A BERNSTEIN TYPE INEQUALITYASSOCIATED WITH WAVELET DECOMPOSITION

Rong-Qing Jia†Department of Mathematics

University of AlbertaEdmonton, Canada T6G 2G1

Abstract

Wavelet decomposition and its related nonlinear approximation problem are investi-gated on the basis of shift invariant spaces of functions. In particular, a Bernstein typeinequality associated with wavelet decomposition is established in such a general setting.Several examples of piecewise polynomial spaces are given to illustrate the general theory.

AMS Subject Classifications: 41 A 17, 41 A 63, 42 C 15, 46 E 35Key Words and Phrases: wavelet decomposition, nonlinear approximation, Bernstein typeinequalities, shift invariant spaces, Besov spaces, multivariate splines

† Supported in part by NSERC Canada under Grant OGP 121336

1. Introduction

In this paper we investigate wavelet decomposition in the Lp spaces (0 < p ≤ ∞) andthe related nonlinear approximation problem. Our results are an extension of the recentwork of DeVore, Jawerth and Popov [8] and that of DeVore, Petrushev and Yu [9].

Given a positive integer d, we denote by IRd the d-dimensional real linear space, andby ZZd the d-dimensional lattice consisting of all d-tuples whose components are integers.For a (Lebesgue) measurable subset E of IRd we denote by |E| its Lebesgue measure.Moreover, we denote by ‖f‖p(E) the quantity (

∫E|f(x)|p dx)1/p for 0 < p < ∞, and by

‖f‖∞(E) the essential supremum of f over E. Thus, for 0 < p ≤ ∞, Lp(E) is a linearspace equipped with the quasi-norm ‖ · ‖p(E). When E = IRd, the affiliate E is oftenomitted. For a positive integer s, let W s

∞ = W s∞(IRd) denote the Sobolev space consisting

of those functions whose derivatives of order ≤ s are all in L∞.Let φ be a function in L∞(IRd) with compact support. Together with φ we have its

dyadic dilates φ(2k·), k ∈ ZZ, and their translates φ(2k · −j), j ∈ ZZd. Following [8], weindex these functions by the dyadic cubes I = j2−k + 2−kΩ with Ω := [0, 1]d. Thus withI = j2−k + 2−kΩ we let

φI(x) := φ(2kx− j), x ∈ IRd.

We also use the notation Dk to denote the set of dyadic cubes 2−k(j + Ω) (j ∈ ZZd) and Dto denote the union of the Dk, k ∈ ZZ.

Let r and s be two positive integers and let φ be a function in W s∞(IRd) with compact

support. Three main assumptions were made in [8] about φ. First, φ was assumed tosatisfy a refinement equation

(1.1) φ =∑

j∈ZZd

b(j)φ(2 · −j)

where b is a sequence on ZZd supported on a finite set. The sequence b is called the mask ofthe refinement equation (1.1). Second, φ was assumed to satisfy the Strang-Fix conditionsof order r, that is, φ(0) 6= 0, and Dν φ(2πγ) = 0 for |ν| < r and γ ∈ ZZd \ 0, whereφ denotes the Fourier transform of φ, ν = (ν1, . . . , νd) is a multi-index with its length|ν| := ν1 + · · · + νd, and Dν := Dν1

1 · · ·Dνd

d with Di being the partial derivative operatorwith respect to the ith coordinate (i = 1, . . . , d). Third, the shifts φ(· − j), j ∈ ZZd,were assumed to be locally linearly independent (see §2 for a discussion of local linearindependence). Under these assumptions, it was shown in [8] and [9] that any functionf ∈ Lp(IRd) (0 < p ≤ ∞) has a representation by means of the functions φI :

(1.2) f =∑I∈D

aIφI .

The decomposition (1.2) is called a wavelet decomposition of f .For a positive integer n, let Σn denote the set of those functions f =

∑I∈D aIφI with

at most n of the coefficients aI 6= 0. Note that the sum of two elements of Σn might notbe in Σn. Thus Σn is a nonlinear manifold. The main results of the papers [8] and [9] are

1

a characterization of the order of Lp approximation (0 < p ≤ ∞) by the elements of Σn interms of Besov spaces.

