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. More-

over, 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 121336

Shift-Invariant Spaces and Linear Operator Equations

§1. Introduction

The purpose of this paper is to investigate the structure of finitely generated shift-

invariant 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 IRs to C is called shift-invariant if it is invariant

under shifts (multi-integer translates), that is,

f ∈ S =⇒ f(· − α) ∈ S ∀α ∈ ZZs.

Let Φ be a set of functions from IRs to C. We denote by S0(Φ) the linear span of the shifts

of the functions in Φ. Then S0(Φ) is the smallest shift-invariant space containing Φ.

Let f be a (Lebesgue) measurable function on IRs. For 1 ≤ p <∞, let

‖f‖p :=(∫

IRs

|f(x)|p dx)1/p

.

For p = ∞, let ‖f‖∞ be the essential supremum of f on IRs. We denote by Lp(IRs) the

Banach space of all measurable functions f on IRs such that ‖f‖p is finite.

If Φ is a subset of Lp(IRs) (1 ≤ p ≤ ∞), we write Sp(Φ) for the closure of S0(Φ)

in Lp(IRs). Thus, Sp(Φ) is the smallest closed shift-invariant subspace of Lp(IRs) that

contains Φ. The functions in Φ are called the generators of Sp(Φ). If Φ is a finite subset

of Lp(IRs), then Sp(Φ) is said to be a finitely generated shift-invariant space. In

particular, if Φ consists of a single function φ, then Sp(φ) is called a principal shift-

invariant 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 ZZs.

1

If f ∈ L1(IRs), the Fourier transform f is defined by

f(ξ) :=∫

IRs

f(x)e−ix·ξ dx, ξ ∈ IRs.

The domain of the Fourier transform can be naturally extended to include L2(IRs).

In [4], de Boor, DeVore, and Ron gave the following characterization of a finitely

generated shift-invariant subspace of L2(IRs) in terms of the Fourier transforms of the

generators. For a finite subset Φ of L2(IRs) and a function f ∈ L2(IRs), f lies in S2(Φ) if

and only if

f =∑φ∈Φ

τφφ

for some 2π-periodic functions τφ, φ ∈ Φ.

The proof of this result given in [4] relies on the known characterization of doubly-

invariant 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 ZZs to C is called a sequence. Let `(ZZs)

denote the linear space of all sequences on ZZs, and let `0(ZZs) denote the linear space of

all finitely supported sequences on ZZs. Given a function φ : IRs → C and a sequence

a ∈ `(ZZs), the semi-convolution φ ∗′ a is the sum∑α∈ZZs

φ(· − α)a(α).

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 IRs to C. We use S(Φ)

to denote the linear space of functions of the form∑φ∈Φ φ∗′aφ, where aφ (φ ∈ Φ) are

sequences on ZZs. 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 Lp(IRs) (1 ≤ p ≤ ∞).

In Section 3 we will prove that S(Φ) ∩ Lp(IRs) is always closed in Lp(IRs); hence Sp(Φ)

is a subspace of S(Φ) ∩ Lp(IRs). In Section 4 we will show that S(Φ) ∩ L2(IRs) = S2(Φ).

Consequently, a function f ∈ L2(IRs) lies in S2(Φ) if and only if

f =∑φ∈Φ

φ∗′aφ

2

for some sequences aφ on ZZs, φ ∈ Φ. For general p, it is a difficult question whether the

two spaces S(Φ) ∩ Lp(IRs) and Sp(Φ) are the same.

When s = 1, it was proved in [11] that Sp(Φ) = S(Φ) ∩ Lp(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 Lp(IRs) whose shifts are stable. Under such

a condition we will show that Sp(Φ) = S(Φ) ∩ Lp(IRs) 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 equa-

tions 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:

n∑k=1

λjkuk = vj , j = 1, . . . ,m,

where λjk ∈ Λ for j = 1, . . . ,m and k = 1, . . . , n, v1, . . . , vm ∈ V , and u1, . . . , un 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 approxima-

tion by shift-invariant spaces. Let Φ be a finite collection of compactly supported functions

in Lp(IRs) (1 ≤ p ≤ ∞). Let r be a positive integer. If Φ consists of a single function φ

with φ(0) 6= 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 = {φ1, . . . , φn}.Suppose the sequences (φk(2πβ))β∈ZZs , 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

§2. Shift-Invariant Subspaces of L2(IRs)

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 L2(IRs) and f ∈ L2(IRs). Then f ∈ S2(Φ) if

and only if

f =∑φ∈Φ

τφφ

for some 2π-periodic functions τφ, φ ∈ Φ.

Recall that L2(IRs) is a Hilbert space with the inner product given by

〈f, g〉 :=∫

IRs

f(x)g(x) dx, f, g ∈ L2(IRs),

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 S2(f) is orthogonal to S2(g) if and only if 〈f(· −α), g〉 = 0 for all α ∈ ZZs.

The following lemma follows from basic properties of the Fourier transform.

Lemma 2.2. If f, g ∈ L2(IRs), then the series

h(ξ) :=∑β∈ZZs

f(ξ + 2πβ) g(ξ + 2πβ)

converges absolutely for almost every ξ ∈ IRs. The Fourier coefficients of the 2π-periodic

function h are 〈f(· − α), g〉, α ∈ ZZs.

The bracket product of two functions f and g in L2(IRs) is defined by

[f, g](eiξ) :=∑β∈ZZs

f(ξ + 2πβ) g(ξ + 2πβ), ξ ∈ IRs.

In particular, if f and g are compactly supported, then

[f, g](eiξ) =∑α∈ZZs

〈f(· − α), g〉 eiα·ξ ∀ ξ ∈ IRs. (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 L2(IRs). The following lemma is an easy consequence of Lemma 2.2.

4

Lemma 2.3. If φ, ψ ∈ L2(IRs), then ψ is orthogonal to S2(φ) if and only if [φ, ψ](eiξ) = 0

for almost every ξ ∈ IRs. Moreover, {φ(· − α) : α ∈ IRs} forms an orthonormal system if

and only if [φ, φ](eiξ) = 1 for almost every ξ ∈ IRs.

Proof of Theorem 2.1. Denote by F (Φ) the linear space of those functions f ∈ L2(IRs)

for which f =∑φ∈Φ τφφ for some 2π-periodic functions τφ (φ ∈ Φ). Suppose f ∈ F (Φ).

Then for every g ∈ S2(Φ)⊥ and almost every ξ ∈ IRs,

[f, g](eiξ) =∑φ∈Φ

τφ(ξ)[φ, g](eiξ) = 0.

By Lemma 2.3, this shows that f is orthogonal to S2(Φ)⊥; hence f ∈ S2(Φ)⊥⊥ = S2(Φ).

In other words, F (Φ) ⊆ S2(Φ).

