s, the equation a 01 u 1 = v 0 is uniquely solvable for u in l ( Z s ). Let u 1 be the solution. By the assumption, the original system of equations in (6.1) is compatible; hence a 01 v j = a j1 v 0. It follows that a 01 a j1 u 1 = a j1 v 0 = a 01 v j. Therefore, a j1 u 1 = v j for j = 1,..., m. This shows that u 1 is the unique solution to the system of equations in (6.1). 20

22 The proof proceeds with induction on n. Suppose n > 1 and the desired result is valid for n 1. Let c j (j = 1,..., m) and a 0k (k = 1,..., n) be the same sequences as in the above. Let w 0 := v 0 = m j=1 c j v j, w j := a 01 v j a j1 w 0 (j = 1,..., m), and b jk := a 01 a jk a j1 a 0k, j = 1,..., m; k = 2,..., n. Then w 0, w 1,..., w m l ( Z s ). Consequently, (6.1) is equivalent to the following system of equations: and a 0k u k = w 0 (6.2) b jk u k = w j, j = 1,..., m. (6.3) k=2 We observe that (6.3) is compatible and the matrix B(z) := ( b jk (z)) 1 j m,2 k n has rank n 1 for every z T s. Thus, by the induction hypothesis, (6.3) is uniquely solvable for u 2,..., u n in l ( Z s ). Once u 2,..., u n are obtained, u 1 is uniquely determined from (6.2). This completes the induction procedure. Finally, if v 1,..., v m lie in l p ( Z s ) for some p, 1 p <, then the above proof shows that the solutions u 1,..., u n also lie in l p ( Z s ). The same conclusion holds true for c 0 ( Z s ). 7. Stable Generators Let Φ be a finite subset of L p (IR s ) (1 p ). In this section, we shall characterize S p (Φ) in terms of the semi-convolutions of the generators with sequences in l p ( Z s ), if the shifts of the functions in Φ are stable. If, in addition, the functions in Φ are compactly supported, we shall prove S p (Φ) = S(Φ) L p (IR s ) for 1 p <. When p =, we denote by L,0 (IR s ) the subspace of L (IR s ) consisting of all functions f L (IR s ) such that f (IR s \ [ k, k] s ) 0 as k. We shall prove S (Φ) = S(Φ) L,0 (IR s ). Let Φ be a finite subset of L p (IR s ) (1 p ). We say that the shifts φ( α) (φ Φ, α Z s ) are L p -stable if there are two positive constants C 1 and C 2 such that p C 1 a φ p φ a φ C 2 a φ p φ Φ φ Φ 21 φ Φ

23 for all sequences a φ l 0 ( Z s ), φ Φ. Under some mild decay conditions on the functions in Φ, it was proved by Jia and Micchelli ([13] and [14]) that the shifts of the functions in Φ are L p -stable if and only if for any ξ IR s, the sequences ( ˆφ(ξ + 2πβ)) β Z s (φ Φ) are linearly independent. When p = 2, their results were generalized by de Boor, DeVore, and Ron in [4]. by Suppose Φ = {φ 1,..., φ n }. Let T Φ be the mapping from (l 0 ( Z s )) n to L p (IR s ) given T Φ (a 1,..., a n ) := φ k a k, a 1,..., a n l 0 ( Z s ). Let X := (l p ( Z s )) n for 1 p < and X := (c 0 ( Z s )) n for p =. The norm on X is defined by (a 1,..., a n ) X := a k p. Suppose that the shifts of the functions in Φ are stable. Then the domain of T Φ can be extended to X, and T Φ is a one-to-one continuous linear operator from X to Y := L p (IR s ). For a = (a 1,..., a n ) X, we write n φ k a k for T Φ (a). In other words, lim φ k a k N α N φ k ( α)a k (α) = 0. p Moreover, there exists a positive constant C such that C a X T Φ (a) Y for all a X. From a well-known result in functional analysis (see, e.g., [18, p. 70]), the range of T Φ is closed. In other words, T Φ (X) = S p (Φ). Thus, we have the following result. Theorem 7.1. Let Φ be a finite subset of L p (IR s ) such that the shifts of the functions in Φ are L p -stable (1 p ). For 1 p <, a function f lies in S p (Φ) if and only if f = φ Φ φ a φ for some sequences a φ in l p ( Z s ). For p =, a function f lies in S (Φ) if and only if f = φ Φ φ a φ for some sequences a φ in c 0 ( Z s ). Theorem 7.1 does not apply to the case in which the stability condition is not satisfied. For example, let φ := χ χ( 1), where χ is the characteristic function of [0, 1). Then χ S 2 (φ) (see [4, Example 2.7]), but χ cannot be written in the form χ = φ a for any 22