Let us recall from [11] that a Besov space measures the smoothness of a functionf ∈ Lp(E), E a domain in IRd, in terms of the modulus of smoothness

ωr(f, t)p := sup|h|≤t

‖∆rhf‖p(E(rh))

where ∆rh is the rth order forward difference operator with step h ∈ IRd, |h| is the Euclidean

length of the vector h, and E(rh) is the set of x such that the line segment [x, x + rh] iscontained in E. If α > 0 and 0 < p, q ≤ ∞, the Besov space Bα

q (Lp(E)) is the collectionof those functions f ∈ Lp(E) for which the following is finite:

(1.3) |f |Bαq (Lp(E)) :=

(∫ ∞

0

[t−αωr(f, t)p]q(1/t) dt)1/q

, 0 < q < ∞;

supt>0

t−αωr(f, t)p

, q = ∞

where r is the least integer larger than α. Of special interest are the spaces Bα := Bασ (Lσ)

where σ := (α/d + 1/p)−1. The expression in (1.3) is a semi-(quasi)norm for Bαq (Lp(E)).

There is another equivalent semi-(quasi)norm for Bαq (Lp(E)) which will be useful in §3. It

is easily seen that

(1.4) |f |Bαq (Lp(E)) ≈

(∑k∈ZZ

[2kαωr(f, 2−k)p]q)1/q

, 0 < q < ∞;

supk∈ZZ

2kαωr(f, 2−k)p

, q = ∞.

This follows by noting that ωr(f, t) is an increasing function of t and then discretizing theintegral in (1.3) at the points 2−k (see [9]).

Given f ∈ Lp(IRd), 0 < p ≤ ∞, the error of approximating f in Lp from Σn is

σn(f)p := infg∈Σn

‖f − g‖p.

The following theorem was proved in [8].

Theorem 1.1. Let φ be a compactly supported function in W s∞(IRd). Suppose φ satisfies

a refinement equation of the type (1.1) with a finite mask, φ satisfies the Strang-Fixconditions of order r, and the shifts of φ are locally linearly independent. Then for 0 <p < ∞ and 0 < β < min(r, s),

(1.5) σn(f)p ≤ Cn−β/d, f ∈ Bβ .

If, in addition, the mask coefficients in the refinement equation (1.1) are nonnegative, thenfor 1 ≤ p < ∞,

(1.6) |g|Bβ ≤ Cnβ/d‖g‖p, g ∈ Σn.

2

In the above inequalities (1.5) and (1.6), Bβ = Bβσ (Lσ) with σ = (β/d + 1/p)−1, and C is

a constant depending only on φ and p.

For the L∞ approximation, a result similar to Theorem 1.1 was established in [9].The inequality (1.5) is a direct theorem of approximation (the Jackson inequality),

and the inequality (1.6) is an inverse theorem (the Bernstein inequality). Once (1.5) and(1.6) are established, then it is well known (see [12]) that the following characterizationresult holds for 0 < α < β:

∞∑n=1

[nα/dσn(f)p]τ/n < ∞ ⇐⇒ f ∈ Bα

where Bα = Bατ (Lτ ) with τ = (α/d + 1/p)−1.

In this paper we consider the problem of wavelet decomposition and approximationin the setting of shift invariant spaces. A linear space S of (measurable) functions on IRd

is said to be shift invariant if it is invariant under integer translation, that is,

f ∈ S =⇒ f(· − j) ∈ S for all j ∈ ZZd.

For instance, if φ is a compactly supported function, then

S(φ) := ∑

j∈ZZd

φ(· − j)a(j) : a is a sequence on ZZd,

the space generated by the shifts of φ, is shift invariant. More generally, if Φ is a finitecollection of compactly supported functions, then

S(Φ) :=∑φ∈Φ

S(φ)

is shift invariant. Such a shift invariant space is said to be finitely generated . We say thata shift invariant space S is refinable, if

f ∈ S =⇒ f(·/2) ∈ S.