For the proof of F (Φ) ⊇ S2(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) = S2(φ), it

suffices to show that F (φ) is closed. Suppose f lies in the closure of F (φ). Then there

exists a sequence (fn)n=1,2,... in F (φ) such that ‖fn − f‖2 → 0 as n→∞. It follows that

‖fn− f‖2 → 0. By passing to a subsequence if necessary, we may assume that fn converges

to f almost everywhere. Since fn ∈ F (φ), fn = τnφ for some 2π-periodic function τn. Let

E := {ξ ∈ IRs : φ(ξ + 2βπ) = 0 ∀β ∈ ZZs}.

Then E is 2π-periodic, i.e., ξ ∈ E implies ξ + 2βπ ∈ E for all β ∈ ZZs. For each n, let

λn(ξ) :={τn(ξ) if ξ /∈ E,0 if ξ ∈ E.

Evidently, λn is 2π-periodic and fn = λnφ. Since fn converges to f almost everywhere,

for almost every ξ ∈ IRs, limn→∞ fn(ξ + 2βπ) = f(ξ + 2βπ) for all β ∈ ZZs. Let ξ be such

a point. If ξ ∈ E, then λn(ξ) = 0 for all n. If ξ /∈ E, then φ(ξ+2βπ) 6= 0 for some β ∈ ZZs.

Consequently,

limn→∞

λn(ξ) = limn→∞

fn(ξ + 2βπ)/φ(ξ + 2βπ) = f(ξ + 2βπ)

/φ(ξ + 2βπ).

This shows that λ(ξ) := limn→∞ λn(ξ) exists for almost every ξ ∈ IRs. Since each λn is 2π-

periodic, the limit function λ is also 2π-periodic. Taking limits of both sides of fn = λnφ,

we obtain f = λφ. This shows that f ∈ F (φ). Therefore F (φ) is closed.

5

Now assume that F (Φ) = S2(Φ) and we wish to prove that F (Φ∪ψ) ⊇ S2(Φ∪ψ) for

any ψ ∈ L2(IRs). Let PΦ denote the orthogonal projection of L2(IRs) onto S2(Φ), and let

1 denote the identity operator on L2(IRs). Let ρ := (1− PΦ)ψ. Then S2(Φ) is orthogonal

to S2(ρ), and hence S2(Φ) + S2(ρ) is closed. With g := PΦψ ∈ S2(Φ) = F (Φ) we have

ρ = ψ− g ∈ F (Φ∪ψ), and so S2(ρ) = F (ρ) ⊆ F (Φ∪ψ). But S2(Φ) + S2(ρ) is closed, and

ψ = g + ρ ∈ S2(Φ) + S2(ρ). Therefore we have

F (Φ ∪ ψ) ⊇ S2(Φ) + S2(ρ) ⊇ S2(Φ ∪ ψ).

This completes the induction procedure.

§3. Local Shift-Invariant Spaces

Let Φ be a finite collection of compactly supported functions in Lp(IRs) (1 ≤ p ≤ ∞).

In this section we shall show that S(Φ) ∩ Lp(IRs) is closed in Lp(IRs).

A measurable function f : IRs → C is called locally integrable if

‖f‖1(K) :=∫K

|f(x)| dx <∞

for every compact subset K of IRs. We denote by Lloc := Lloc(IRs) the linear space of all

locally integrable functions on IRs. For k = 1, 2, . . ., the functional pk given by

pk(f) :=∫

[−k,k]s|f(x)| dx

is a semi-norm on Lloc. The family of semi-norms {pk : k = 1, 2, . . .} induces a topology on

Lloc so that Lloc becomes a complete, metrizable, locally convex topological vector space.

In other words, Lloc is a Frechet space (see, e.g., [7, p. 160]). Let (fn)n=1,2,... be a sequence

in Lloc. Then fn converges to a function f ∈ Lloc if and only if for every compact subset

K of IRs, ‖fn − f‖1(K) → 0 as n→∞.

The following theorem shows that a finitely generated local shift-invariant subspace

of Lloc(IRs) is closed in it.

Theorem 3.1. Let Φ be a finite collection of compactly supported integrable functions

on IRs. Then S(Φ) is a closed subspace of Lloc(IRs). If Φ is a subset of Lp(IRs) for some

p, 1 ≤ p ≤ ∞, then S(Φ) ∩ Lp(IRs) is closed in Lp(IRs).

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

Let S = S(Φ), where Φ is a finite collection of compactly supported integrable func-

tions on IRs. The restriction of S to the cube [0, 1)s is finite dimensional. Thus, we can

find a finite collection {ψi : i ∈ I} of integrable functions on IRs such that each ψi vanishes

outside [0, 1)s and {ψi|[0,1)s : i ∈ I} forms a basis for S|[0,1)s . The shifts of the functions

ψi (i ∈ I) are linearly independent; that is, for sequences ai ∈ `(ZZs) (i ∈ I),∑i∈I

ψi∗′ai = 0 =⇒ ai = 0 ∀ i ∈ I.

Let `(ZZs)I denote the linear space of all mappings from I to `(ZZs). We define the

linear mapping T from `(ZZs)I to Lloc(IRs) as follows:

T (a) :=∑i∈I

ψi∗′ai for a = (ai)i∈I ∈ `(ZZs)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 =∑i∈I

ψi∗′fi, (3.1)

where fi ∈ `(ZZs), i ∈ I. Suppose Φ = {φj : j ∈ J}. Then f lies in S(Φ) if and only if

there exist sequences uj (j ∈ J) on ZZs such that

f =∑j∈J

φj∗′uj . (3.2)

Since each φj is compactly supported and belongs to V , we can find finitely supported

sequences cij on ZZs (i ∈ I, j ∈ J) such that

φj =∑i∈I

ψi∗′cij , j ∈ J. (3.3)

It follows from (3.2) and (3.3) that

f =∑j∈J

∑α∈ZZs

φj(· − α)uj(α)

=∑j∈J

∑α∈ZZs

∑i∈I

∑β∈ZZs

ψi(· − α− β)cij(β)uj(α)

=∑i∈I

∑α∈ZZs

[∑j∈J

∑β∈ZZs

cij(β)uj(α− β)]ψi(· − α).

7

Comparing this with (3.1), we conclude that f lies in S(Φ) if and only if∑j∈J

∑β∈ZZs

cij(β)uj(α− β) = fi(α) ∀α ∈ ZZs and i ∈ I. (3.4)

Given α ∈ ZZs, we denote by τα the difference operator on `(ZZs) given by

ταa := a(·+ α), a ∈ `(ZZs).

If p =∑α∈ZZs cαz

α is a Laurent polynomial, where cα = 0 except for finitely many α, then

p induces the difference operator

p(τ) :=∑α∈ZZs

cατα.

For i ∈ I and j ∈ J , let gij denote the Laurent polynomial given by

gij(z) =∑β∈ZZs

cij(β)z−β , z ∈ (C \ {0})s.