24 a l 2 ( Z). Indeed, if a is an element of l 2 ( Z), then the 2π-periodic function ξ ã(e iξ ) is square integrable on [0, 2π) and φ a(ξ) = ˆφ(ξ)ã(e iξ ) = ˆχ(ξ)(1 e iξ )ã(e iξ ). Thus, χ = φ a implies that ã(e iξ ) = 1 / (1 e iξ ) for a.e. ξ IR. But the function ξ 1/(1 e iξ ) is not square integrable on [0, 2π). This contradiction verifies our claim. Moreover, we have IR χ(x) dx = 1 and χ = j=0 φ( j) S(φ). However, any function f in S 0 (φ) satisfies IR f(x)dx = 0. Since S 1(φ) is the closure of S 0 (φ) in L 1 (IR), we also have IR f(x)dx = 0 for all f S 1(φ). This shows that χ / S 1 (φ). Therefore S 1 (φ) S(φ) L 1 (IR). When Φ is a finite collection of compactly supported functions in L p (IR), it was shown in [11] that S p (Φ) = S(Φ) L p (IR) for 1 < p < and S (Φ) = S(Φ) L,0 (IR). The following theorem gives a similar result for s > 1 if the shifts of the functions in Φ are stable. Theorem 7.2. Let Φ be a finite collection of compactly supported functions in L p (IR s ) (1 p ). If the shifts of the functions in Φ are L p -stable, then S p (Φ) = S(Φ) L p (IR s ) for 1 p <, and S (Φ) = S(Φ) L,0 (IR s ). Proof. By Theorem 3.1, S(Φ) L p (IR s ) is closed in L p (IR s ) (1 p ). Hence S p (Φ) is contained in S(Φ) L p (IR s ). For p =, we also have S (Φ) S(Φ) L,0 (IR s ). Suppose Φ = {φ 1,..., φ n }. We can find functions ψ 1,..., ψ m L p (IR s ) such that they vanish outside the unit cube [0, 1) s and {ψ j [0,1) s S(Φ) [0,1) s. Then each φ k (k = 1,..., n) can be represented as φ k = : j = 1,..., m} forms a basis for m ψ j a jk, (7.1) where a jk (j = 1,..., m; k = 1,..., n) are finitely supported sequences on Z s. j=1 A function f S(Ψ) has the following representation: f = m ψ j v j, (7.2) j=1 23

25 where v 1,..., v m are sequences on Z s. If f lies in L p (IR s ) for 1 p <, then the sequences v 1,..., v m lie in l p ( Z s ). To see this, we observe that, for β Z s, f(x) = m v j (β)ψ j (x β) for x β + [0, 1) s. j=1 Hence, there exists a constant C > 0 such that vj (β) p C p β+[0,1) s f(x) p dx j = 1,..., m and β Z s. It follows that v j p C f p for j = 1,..., m. Thus, v 1,..., v m lie in l p ( Z s ). Similarly, if f L,0 (IR s ), then v 1,..., v m lie in c 0 ( Z s ). Now assume that f S(Φ). Then there exist sequences u 1,..., u n on Z s such that f = n φ k u k. This in connection with (7.1) and (7.2) tells us that u 1,..., u n satisfy the following system of discrete convolution equations: a jk u k = v j, j = 1,..., m. (7.3) Consequently, this system of equations is compatible. We shall show that the matrix A(z) := ( ã jk (z) ) 1 j m,1 k n has rank n for every z T s, provided that the shifts of φ 1,..., φ n are stable. For this purpose, we deduce from (7.1) that for k = 1,..., n, ˆφ k (ξ + 2πβ) = m ã jk (e iξ ) ˆψ j (ξ + 2πβ), ξ IR s, β Z s. j=1 If A(e iξ ) had rank less than n for some ξ IR s, then the sequences ( ˆφ k (ξ + 2πβ)) β Z s, k = 1,..., n, would be linearly dependent, which contradicts the assumption on stability. Since A(z) has rank n for every z T s and the system of equations in (7.3) is compatible, we conclude that (7.3) is uniquely solvable for u 1,..., u n in l p ( Z s ), by Theorem 6.1. Let (u 1,..., u n ) be the solution. Then f = n φ k u k lies in S p (Φ). This shows that S p (Φ) = S(Φ) L p (IR s ) for 1 p <. If f S(Φ) L,0 (IR s ), then the sequences v 1,..., v m lie in c 0 ( Z s ); hence u 1,..., u n lie in c 0 ( Z s ). This shows that S (Φ) = S(Φ) L,0 (IR s ). 24