Evidently, if φ is refinable with a finite mask, then S(φ) is refinable.We will report elsewhere our study of wavelet decomposition based on refinable shift

invariant spaces. In this paper we shall focus our attention on Bernstein inequalitiesassociated with wavelet decomposition. The problem can be formulated as follows. Let Φbe a finite collection of compactly supported functions in W s

∞(IRd), where s is a positiveinteger. Let S(Φ) be the space generated by the shifts of the functions in Φ. For a positiveinteger n, let Σn denote the set of all functions

(1.7) f =∑

φ∈Φ,I∈D

aφ,IφI

3

with at most n of the coefficients aφ,I 6= 0. The main result of this paper says that if S(Φ)is refinable, and if the shifts of the functions in Φ are locally linearly independent, thenfor 0 < α < s and 0 < p ≤ ∞,

|f |Bα ≤ Cnα/d‖f‖p for all f ∈ Σn,

where Bα = Bασ (Lσ) with σ = (α/d + 1/p)−1, and C is a constant depending only on Φ

and p when p is small. This will be proved in §4. In order to prove the Bernstein inequalitystated above we discuss certain properties of shift invariant spaces and Besov spaces in§2 and §3, respectively. The role of local linear independence will also be discussed in §2.Finally, in §5, we give several examples to illustrate the general theory.

Before concluding this introductory section, we wish to compare our result with thosein [8] and [9]. First, we allow Φ to consist of finitely many functions rather than just asingle function. Second, our result applies to the full range of p, i.e., 0 < p ≤ ∞, whiletheir results were established for the case 1 ≤ p ≤ ∞. Third, we do not assume that themask in the refinement equation is nonnegative. Fourth, we make no assumptions aboutthe Strang-Fix conditions. Indeed, in my opinion, the Strang-Fix conditions are irrelevantto the Bernstein inequality. Fifth, as will be demonstrated in Section 5, in our study thecondition of local linear independence can be relaxed in many cases. Last but not theleast, our proof is less complicated than the proof given in [8].

2. Shift Invariant Spaces

In this section we establish some useful properties of shift invariant spaces of functionsand discuss the local linear independence of a finite number of functions.

Let S be a shift invariant subspace of L∞(IRd). We say that S has finite local di-mension, if S|[0,1]d is finite dimensional. Evidently, if Φ is a finite collection of compactlysupported functions in L∞(IRd), then S(Φ) has finite local dimension. By dilation, weform the dyadic scale of spaces

Sk := f(2k·) : f ∈ S, k ∈ ZZ.

Also recall that for an integer k, Dk denotes the collection of all dyadic cubes of the form2−k(j + [0, 1]d), j ∈ ZZd.

The following lemma is a modification of [9, Lemma 6.1].

Lemma 2.1. Let S be a shift invariant subspace of L∞(IRd) having finite local dimension.Given 0 < p ≤ ∞, there exist constants ε > 0 and C > 0 such that for each f ∈ Sk andI ∈ Dk, one has for any measurable subset E of I with |E| ≤ ε|I|,

(2.1) ‖f‖p(I) ≤ |I|1/p‖f‖∞(I) ≤ C‖f‖p(I \ E)

where the constants ε and C depend only on S, and p when p is small.

Proof. Suppose I = 2−k(j + Ω) where k ∈ ZZ and j ∈ ZZd. By considering f(2−k(·+ j))we may assume without loss of generality that k = 0 and j = 0, that is , I = Ω = [0, 1]d.

4

Since S|[0,1]d is finite dimensional, there exists a basis, say φ1, . . . , φm, for it. We maychoose a basis in such a way that ‖φi‖∞ = 1 for i = 1, . . . ,m.

Let γ > 0 be a fixed real number. For γ ≤ p ≤ ∞ and G ⊆ I we have

(2.2) ‖f‖γ(G) ≤ ‖f‖p(G) for all f ∈ Lp(I).