Then (3.4) can be rewritten as∑j∈J

gij(τ)uj = fi, i ∈ I. (3.5)

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 (uj)j∈J . It is said to be compatible if the following compatibility conditions

are satisfied: For Laurent polynomials qi (i ∈ I),∑i∈I

qigij = 0 ∀ j ∈ J =⇒∑i∈I

qi(τ)fi = 0.

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 :={

(qi)i∈I :∑i∈I

qigij = 0 ∀ j ∈ J}.

Thus, we arrive at the following conclusion.

8

Lemma 3.2. There exists a subset Q of P I such that a function f =∑i∈I ψi∗′fi lies in

S(Φ) if and only if ∑i∈I

qi(τ)fi = 0 ∀ (qi)i∈I ∈ Q. (3.6)

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 Lloc(IRs).

Since {ψi|[0,1)s : i ∈ I} is linearly independent, there exist two positive constants C1 and

C2 such that for f = T (a) ∈ V and for all β ∈ ZZs,

C1

∑i∈I

|ai(β)| ≤ ‖f‖1(β + [0, 1)s) ≤ C2

∑i∈I

|ai(β)|. (3.7)

Let (f (n))n=1,2,... be a sequence in V converging to a function f in Lloc(IRs). Suppose

f (n) = T (a(n)). Then by (3.7) we have∣∣a(m)i (β)− a

(n)i (β)

∣∣ ≤ C−11 ‖f (m) − f (n)‖1(β + [0, 1)s)

for all i ∈ I and β ∈ ZZs. This shows that (a(n)i (β))n=1,2,... is a Cauchy sequence of complex

numbers. Let ai(β) := limn→∞ a(n)i (β) and a := (ai)i∈I . Using (3.7) again, we see that

for all β ∈ ZZs

‖T (a)− T (a(n))‖1(β + [0, 1)s) ≤ C2

∑i∈I

∣∣ai(β)− a(n)i (β)

∣∣.Hence T (a(n)) converges to T (a) in Lloc(IRs). In other words, f = T (a), thereby proving

that V is closed in Lloc(IRs).

Next, we show that S(Φ) is closed in V . Let(f (n)

)n=1,2,...

be a sequence in S(Φ)

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 ∈ ZZs, a(n)i (β) converges to

ai(β) as n → ∞. In other words, a(n)i converges to ai pointwise. Since each f (n) lies in

S(Φ), by Lemma 3.2 we have∑i∈I

qi(τ)a(n)i = 0 ∀ (qi)i∈I ∈ Q. (3.8)

For a fixed element (qi)i∈I ∈ Q and a fixed β ∈ ZZs, qi(τ)ai(β) only involves finitely many

ai(α), α ∈ ZZs. Letting n→∞ in (3.8) we conclude that∑i∈I

qi(τ)ai = 0 ∀ (qi)i∈I ∈ Q.

9

This shows that f = T (a) lies in S(Φ), by Lemma 3.2. Therefore, S(Φ) is a closed subspace

of Lloc(IRs).

Finally, suppose that Φ is a subset of Lp(IRs) for some p, 1 ≤ p ≤ ∞. If (f (n))n=1,2,...

is a sequence in S(Φ) ∩ Lp(IRs) converging to f in Lp(IRs), then f (n) converges to f in

the topology of Lloc(IRs). Hence f lies in S(Φ) by what has been proved. This shows that

S(Φ) ∩ Lp(IRs) is closed in Lp(IRs).

§4. Local Shift-Invariant Subspaces of L2(IRs)

Let Φ be a finite collection of compactly supported functions in L2(IRs). In [3] de Boor,

DeVore, and Ron demonstrated that the two spaces S(Φ) ∩ L2(IRs) and S2(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 S2(Φ) in terms of the semi-convolutions of

the generators with sequences on ZZs.

Theorem 4.1. Let Φ be a finite collection of compactly supported functions in L2(IRs).

Then S(Φ) ∩ L2(IRs) = S2(Φ). Consequently, a function f ∈ L2(IRs) lies in S2(Φ) if and

only if

f =∑φ∈Φ

φ∗′aφ

for some sequences aφ on ZZs, φ ∈ Φ.

In our proof we use the following two basic facts. First, if f ∈ L2(IRs) and a ∈ `0(ZZs),then

f∗′a(ξ) = f(ξ)a(e−iξ), ξ ∈ IRs, (4.1)

where a(z) :=∑α∈ZZs a(α)zα is the symbol of a. Second, if f ∈ L2(IRs) and g = f∗′a for

some nontrivial sequence a ∈ `0(ZZs), then S2(f) = S2(g) (see [4, Corollary 2.5]). Indeed,

g = f∗′a implies

g(ξ) = f(ξ)a(e−iξ) and f(ξ) = g(ξ)/a(e−iξ), for a.e. ξ ∈ IRs,

where a(e−iξ) is a 2π-periodic trigonometric polynomial. So f ∈ S2(g) and g ∈ S2(f) by

Theorem 2.1.

We also need the following lemma (cf. [14, Theorem 4.4] and [4, Theorem 3.38]).

10

Lemma 4.2. Let Φ be a finite collection of compactly supported functions in L2(IRs), and

let PΦ denote the orthogonal projection of L2(IRs) onto S2(Φ). Then there exists a non-

trivial sequence b ∈ `0(ZZs) such that for every compactly supported function g ∈ L2(IRs),

PΦ(g∗′b) is compactly supported.

Proof. The proof proceeds by induction on #Φ. Suppose Φ consists of a single function

φ 6= 0. For a compactly supported function g ∈ L2(IRs), let h and u be the functions

determined by

h(ξ) = [φ, φ](eiξ)g(ξ) and u(ξ) = [g, φ](eiξ)φ(ξ), ξ ∈ IRs.

Let b and c be the sequences such that b(e−iξ) = [φ, φ](eiξ) and c(e−iξ) = [g, φ](eiξ),

ξ ∈ IRs. 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 = Pφh = Pφ(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 L2(IRs). We wish to prove that it is also true for Φ ∪ ψ, where ψ is a compactly sup-

ported function in L2(IRs). By the induction hypothesis, there exists a nontrivial sequence

b ∈ `0(ZZs) such that for every compactly supported function g ∈ L2(IRs), PΦ(g∗′b) is com-

pactly supported. Let ρ := ψ∗′b − PΦ(ψ∗′b). Then ρ is compactly supported. Moreover,

since S2(ψ) = S2(ψ∗′b), the space

S2(Φ ∪ ψ) = S2

(Φ ∪ (ψ∗′b)

)= S2(Φ ∪ ρ)

is the orthogonal sum of S2(Φ) and S2(ρ). By what has been proved, there exists a non-

trivial sequence c such that for every compactly supported function g ∈ L2(IRs), Pρ(g∗′c)is compactly supported. Note that g∗′(b ∗ c) = (g∗′b)∗′c = (g∗′c)∗′b. Therefore, for every

compactly supported function g ∈ L2(IRs),

PΦ∪ψ(g ∗′ (b ∗ c)) = PΦ(g ∗′ (b ∗ c)) + Pρ(g ∗′ (b ∗ c))

is compactly supported.