26 8. Approximation Order In this section we shall apply the results on linear operator equations to a study of approximation by shift-invariant spaces. See [10] for a recent survey on this topic. to E by For a subset E of L p (IR s ) (1 p ) and f L p (IR s ), define the distance from f dist (f, E) p := inf g E { f g p}. Let S be a closed shift-invariant subspace of L p (IR s ). For h > 0, let σ h be the scaling operator given by the equation σ h f := f( /h) for functions f on IR s. Let S h := σ h (S). For a real number r > 0, we say that S provides L p -approximation order r if, for every sufficiently smooth function f in L p (IR s ), dist (f, S h ) p C f h r h > 0, where C f is a constant independent of h. We say that S provides L p -density order r if lim dist (f, h 0 Sh ) p /h r = 0. + Let Φ be a finite collection of compactly supported functions in L p (IR s ) (1 p ). We say that S(Φ) provides approximation order r (resp. density order r) if S(Φ) L p (IR s ) does. Let r be a positive integer, and let φ be a compactly supported function in L p (IR s ) (1 p ) with ˆφ(0) 0. Then S(φ) provides approximation order r if and only if S(φ) contains Π r 1. This result was established by Ron [17] for the case p =, and by Jia [9] for the general case 1 p. The following theorem extends their results to finitely generated shift-invariant spaces. Theorem 8.1. Let Φ = {φ 1,..., φ n } be a finite collection of compactly supported functions in L p (IR s ) (1 p ). Suppose that the sequences ( ˆφ k (2πβ)) β Z s, k = 1,..., n, are linearly independent. For a positive integer r, the following statements are equivalent: (a) S(Φ) provides L p -approximation order r. (b) S(Φ) provides L p -density order r 1. (c) S(Φ) Π r 1. (d) There exists a function ψ S 0 (Φ) such that α Z s q(α)ψ( α) = q q Π r 1. (8.1) 25

27 It is known that (8.1) is true if and only if for all ν IN s 0 with ν < r and all β Z s D ν ˆψ(2πβ) = δ0ν δ 0β, where δ stands for the Kronecker sign. This result was first established by Schoenberg [19] for the univariate case, and then extended by Strang and Fix [20] to the multivariate case. If the shifts of the functions in Φ are stable, then, for each ξ IR s, the sequences ( ˆφ k (ξ + 2πβ)) β Z s (k = 1,..., n) are linearly independent. Thus, the conclusion of Theorem 8.1 is valid if the shifts of the functions in Φ are stable. This weaker form of Theorem 8.1 was first established by Lei, Jia, and Cheney [16]. Suppose Φ is contained in L 2 (IR s ). Recall that the bracket product [φ j, φ k ] is given by [φ j, φ k ](e iξ ) = ˆφj (ξ + 2πβ) ˆφ k (ξ + 2πβ), ξ IR s. β Z s Define the Gram matrix G Φ by G Φ (ξ) := ( [φ j, φ k ](e iξ ) ) 1 j,k n, ξ IRs. Then the sequences ( ˆφ k (2πβ)) β Z s, k = 1,..., n, are linearly independent if and only if det G Φ (0) 0. In order to prove Theorem 8.1 we observe that (a) implies (b) trivially. It was proved in [9] that (b) implies (c). The implication (d) (a) is well known. See [12] for an explicit L p -approximation scheme. It remains to prove (c) (d). This was proved by de Boor [2] for the case where Φ consists of a single function. For the general case, we need some auxiliary results about polynomials and polynomial sequences. Let T Φ be the mapping given by T Φ (q 1,..., q n ) := φ k ( α)q k (α), for (q 1,..., q n ) Π n. α Z s Lemma 8.2. Let Φ = {φ 1,..., φ n } be a collection of integrable functions on IR s with compact support. Then the following conditions are equivalent. (a) The sequences ( ˆφ k (2πβ)) β Z s, k = 1,..., n, are linearly independent. (b) T Φ (q 1,..., q n ) = 0 for polynomials q 1,..., q n implies q 1 = = q n = 0. (c) Any polynomial q S(Φ) can be uniquely represented as T Φ (q 1,..., q n ) for some polynomials q 1,..., q n. 26