This can be proved by applying Holder’s inequality to the integral∫

G|f(x)|γ dx and noting

that |G| ≤ 1. Obviously, (2.2) is also valid for p = ∞.There exists a constant c > 0 such that

(2.3) ‖m∑

i=1

aiφi‖γ(I) ≥ cm∑

i=1

|ai| for all (a1, . . . , am) ∈ IRm.

Now let f be an arbitrary function in S. Then f |I is a linear combination of φ1, . . . , φm:f |I =

∑mi=1 aiφi. Since ‖φi‖∞ = 1 for i = 1, . . . ,m, we have

(2.4) ‖f‖∞(I) ≤m∑

i=1

|ai|.

Let E be a subset of I. It follows from (2.2) that

‖f‖γγ(I) = ‖f‖γ

γ(E) + ‖f‖γγ(I \ E) ≤ |E|‖f‖γ

∞(I) + ‖f‖γp(I \ E).

This together with (2.3) and (2.4) implies that

(c

m∑i=1

|ai|)γ ≤ |E|

( m∑i=1

|ai|)γ + ‖f‖γ

p(I \ E),

from which we findm∑

i=1

|ai| ≤ (21/γ/c)‖f‖p(I \ E)

provided that |E| ≤ ε := cγ/2. Finally, with C := 21/γ/c we obtain

‖f‖p(I) ≤ ‖f‖∞(I) ≤m∑

i=1

|ai| ≤ C‖f‖p(I \ E).

A similar argument can be used to prove the following Markov type inequality.

Lemma 2.2. Let S be a shift invariant subspace of W s∞(IRd) which has finite local dimen-

sion. Let λ = (λ1, . . . , λd) be a multi-index with its length |λ| = s. Then for 0 < p ≤ ∞,

(2.5) ‖Dλf‖p(I) ≤ C2ks‖f‖p(I) for all f ∈ Sk and I ∈ Dk,

5

where C is a constant depending only on S, and p when p is small.

Proof. First we prove (2.5) for the case k = 0 and I = [0, 1]d. Let f ∈ S. Then f |[0,1]d

can be represented as a sum∑m

i=1 aiφi, where φ1, . . . , φm is a basis for S|[0,1]d with‖φi‖ = 1 for i = 1, . . . ,m. We have

‖Dλf‖p(I) ≤ ‖Dλf‖∞(I) = ‖m∑

i=1

ai(Dλφi)‖∞(I) ≤ C1

m∑i=1

|ai|

where C1 := max1≤i≤m ‖Dλφi‖∞(I). But, by means of (2.3),∑m

i=1 |ai| ≤ ‖f‖p(I)/c forsome constant c > 0. This proves (2.5) for the case k = 0 with C = C1/c.

Now let f ∈ Sk and I = 2−k([0, 1]d + j), where k ∈ ZZ and j ∈ ZZd. Then the functiong := f(2−k(·+ j)) is in S and f = g(2k · −j). Thus we have

‖Dλf‖p(I) = ‖2ks(Dλg)(2k · −j)‖p(I) = 2ks−kd/p‖Dλg‖p([0, 1]d)

≤ 2ks−kd/pC‖g‖p([0, 1]d) ≤ C2ks‖f‖p(I).

The rest of this section will be devoted to a discussion of the local linear independenceof a finite number of functions.

Given an integer k and a function g ∈ Sk, a cube I ∈ Dk is said to be a support cubeof g, if the measure of the set x ∈ I : g(x) 6= 0 is positive. When g = φI for some φ ∈ Φand I ∈ Dk, a support cube of φI always means a support cube in Dk. Let Φ be a finitecollection of compactly supported functions in L∞(IRd). For a given cube J ∈ Dk, wedenote by ΛJ the set of all pairs (φ, I) ∈ Φ×Dk for which J is a support cube of φI . Wesay that the shifts of the functions in Φ are locally linearly independent if for any J ∈ D0,the functions φI , (φ, I) ∈ ΛJ , are linearly independent over J . In such a case, if

f =∑

φ∈Φ,I∈Dk

aφ,IφI ,

then the coefficients aφ,I are uniquely determined by f .