11

Proof of Theorem 4.1. Theorem 3.1 shows S(Φ) ∩ L2(IRs) ⊇ S2(Φ), so we only have

to show S2(Φ) ⊇ S(Φ) ∩ L2(IRs). 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∗′aφ ∈ S(Φ) ∩ L2(IRs).

We wish to prove f ∈ S2(Φ). For this purpose, we observe that for every compactly

supported function g ∈ S2(Φ)⊥,

〈f, g〉 =∑φ∈Φ

∑α∈ZZs

〈φ(· − α), g〉 aφ(α) = 0.

By Lemma 4.2 we can find a nontrivial sequence b ∈ `0(ZZs) such that for every function

h ∈ L2(IRs) with compact support, PΦ(h∗′b) is compactly supported. Let g ∈ S2(Φ)⊥.

Then PΦ(g∗′b) = 0. There exists a sequence (gn)n=1,2,... of compactly supported functions

in L2(IRs) such that ‖gn − g‖2 → 0 as n→∞. Let

hn := gn∗′b− PΦ(gn∗′b).

Then each hn is compactly supported and hn ∈ S2(Φ)⊥. Hence 〈f, hn〉 = 0 for n = 1, 2, . . ..

Furthermore,

limn→∞

〈f, hn〉 = 〈f, g∗′b− PΦ(g∗′b)〉 = 〈f, g∗′b〉.

This shows 〈f, g∗′b〉 = 0. Let c be the sequence given by c(α) = b(−α) for all α ∈ ZZs.

Then 〈f, g∗′b〉 = 0 implies

〈f∗′c, g〉 =∑α∈ZZs

〈f(· − α), g〉c(α)

=∑α∈ZZs

〈f, g(·+ α)c(α)〉 = 〈f, g∗′b〉 = 0.

This is true for all g ∈ S2(Φ)⊥; hence f∗′c ∈ S2(Φ)⊥⊥ = S2(Φ). Therefore we have

f ∈ S2(f∗′c) ⊆ S2(Φ), as desired.

12

§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 kerλ to denote its kernel {u ∈ V : λu = 0}. Thus, λ is one-to-one if kerλ = {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.

We are interested in commutative subrings of L(V ) with identity. Let Λ be such a

subring. The ideal generated by finitely many elements λ1, . . . , λm in Λ is denoted by

(λ1, . . . , λm). 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 kerλ, λ ∈ I.Consider the following system of linear operator equations:

n∑k=1

λjkuk = vj , j = 1, . . . ,m, (5.1)

where λjk ∈ Λ for j = 1, . . . ,m and k = 1, . . . , n, v1, . . . , vm ∈ V , and u1, . . . , un 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 u1, . . . , un ∈ 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 µ1, . . . , µm ∈ Λ

with∑mj=1 µjλjk = 0, k = 1, . . . , n, one must have

∑mj=1 µjvj = 0. Evidently, if (5.1) is

consistent, then it is compatible.

If we replace every vector vj (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

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 u1, . . . , un 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

λju = vj , j = 1, . . . ,m. (5.2)

By the assumption, the associated homogeneous system

λju = 0, j = 1, . . . ,m,

only has the trivial solution. In other words, ker (λ1, . . . , λm) = {0}; hence (λ1, . . . , λm) is

invertible. Thus, there exist µ1, . . . , µm ∈ Λ such that ν := µ1λ1 + · · ·+µmλm is invertible.

Let

u := ν−1(µ1v1 + · · ·+ µmvm).

We claim that u satisfies the equations in (5.2). Indeed, since (5.2) is compatible, we have

λjvk = λkvj for j, k ∈ {1, . . . ,m}. Therefore

νλju = λj(νu) =m∑k=1

λjµkvk =m∑k=1

µk(λjvk) =m∑k=1

µkλkvj = νvj .

But ν is invertible, so it follows that λju = vj 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 (λ11, . . . , λm1) is trivial, for otherwise the associated homogeneous

system would have nontrivial solutions. Thus, there exist µ1, . . . , µm ∈ Λ such that the

linear mapping ν := µ1λ11 + · · · + µmλm1 is invertible. Apply ν to both sides of each

equation in (5.1):n∑k=1

νλjkuk = νvj , j = 1, . . . ,m. (5.3)

Since ν is invertible, two systems (5.1) and (5.3) are equivalent. Let

v0 :=m∑j=1

µjvj and λ0k :=m∑j=1

µjλjk, k = 1, . . . , n.

14

Then λ01 = ν, and the equationn∑k=1

λ0kuk = v0 (5.4)

is a consequence of (5.1). For each j = 1, . . . ,m, apply λj1 to both sides of (5.4) and

subtract the resulting equation from (5.3). In this way we obtain

n∑k=2

(λ01λjk − λj1λ0k

)uk = λ01vj − λj1v0, j = 1, . . . ,m. (5.5)

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 (u2, . . . , un). 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 (u2, . . . , un) is a nontrivial

solution of the homogeneous system, then we can find u1 ∈ V such that

λ01u1 = −(λ02u2 + · · ·+ λ0nun),

because λ01 = ν is invertible. Thus, (u1, u2, . . . , un) 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 (u2, . . . , un) be the solution. Since

λ01 = ν is invertible, we can find u1 ∈ V such that

νu1 = v0 −n∑k=2

λ0kuk.

Consequently, (u1, u2, . . . , un) is the unique solution of (5.1).

Next, we discuss two special linear operator equations: linear partial difference equa-

tions and linear partial differential equations. Theorem 5.1 will be used to give criteria for

solvability of those equations.

Let Π(Cs) denote the linear space of all polynomials of s variables with coefficients

in C. For a nonnegative integer d, we denote by Πd(Cs) the subspace of all polynomials

of (total) degree less than or equal to d. If no ambiguity arises, we write Π for Π(Cs) and

Πd for Πd(Cs), respectively.

15

A mapping a from ZZs 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 ∈ ZZs. The degree

of a is the same as the degree of q. Let IP(ZZs) denote the linear space of all polynomial

sequences on ZZs.

Suppose p(z) =∑α∈ZZs cαz

α is a Laurent polynomial of s variables with coefficients

in C, where cα = 0 except for finitely many α. Let e denote the s-tuple (1, . . . , 1). Then

p(e) =∑α∈ZZs cα. The polynomial p induces the difference operator p(τ) =

∑α∈ZZs cατ

α.

It is easily seen that p(τ) maps IP(ZZs) to itself. For a sequence a on ZZs we have

(τα − 1)a = a(·+ α)− a.

Hence (τα − 1)a = 0 if a is a constant sequence. Moreover, if a is a polynomial sequence,

then (τα − 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 poly-

nomial sequence a, p(τ)a is a polynomial sequence of degree less than the degree of a.