28 Proof. As in the proof of Theorem 7.2, there exist functions ψ 1,..., ψ m L 1 (IR s ) such that they vanish outside the unit cube [0, 1) s and {ψ j [0,1) s for S(Φ) [0,1) s. Then each φ k (k = 1,..., n) can be represented as φ k = : j = 1,..., m} forms a basis m ψ j a jk, (8.2) where a jk (j = 1,..., m; k = 1,..., n) are finitely supported sequences on Z s. Let g jk (z) := j=1 β Z s a jk (β)z β, z (C \ {0}) s. For given v 1,..., v m l( Z s ), the function f := m j=1 ψ j v j lies in S(Φ) if and only if the following system of linear partial difference equations g jk (τ)u k = v j, j = 1,..., m, (8.3) is solvable for (u 1,..., u n ) (l( Z s )) n. Now we restrict the difference operators g jk (τ) to the space IP( Z s ). From (8.2) we deduce that ˆφ k (2πβ) = m g jk (e) ˆψ j (2πβ), k = 1,..., n, j=1 where e is the s-tuple (1,..., 1). Since the shifts of ψ 1,..., ψ m are linearly independent, the sequences ( ˆψ j (2πβ)) β Z s, j = 1,..., m, are linearly independent (see [13]). Thus, the sequences ( ˆφ k (2πβ)) β Z s (k = 1,..., n) are linearly independent if and only if the matrix G := (g jk (e)) 1 j m,1 k n has rank n. We observe that T Φ (q 1,..., q n ) = 0 if and only if g jk (τ)q k = 0. By Theorem 5.2, we conclude that conditions (a) and (b) are equivalent. Obviously, (c) implies (b). It remains to prove (b) implies (c). To this end, let e 1,..., e s be the unit coordinate vectors in IR s, and let t (t = 1,..., s) be the difference operator given by t f = f f( e t ). Let q S(Φ) Π and assume that q = m j=1 ψ j v j for some sequences v 1,..., v m. We claim that v 1,..., v m are polynomial sequences. Indeed, if q is a polynomial of degree less than r, then m ψ j ( r ) t v j = r t q = 0, t = 1,..., s. j=1 27

29 Since the shifts of ψ 1,..., ψ m are linearly independent, we have r t v j = 0 for t = 1,..., s and j = 1,..., m. This shows that v 1,..., v m are polynomial sequences. Since q lies in S(Φ), there exist sequences u 1,..., u n satisfying the system (8.3) of linear partial difference equations; hence (8.3) is compatible. Moreover, condition (b) tells us that the associated homogeneous system of (8.3) only has the trivial solution. Thus, by Theorem 5.1, the system (8.3) is uniquely solvable for (u 1,..., u n ) IP( Z s ) n. This shows that q can be uniquely represented as T Φ (q 1,..., q n ) for some polynomials q 1,..., q n. Lemma 8.3. Let F be a linear mapping from Π r to Π. Suppose F commutes with the shift operators, that is, F ( q( α) ) = (F q)( α) q Π r and α Z s. Then there exists a polynomial f Π r such that F (q) = f(τ)q q Π r. Proof. We use r to denote the set {α IN s 0 : α r}. For β IN s 0, let q β denote the monomial given by q β (z) = z β. We wish to find a polynomial f Π r such that f(τ)q β (0) = c β := F q β (0) β r. (8.4) Suppose f(z) = α r a α z α. Then the above equation is equivalent to the following: a α α β = c β, β r. (8.5) α r The matrix (α β ) α,β r is nonsingular. Indeed, if b β (β Z s ) are complex numbers such that β r b β α β = 0 for all α r, then b β = 0 for all β r (see, e.g., [1, 4]). Thus, there exists a unique vector (a α ) α r satisfying (8.5). With a α chosen in this way, the polynomial f(z) = α r a α z α satisfies (8.4). Since the monomials q β (β r ) span Π r, it follows that F q(0) = f(τ)q(0) for all q Π r. For any γ Z s, we have F q(γ) = F (q( + γ))(0) = f(τ)q( + γ)(0) = f(τ)q(γ). Thus, the two polynomials F q and f(τ)q agree on Z s. Hence F q = f(τ)q for all q Π r. This completes the proof. 28

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