Lemma 2.3. Let Φ be a finite collection of compactly supported functions in L∞(IRd)such that the shifts of the functions in Φ are locally linearly independent. Let k be aninteger and E a subset of IRd. If f =

∑φ∈Φ,I∈Dk

aφ,IφI has the property that any φI withaI,φ 6= 0 has a support cube contained in E, then for 0 < p ≤ ∞,

(2.6) ‖f‖p ≤ C‖f‖p(E)

where C is a constant depending only on Φ, and p when p is small.

Proof. After a suitable scaling, we may assume without loss of generality that k = 0.We may also assume that ‖φ‖∞ = 1 for all φ ∈ Φ. Observe that for any fixed x ∈ IRd,the number of nonzero terms in the sum

∑φ∈Φ,I∈D0

aφ,IφI(x) does not exceed a constant

6

depending only on Φ. Hence there exists a positive constant C1 depending on Φ and p forsmall p such that for all sequences (aφ,I)φ∈Φ,I∈D0

(2.7) ‖∑

φ∈Φ,I∈D0

aφ,IφI‖p ≤ C1

( ∑φ∈Φ,I∈D0

|aφ,I |p)1/p

with the usual change when p = ∞.For a given J ∈ D0, by assumption the functions φI , (φ, I) ∈ ΛJ , are linearly inde-

pendent over J . Consider the linear space spanned by φI |J : (φ, I) ∈ ΛJ. Any twoquasi-norms on this finite dimensional space are equivalent, so we have

(2.8) max(φ,I)∈ΛJ

|aφ,I | ≤ C2‖∑

(φ,I)∈ΛJ

aφ,IφI‖p(J)

where C2 is a constant depending only on Φ and p when p is small. Note that (φ, I) ∈ ΛJ

if and only if J is a support cube of φI . Now let f =∑

φ∈Φ,I∈D0aφ,IφI . By hypothesis,

any φI with aφ,I 6= 0 has a support cube J ⊆ E, hence we can deduce from (2.7) and (2.8)that

‖f‖∞ ≤ C1C2‖f‖∞(E),

which proves (2.6) for the case p = ∞. Let 0 < p < ∞. Then by (2.7) and (2.8) we have

‖f‖pp ≤ Cp

1

∑φ∈Φ,I∈D0

|aφ,I |p ≤ Cp1Cp

2

∑J⊆E,J∈D0

∫J

|f |p dx( ∑(φ,I)∈ΛJ

1).

But for a fixed J , the number of pairs (φ, I) such that (φ, I) ∈ ΛJ is bounded from aboveby a constant depending only on Φ. This proves (2.6) for the case 0 < p < ∞.

3. Besov Spaces

This section is devoted to an estimation of the semi-norm of functions in Besov spaces.In the following theorem, s is a positive integer, and S is a shift invariant subspace ofW s∞(IRd) having finite local dimension. Recall that Sk = f(2k·) : f ∈ S for k ∈ ZZ.

Theorem 3.1. Let 0 < α < s, 0 < p ≤ ∞, σ := (α/d + 1/p)−1, and Bα := Bασ (Lσ(IRd)).

Suppose fk ∈ Sk (k ∈ ZZ) and the sum∑

k∈ZZ fk converges to a function f ∈ Lσ(IRd)almost everywhere. Then

(3.1) |f |Bα ≤ C(∑k∈ZZ

2kασ‖fk‖σσ

)1/σ

where C is a constant depending only on the space S, and p if p is small.

A special case of Theorem 3.1 was proved in [8, Lemma 4.2], where S was assumed tobe generated by a single compactly supported function φ ∈ W s

∞, and φ was assumed to berefinable and satisfy the Strang-Fix conditions of some order r > α. Moreover, the shiftsof φ were assumed to be locally linearly independent. These assumptions are not neededin Theorem 3.1 except that S has finite local dimension.