Consequently, p(τ) is invertible on IP(ZZs) if and only if p(e) 6= 0. Indeed, if p(e) = 0,

then p(τ)a = 0 for any constant sequence a. If p(e) 6= 0, then we can write p = c − p0,

where c = p(e) and p0(e) = 0. Thus, p0(τ) is degree-reducing, and so pn+10 (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+ p0(τ)/c2 + · · ·+ pn0 (τ)/cn+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 p1(τ), . . . , pm(τ). If ker I = {0}, then for at

least one j, pj(e) 6= 0, for otherwise the constant sequences would lie in the kernel of I.

But pj(e) 6= 0 implies that pj(τ) is invertible. This shows that I is invertible.

Theorem 5.2. Let pjk (j = 1, . . . ,m; k = 1, . . . , n) be Laurent polynomials of s variables

with coefficients in C. The homogeneous system of linear partial difference equations

n∑k=1

pjk(τ)uk = 0, j = 1, . . . ,m, (5.6)

16

only has the trivial solution if and only if the matrix

P :=(pjk(e)

)1≤j≤m,1≤k≤n

has rank n. Consequently, for given polynomial sequences v1, . . . , vm, the system of equa-

tionsn∑k=1

pjk(τ)uk = vj , j = 1, . . . ,m,

is uniquely solvable for (u1, . . . , un) ∈ IP(ZZs)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 (a1, . . . , an) in

Cn \ {0} such thatn∑k=1

pjk(e)ak = 0 for j = 1, . . . ,m. (5.7)

For each k, let uk be the constant sequence α 7→ ak, α ∈ ZZs. Then (u1, . . . , un) is a

nontrivial solution to the homogeneous system (5.6).

Conversely, suppose that the homogeneous system (5.6) has a nontrivial solution

(u1, . . . , un). We observe that for any polynomial q, (q(τ)u1, . . . , q(τ)un) is also a so-

lution of (5.6). We can find a polynomial q such that q(τ)u1, . . . , q(τ)un are constant

sequences but q(τ)uk 6= 0 for at least one k. Let ak = q(τ)uk(0) for k = 1, . . . , n. Then

the complex vector (a1, . . . , an) 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 IN0 := IN ∪ {0}. An element in INs0 is called a multi-index. If α = (α1, . . . , αs)

is a multi-index, then its length |α| is defined by |α| := α1 + · · · + αs, and its factorial is

defined by α! := α1! · · ·αs!. For two multi-indices α = (α1, . . . , αs) and β = (β1, . . . , βs),

by α ≤ β, or β ≥ α, we mean αj ≤ βj for j = 1, . . . , s.

Let α ∈ INs0 be a multi-index. The differential operator Dα on Π(Cs) is defined by

Dα

( ∑β∈INs

0

bβzβ

):=

∑β≥α

bββ!

(β − α)!zβ−α.

17

A polynomial p =∑α∈INs

0aαz

α induces the differential operator p(D) :=∑α∈INs

0aαD

α.

The differential operator p(D) is invertible on Π if and only if p(0) 6= 0. Indeed, if

p(0) = 0, then p(D) 1 = 0. Conversely, if p(0) 6= 0, then we may write p = c − p0, where

c = p(0) and p0 is a polynomial with p0(0) = 0. Then for any polynomial q of degree n,

the equation p(D)r = q has a unique solution

r =[1/c+ p0(D)/c2 + · · ·+ pn0 (D)/cn+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 p1(D), . . . , pm(D). If ker I = {0}, then pj(0) 6= 0 for

at least one j, for otherwise the constants would lie in ker I. But pj(0) 6= 0 implies that

pj(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 pjk (j = 1, . . . ,m; k = 1, . . . , n) be polynomials of s variables with

coefficients in C. The homogeneous system of linear partial differential equations

n∑k=1

pjk(D)uk = 0, j = 1, . . . ,m,

only has the trivial solution if and only if the matrix

P :=(pjk(0)

)1≤j≤m,1≤k≤n

has rank n. Consequently, for given polynomials v1, . . . , vm, the system of equations

n∑k=1

pjk(D)uk = vj , j = 1, . . . ,m,

is uniquely solvable for (u1, . . . , un) ∈ Πn if and only if the matrix P has rank n and the

system is compatible.

18

§6. Discrete Convolution Equations

In this section we shall give a criterion for solvability of discrete convolution equations.

Recall that `(ZZs) is the linear space of complex-valued sequences on ZZs, and `0(ZZs)

is the linear space of all finitely supported sequences on ZZs. Moreover, we use c0(ZZs)

to denote the linear space of all sequences a on ZZs such that lim|α|→∞ a(α) = 0, where

|α| := |α1|+ · · ·+ |αs| for α = (α1, . . . , αs) ∈ ZZs. Given a sequence a on ZZs, we define

‖a‖p :=( ∑α∈ZZs

|a(α)|p)1/p

, 1 ≤ p <∞.

For p = ∞, we define ‖a‖∞ to be the supremum of {|a(α)| : α ∈ ZZs}. For 1 ≤ p ≤ ∞ we

denote by `p(ZZs) the Banach space of all sequences a on ZZs such that ‖a‖p <∞.

Given a ∈ `(ZZs), the formal Laurent series∑α∈ZZs a(α)zα is called the symbol of a,

and denoted by a(z). If a ∈ `1(ZZs), then the symbol a is a continuous function on the

torus

Ts := {(z1, . . . , zs) ∈ Cs : |z1| = . . . = |zs| = 1}.

If a ∈ `0(ZZs), then a is a Laurent polynomial.

For a, b ∈ `(ZZs), we define the convolution of a and b by

a∗b(α) :=∑β∈ZZs

a(α− β)b(β), α ∈ ZZs,

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 ∈ ZZs \ {0}, then a∗δ = a for all a ∈ `(ZZs).

Evidently, for a ∈ `0(ZZs) and b ∈ `(ZZs), the convolution a∗b is well defined.

Let a be an element in `0(ZZs) such that a(z) 6= 0 for all z ∈ Ts. For given v ∈ `∞(ZZs),

the discrete convolution equation

a∗u = v

has a unique solution for u ∈ `∞(ZZs). To see this, let

c(α) :=1

(2π)s

∫[0,2π)s

1a(eiξ)

e−iα·ξ dξ, α ∈ ZZs.

Then the sequence c decays exponentially fast, and c(z)a(z) = 1 for all z ∈ Ts. Hence

c∗a = δ. If a∗u = v, then it follows that

u = δ∗u = (c∗a)∗u = c∗(a∗u) = c∗v.

19

This proves uniqueness of the solution. Moreover, if v lies in `p(ZZs) for some p, 1 ≤ p ≤ ∞,

then the solution u lies in `p(ZZs); if v ∈ c0(ZZs), then the solution u also lies in c0(ZZs).

Consider the system of discrete convolution equations

n∑k=1

ajk∗uk = vj , j = 1, . . . ,m, (6.1)

where ajk ∈ `0(ZZs) (j = 1, . . . ,m; k = 1, . . . , n) and vj ∈ `(ZZs) (j = 1, . . . ,m). We

say that this system of equations is compatible if for any c1, . . . , cm ∈ `0(ZZs) with∑mj=1 cj∗ajk = 0, k = 1, . . . , n, one must have

∑mj=1 cj∗vj = 0.