We shall use (1.4) to estimate |f |Bα . To this end we need to estimate the moduliof smoothness ωr(f, 2−k)σ for k ∈ ZZ. This will be done in Lemma 3.2. Then a discreteversion of Hardy type inequalities will be employed to give a proof of Theorem 3.1.

7

Lemma 3.2. Let r ≤ s be a positive integer, and j, k ∈ ZZ. Then for f ∈ Sj and0 < σ ≤ ∞,

(3.2) ωr(f, 2−k)σ ≤ Cmin(1, 2j−k)

r‖f‖σ

where C is a constant depending only on the space S, and σ when σ is small.

Proof. It is easy to prove (3.2) for the case j ≥ k. Let h be a nonzero vector in IRd.From the well-known expression

∆rhf =

r∑m=0

(−1)r−m

(r

m

)f(·+ mh)

we find that‖∆r

hf‖σ ≤ 2r‖f‖σ for 1 ≤ σ ≤ ∞

and‖∆r

hf‖σσ ≤ 2r‖f‖σ

σ for 0 < σ < 1.

This proves that for any t > 0

ωr(f, t)σ = sup|h|≤t

‖∆rhf‖σ ≤ C‖f‖σ

with C := max(2r, 2r/σ).Now consider the case j < k. In what follows, C1, C2 and C3 denote constants

depending only on S, and σ when σ is small. Let x ∈ IRd and let h be a nonzero vector inIRd. Then by a well-known property of finite difference, we have

(3.3) ∆rhf(x) = r!Dr

hf(ξ)

for some ξ in the line segment [x, x + rh], where Dhf denotes the directional derivative off in the direction h. It follows from (3.3) that for |h| ≤ 2−k ≤ 2−j

(3.4) |∆rhf(x)| ≤ C12−kr‖f (r)‖∞(x + r2−j [−1, 1]d)

where f (r) :=∑

|λ|=r |Dλf |. Suppose now f ∈ Sj . Then by Lemma 2.2,

(3.5) ‖f (r)‖∞(J) ≤ C22jr‖f‖∞(J) for all J ∈ Dj .

On the other hand, (2.1) gives

(3.6) |J |1/σ‖f‖∞(J) ≤ C3‖f‖σ(J) for all J ∈ Dj .

Combining (3.4), (3.5) and (3.6) together, we obtain

ωr(f, 2−k)∞ = sup|h|≤2−k

‖∆rhf‖∞ ≤ C1C22(j−k)r‖f‖∞,

8

and for 0 < σ < ∞,

ωr(f, 2−k)σσ = sup

|h|≤2−k

∑I∈Dj

∫I

|∆rhf(x)|σ dx

≤ (C2(j−k)r)σ∑

I∈Dj

∑J∈Dj ,J⊆I+r2−j [−1,1]d

∫J

|f(x)|σ dx

where C = C1C2C3. But for a fixed I ∈ Dj , the number of dyadic cubes J ∈ Dj containedin I + r2−j [−1, 1]d does not exceed (2r + 1)d. This proves (3.2) for the case j < k.

The following Hardy type inequalities appeared in [11, (5.2) and (5.3)]. For thereader’s convenience we provide a proof for these inequalities, since no proof was giventhere.

Lemma 3.3. Let (ak)k∈ZZ be a sequence of nonnegative real numbers.(i) If bk =

∑∞j=k aj for all k ∈ ZZ, then for α > 0 and 1 ≤ τ ≤ ∞

(3.7)[ ∞∑k=−∞

(2kαbk)τ]1/τ ≤ C

[ ∞∑k=−∞

(2kαak)τ]1/τ

with the usual change when τ = ∞, where the constant C = 1/(1− 2−α).(ii) If bk = 2−kλ

(∑kj=−∞ 2jλaj

), then (3.7) holds for α < λ and 1 ≤ τ ≤ ∞ with the

constant C = 1/(1− 2α−λ).