Theorem 6.1. Let v1, . . . , vm ∈ `∞(ZZs). Suppose that the system of discrete convolution

equations in (6.1) is compatible. If the matrix

A(z) :=(ajk(z)

)1≤j≤m,1≤k≤n

has rank n for every z ∈ Ts, then the system of equations in (6.1) is uniquely solvable for

(u1, . . . , un) ∈ (`∞(ZZs))n. Furthermore, if v1, . . . , vm lie in `p(ZZs) for some p, 1 ≤ p <∞,

then the solutions u1, . . . , un also lie in `p(ZZs); if v1, . . . , vm lie in c0(ZZs), then the solutions

u1, . . . , un also lie in c0(ZZs).

Proof. For j = 1, . . . ,m, let cj be the sequence given by cj(α) = aj1(−α), α ∈ ZZs. Then

cj(z) = aj1(z) for z ∈ Ts. Let a0k :=∑mj=1 cj∗ajk, k = 1, . . . , n. Since A(z) has rank n,

the Laurent polynomials a11(z), . . . , am1(z) do not have common zeros in Ts. Hence

a01(z) =m∑j=1

cj(z)aj1(z) =m∑j=1

∣∣aj1(z)∣∣2 > 0 ∀ z ∈ Ts.

Let us consider the case n = 1 first. In this case, (6.1) implies

a01∗u1 =m∑j=1

cj∗vj =: v0.

Since a01(z) > 0 for all z ∈ Ts, the equation a01∗u1 = v0 is uniquely solvable for u in

`∞(ZZs). Let u1 be the solution. By the assumption, the original system of equations in

(6.1) is compatible; hence a01∗vj = aj1∗v0. It follows that a01∗aj1∗u1 = aj1∗v0 = a01∗vj .Therefore, aj1∗u1 = vj for j = 1, . . . ,m. This shows that u1 is the unique solution to the

system of equations in (6.1).

20

The proof proceeds with induction on n. Suppose n > 1 and the desired result is valid

for n − 1. Let cj (j = 1, . . . ,m) and a0k (k = 1, . . . , n) be the same sequences as in the

above. Let w0 := v0 =∑mj=1 cj∗vj , wj := a01∗vj − aj1∗w0 (j = 1, . . . ,m), and

bjk := a01∗ajk − aj1∗a0k, j = 1, . . . ,m; k = 2, . . . , n.

Then w0, w1, . . . , wm ∈ `∞(ZZs). Consequently, (6.1) is equivalent to the following system

of equations:n∑k=1

a0k∗uk = w0 (6.2)

andn∑k=2

bjk∗uk = wj , j = 1, . . . ,m. (6.3)

We observe that (6.3) is compatible and the matrix B(z) := (bjk(z))1≤j≤m,2≤k≤n has rank

n − 1 for every z ∈ Ts. Thus, by the induction hypothesis, (6.3) is uniquely solvable for

u2, . . . , un in `∞(ZZs). Once u2, . . . , un are obtained, u1 is uniquely determined from (6.2).

This completes the induction procedure.

Finally, if v1, . . . , vm lie in `p(ZZs) for some p, 1 ≤ p <∞, then the above proof shows

that the solutions u1, . . . , un also lie in `p(ZZs). The same conclusion holds true for c0(ZZs).

§7. Stable Generators

Let Φ be a finite subset of Lp(IRs) (1 ≤ p ≤ ∞). In this section, we shall characterize

Sp(Φ) in terms of the semi-convolutions of the generators with sequences in `p(ZZs), if the

shifts of the functions in Φ are stable. If, in addition, the functions in Φ are compactly

supported, we shall prove Sp(Φ) = S(Φ)∩Lp(IRs) for 1 ≤ p <∞. When p = ∞, we denote

by L∞,0(IRs) the subspace of L∞(IRs) consisting of all functions f ∈ L∞(IRs) such that

‖f‖∞(IRs \ [−k, k]s) → 0 as k →∞. We shall prove S∞(Φ) = S(Φ) ∩ L∞,0(IRs).

Let Φ be a finite subset of Lp(IRs) (1 ≤ p ≤ ∞). We say that the shifts φ(· − α)

(φ ∈ Φ, α ∈ ZZs) are Lp-stable if there are two positive constants C1 and C2 such that

C1

∑φ∈Φ

‖aφ‖p ≤∥∥∥∥∑φ∈Φ

φ∗′aφ∥∥∥∥p

≤ C2

∑φ∈Φ

‖aφ‖p

21

for all sequences aφ ∈ `0(ZZs), 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 Lp-stable if and only if for any ξ ∈ IRs, the sequences (φ(ξ + 2πβ))β∈ZZs (φ ∈ Φ) are

linearly independent. When p = 2, their results were generalized by de Boor, DeVore, and

Ron in [4].

Suppose Φ = {φ1, . . . , φn}. Let TΦ be the mapping from (`0(ZZs))n to Lp(IRs) given

by

TΦ(a1, . . . , an) :=n∑k=1

φk∗′ak, a1, . . . , an ∈ `0(ZZs).

Let X := (`p(ZZs))n for 1 ≤ p < ∞ and X := (c0(ZZs))n for p = ∞. The norm on X is

defined by

‖(a1, . . . , an)‖X :=n∑k=1

‖ak‖p.

Suppose that the shifts of the functions in Φ are stable. Then the domain of TΦ can be

extended to X, and TΦ is a one-to-one continuous linear operator from X to Y := Lp(IRs).

For a = (a1, . . . , an) ∈ X, we write∑nk=1 φk∗′ak for TΦ(a). In other words,

limN→∞

∥∥∥∥ n∑k=1

φk∗′ak −n∑k=1

∑|α|≤N

φk(· − α)ak(α)∥∥∥∥p

= 0.

Moreover, there exists a positive constant C such that C‖a‖X ≤ ‖TΦ(a)‖Y for all a ∈ X.

From a well-known result in functional analysis (see, e.g., [18, p. 70]), the range of TΦ is

closed. In other words, TΦ(X) = Sp(Φ). Thus, we have the following result.

Theorem 7.1. Let Φ be a finite subset of Lp(IRs) such that the shifts of the functions in

Φ are Lp-stable (1 ≤ p ≤ ∞). For 1 ≤ p <∞, a function f lies in Sp(Φ) if and only if

f =∑φ∈Φ

φ∗′aφ

for some sequences aφ in `p(ZZs). For p = ∞, a function f lies in S∞(Φ) if and only if

f =∑φ∈Φ φ∗′aφ for some sequences aφ in c0(ZZs).

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 ∈ S2(φ) (see [4, Example 2.7]), but χ cannot be written in the form χ = φ∗′a for any

22

a ∈ `2(ZZ). Indeed, if a is an element of `2(ZZ), then the 2π-periodic function ξ 7→ a(eiξ) is

square integrable on [0, 2π) and

φ∗′a(ξ) = φ(ξ)a(e−iξ) = χ(ξ)(1− e−iξ)a(e−iξ).

Thus, χ = φ∗′a implies that

a(eiξ) = 1/

(1− eiξ) for a.e. ξ ∈ IR.

But the function ξ 7→ 1/(1 − eiξ) is not square integrable on [0, 2π). This contradiction

verifies our claim. Moreover, we have∫IRχ(x) dx = 1 and χ =

∑∞j=0 φ(· − j) ∈ S(φ).

However, any function f in S0(φ) satisfies∫IRf(x)dx = 0. Since S1(φ) is the closure of

S0(φ) in L1(IR), we also have∫IRf(x)dx = 0 for all f ∈ S1(φ). This shows that χ /∈ S1(φ).

Therefore S1(φ) 6= S(φ) ∩ L1(IR).

When Φ is a finite collection of compactly supported functions in Lp(IR), it was shown

in [11] that Sp(Φ) = S(Φ) ∩ Lp(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 Lp(IRs)

(1 ≤ p ≤ ∞). If the shifts of the functions in Φ are Lp-stable, then Sp(Φ) = S(Φ)∩Lp(IRs)for 1 ≤ p <∞, and S∞(Φ) = S(Φ) ∩ L∞,0(IRs).

Proof. By Theorem 3.1, S(Φ) ∩ Lp(IRs) is closed in Lp(IRs) (1 ≤ p ≤ ∞). Hence Sp(Φ)

is contained in S(Φ) ∩ Lp(IRs). For p = ∞, we also have S∞(Φ) ⊆ S(Φ) ∩ L∞,0(IRs).

Suppose Φ = {φ1, . . . , φn}. We can find functions ψ1, . . . , ψm ∈ Lp(IRs) such that

they vanish outside the unit cube [0, 1)s and {ψj |[0,1)s : j = 1, . . . ,m} forms a basis for

S(Φ)|[0,1)s . Then each φk (k = 1, . . . , n) can be represented as

φk =m∑j=1

ψj∗′ajk, (7.1)

where ajk (j = 1, . . . ,m; k = 1, . . . , n) are finitely supported sequences on ZZs.

A function f ∈ S(Ψ) has the following representation:

f =m∑j=1

ψj∗′vj , (7.2)

23

where v1, . . . , vm are sequences on ZZs. If f lies in Lp(IRs) for 1 ≤ p < ∞, then the

sequences v1, . . . , vm lie in `p(ZZs). To see this, we observe that, for β ∈ ZZs,

f(x) =m∑j=1

vj(β)ψj(x− β) for x ∈ β + [0, 1)s.

Hence, there exists a constant C > 0 such that∣∣vj(β)∣∣p ≤ Cp

∫β+[0,1)s

|f(x)|p dx ∀ j = 1, . . . ,m and β ∈ ZZs.

It follows that ‖vj‖p ≤ C‖f‖p for j = 1, . . . ,m. Thus, v1, . . . , vm lie in `p(ZZs). Similarly,

if f ∈ L∞,0(IRs), then v1, . . . , vm lie in c0(ZZs).

Now assume that f ∈ S(Φ). Then there exist sequences u1, . . . , un on ZZs such that

f =∑nk=1 φk∗′uk. This in connection with (7.1) and (7.2) tells us that u1, . . . , un satisfy

the following system of discrete convolution equations:

n∑k=1

ajk∗uk = vj , j = 1, . . . ,m. (7.3)

Consequently, this system of equations is compatible. We shall show that the matrix

A(z) :=(ajk(z)

)1≤j≤m,1≤k≤n

has rank n for every z ∈ Ts, provided that the shifts of φ1, . . . , φn are stable. For this

purpose, we deduce from (7.1) that for k = 1, . . . , n,

φk(ξ + 2πβ) =m∑j=1

ajk(e−iξ)ψj(ξ + 2πβ), ξ ∈ IRs, β ∈ ZZs.

If A(e−iξ) had rank less than n for some ξ ∈ IRs, then the sequences (φk(ξ + 2πβ))β∈ZZs ,

k = 1, . . . , n, would be linearly dependent, which contradicts the assumption on stability.

Since A(z) has rank n for every z ∈ Ts and the system of equations in (7.3) is compatible,

we conclude that (7.3) is uniquely solvable for u1, . . . , un in `p(ZZs), by Theorem 6.1. Let

(u1, . . . , un) be the solution. Then f =∑nk=1 φk∗′uk lies in Sp(Φ). This shows that

Sp(Φ) = S(Φ) ∩ Lp(IRs) for 1 ≤ p <∞.

If f ∈ S(Φ)∩L∞,0(IRs), then the sequences v1, . . . , vm lie in c0(ZZs); hence u1, . . . , un

lie in c0(ZZs). This shows that S∞(Φ) = S(Φ) ∩ L∞,0(IRs).

24

§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.

For a subset E of Lp(IRs) (1 ≤ p ≤ ∞) and f ∈ Lp(IRs), define the distance from f

to E by

dist (f,E)p := infg∈E

{‖f − g‖p}.

Let S be a closed shift-invariant subspace of Lp(IRs). For h > 0, let σh be the scaling

operator given by the equation σhf := f(·/h) for functions f on IRs. Let Sh := σh(S).

For a real number r > 0, we say that S provides Lp-approximation order r if, for every

sufficiently smooth function f in Lp(IRs),

dist (f, Sh)p ≤ Cfhr ∀h > 0,

where Cf is a constant independent of h. We say that S provides Lp-density order r if

limh→0+

dist (f, Sh)p/hr = 0.

Let Φ be a finite collection of compactly supported functions in Lp(IRs) (1 ≤ p ≤ ∞).

We say that S(Φ) provides approximation order r (resp. density order r) if S(Φ)∩Lp(IRs)does.

Let r be a positive integer, and let φ be a compactly supported function in Lp(IRs)

(1 ≤ p ≤ ∞) with φ(0) 6= 0. Then S(φ) provides approximation order r if and only if S(φ)

contains Πr−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 = {φ1, . . . , φn} be a finite collection of compactly supported func-

tions in Lp(IRs) (1 ≤ p ≤ ∞). Suppose that the sequences (φk(2πβ))β∈ZZs , k = 1, . . . , n,

are linearly independent. For a positive integer r, the following statements are equivalent:

(a) S(Φ) provides Lp-approximation order r.

(b) S(Φ) provides Lp-density order r − 1.

(c) S(Φ) ⊇ Πr−1.

(d) There exists a function ψ ∈ S0(Φ) such that∑α∈ZZs

q(α)ψ(· − α) = q ∀ q ∈ Πr−1. (8.1)

25

It is known that (8.1) is true if and only if for all ν ∈ INs0 with |ν| < r and all β ∈ ZZs

Dνψ(2πβ) = δ0νδ0β ,

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 ∈ IRs, the sequences

(φk(ξ + 2πβ))β∈ZZs (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 L2(IRs). Recall that the bracket product [φj , φk] is given

by

[φj , φk](eiξ) =∑β∈ZZs

φj(ξ + 2πβ)φk(ξ + 2πβ), ξ ∈ IRs.

Define the Gram matrix GΦ by

GΦ(ξ) :=([φj , φk](eiξ)

)1≤j,k≤n, ξ ∈ IRs.

Then the sequences (φk(2πβ))β∈ZZs , k = 1, . . . , n, are linearly independent if and only if

detGΦ(0) 6= 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

Lp-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 TΦ be the mapping

given by

TΦ(q1, . . . , qn) :=n∑k=1

∑α∈ZZs

φk(· − α)qk(α), for (q1, . . . , qn) ∈ Πn.

Lemma 8.2. Let Φ = {φ1, . . . , φn} be a collection of integrable functions on IRs with

compact support. Then the following conditions are equivalent.

(a) The sequences (φk(2πβ))β∈ZZs , k = 1, . . . , n, are linearly independent.

(b) TΦ(q1, . . . , qn) = 0 for polynomials q1, . . . , qn implies q1 = · · · = qn = 0.

(c) Any polynomial q ∈ S(Φ) can be uniquely represented as TΦ(q1, . . . , qn) for some

polynomials q1, . . . , qn.

26

Proof. As in the proof of Theorem 7.2, there exist functions ψ1, . . . , ψm ∈ L1(IRs) such

that they vanish outside the unit cube [0, 1)s and {ψj |[0,1)s : j = 1, . . . ,m} forms a basis

for S(Φ)|[0,1)s . Then each φk (k = 1, . . . , n) can be represented as

φk =m∑j=1

ψj∗′ajk, (8.2)

where ajk (j = 1, . . . ,m; k = 1, . . . , n) are finitely supported sequences on ZZs.

Let

gjk(z) :=∑β∈ZZs

ajk(β)z−β , z ∈ (C \ {0})s.

For given v1, . . . , vm ∈ `(ZZs), the function f :=∑mj=1 ψj∗′vj lies in S(Φ) if and only if the

following system of linear partial difference equationsn∑k=1

gjk(τ)uk = vj , j = 1, . . . ,m, (8.3)

is solvable for (u1, . . . , un) ∈ (`(ZZs))n.

Now we restrict the difference operators gjk(τ) to the space IP(ZZs). From (8.2) we

deduce that

φk(2πβ) =m∑j=1

gjk(e)ψj(2πβ), k = 1, . . . , n,

where e is the s-tuple (1, . . . , 1). Since the shifts of ψ1, . . . , ψm are linearly independent,

the sequences (ψj(2πβ))β∈ZZs , j = 1, . . . ,m, are linearly independent (see [13]). Thus, the

sequences (φk(2πβ))β∈ZZs (k = 1, . . . , n) are linearly independent if and only if the matrix

G := (gjk(e))1≤j≤m,1≤k≤n has rank n. We observe that TΦ(q1, . . . , qn) = 0 if and only if

n∑k=1

gjk(τ)qk = 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

e1, . . . , es be the unit coordinate vectors in IRs, and let ∇t (t = 1, . . . , s) be the difference

operator given by ∇tf = f −f(·−et). Let q ∈ S(Φ)∩Π and assume that q =∑mj=1 ψj∗′vj

for some sequences v1, . . . , vm. We claim that v1, . . . , vm are polynomial sequences. Indeed,

if q is a polynomial of degree less than r, thenm∑j=1

ψj∗′(∇rtvj

)= ∇r

t q = 0, t = 1, . . . , s.

27

Since the shifts of ψ1, . . . , ψm are linearly independent, we have ∇rtvj = 0 for t = 1, . . . , s

and j = 1, . . . ,m. This shows that v1, . . . , vm are polynomial sequences. Since q lies in

S(Φ), there exist sequences u1, . . . , un 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 (u1, . . . , un) ∈ IP(ZZs)n. This shows that q can be

uniquely represented as TΦ(q1, . . . , qn) for some polynomials q1, . . . , qn.

Lemma 8.3. Let F be a linear mapping from Πr to Π. Suppose F commutes with the

shift operators, that is,

F(q(· − α)

)= (Fq)(· − α) ∀ q ∈ Πr and α ∈ ZZs.

Then there exists a polynomial f ∈ Πr such that

F (q) = f(τ)q ∀ q ∈ Πr.

Proof. We use ∆r to denote the set {α ∈ INs0 : |α| ≤ r}. For β ∈ INs

0, let qβ denote the

monomial given by qβ(z) = zβ . We wish to find a polynomial f ∈ Πr such that

f(τ)qβ(0) = cβ := Fqβ(0) ∀β ∈ ∆r. (8.4)

Suppose f(z) =∑α∈∆r

aαzα. Then the above equation is equivalent to the following:

∑α∈∆r

aααβ = cβ , β ∈ ∆r. (8.5)

The matrix (αβ)α,β∈∆ris nonsingular. Indeed, if bβ (β ∈ ZZs) are complex numbers such

that∑β∈∆r

bβαβ = 0 for all α ∈ ∆r, then bβ = 0 for all β ∈ ∆r (see, e.g., [1, §4]). Thus,

there exists a unique vector (aα)α∈∆rsatisfying (8.5). With aα chosen in this way, the

polynomial f(z) =∑α∈∆r

aαzα satisfies (8.4). Since the monomials qβ (β ∈ ∆r) span Πr,

it follows that Fq(0) = f(τ)q(0) for all q ∈ Πr. For any γ ∈ ZZs, we have

Fq(γ) = F (q(·+ γ))(0) = f(τ)q(·+ γ)(0) = f(τ)q(γ).

Thus, the two polynomials Fq and f(τ)q agree on ZZs. Hence Fq = f(τ)q for all q ∈ Πr.

This completes the proof.

28

Proof of Theorem 8.1. It remains to prove (c) ⇒ (d). Suppose q ∈ Πr−1. Then

q ∈ S(Φ), and by Lemma 8.2 there exist unique polynomials q1, . . . , qn such that

q =∑α∈ZZs

n∑k=1

φk(· − α)qk(α).

For each k, the mapping Fk : q 7→ qk is a linear mapping from Πr−1 to Π which commutes

with shift operators. By Lemma 8.3 we can find a polynomial fk ∈ Πr−1 such that

Fkq = fk(τ)q for all q ∈ Πr−1. It follows that, for each q ∈ Πr−1,

q =∑α∈ZZs

n∑k=1

φk(· − α)fk(τ)q(α) =∑α∈ZZs

n∑k=1

(fk(τ)φk

)(· − α)q(α) =

∑α∈ZZs

ψ(· − α)q(α),

where ψ :=∑nk=1 fk(τ)φk belongs to S0(Φ). This shows that (c) implies (d).

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