Proof. We first prove part (i) for two special cases: τ = 1 and τ = ∞, and then applythe Riesz-Thorin interpolation theorem (see e.g., [13, p. 193]) to prove the general case.

For the case τ = 1, we have

∞∑k=−∞

2kαbk =∞∑

k=−∞

2kα∞∑

j=k

aj =∞∑

j=−∞aj(

j∑k=−∞

2kα) =∞∑

j=−∞aj2jα/(1− 2−α).

Let M := supk∈ZZ2kαak. Then ak ≤ 2−kαM for all k ∈ ZZ, and consequently

2kαbk = 2kα( ∞∑

j=k

aj

)≤ M

∞∑j=k

2(k−j)α = M/(1− 2−α) for all k ∈ ZZ.

This proves (3.7) for the case τ = ∞.Let uk := 2kαak and vk = 2kαbk for k ∈ ZZ. Then vk = 2kα(

∑∞j=k 2−jαuj). The

mapping Tα : u = (uk)k∈ZZ 7→ v = (vk)k∈ZZ is a linear mapping from `∞(ZZ) to itself.Recall that for 1 ≤ τ ≤ ∞, `τ (ZZ) is the linear space of those sequences c = (ck)k∈ZZ forwhich ‖c‖`τ (ZZ) := (

∑k∈ZZ |ck|τ )1/τ < ∞. We have proved that

(3.8) ‖Tαu‖`τ (ZZ) ≤ ‖u‖`τ (ZZ)/(1− 2−α)

9

for τ = 1 and τ = ∞. By the Riesz-Thorin interpolation theorem we conclude that (3.8)holds true for all τ , 1 ≤ τ ≤ ∞. This proves part (i).

Part (ii) can be reduced to part (i). Indeed, let uk := 2−kαa−k and vk = 2−kαb−k fork ∈ ZZ in part (ii). Then

vk = 2−kα2kλ−k∑

j=−∞(2jλ2−jαu−j) = 2k(λ−α)

∞∑j=k

2−j(λ−α)uj ,

hence v = Tλ−αu. By (3.8) we have

‖v‖`τ (ZZ) ≤ ‖u‖`τ (ZZ)/(1− 2α−λ) for 1 ≤ τ ≤ ∞.

This finishes the proof for part (ii).

Proof of Theorem 3.1. Consider the case 1 ≤ σ ≤ ∞ first. In this case, we have

ωr(f, 2−k)σ ≤∞∑

j=−∞ωr(fj , 2−k)σ ≤ C

( ∞∑j=k+1

‖fj‖σ +k∑

j=−∞2(j−k)r‖fj‖σ

)where we have used Lemma 3.2 to derive the second inequality. By Lemma 3.3 we find

( ∞∑k=−∞

[2kαωr(f, 2−k)σ]σ)1/σ ≤ C

( ∞∑k=−∞

[2kα‖fk‖σ]σ)1/σ

.

This together with (1.4) proves (3.1) for the case 1 ≤ σ ≤ ∞.Now let 0 < σ < 1. Then by Lemma 3.2 we have

ωr(f, 2−k)σσ ≤

∞∑j=−∞

ωr(fj , 2−k)σσ ≤ C

( ∞∑j=k+1

‖fj‖σσ +

k∑j=−∞

2(j−k)rσ‖fj‖σσ

).

Applying Lemma 3.3 with τ = 1 to the sequence (ωr(f, 2−k)σσ)k∈ZZ, we obtain

∞∑k=−∞

2kασωr(f, 2−k)σσ ≤ C

∞∑k=−∞

2kασ‖fk‖σσ.

This proves (3.1) for the case 0 < σ < 1.

10

4. The Bernstein Inequality

This section is devoted to a proof of our main result, the Bernstein inequality statedas follows.

Theorem 4.1. Let Φ be a finite collection of compactly supported functions in W s∞(IRd),

where s is a positive integer. Suppose that S(Φ) is refinable and the shifts of the functionsin Φ are locally linearly independent. Then for each α, 0 < α < s, and each p, 0 < p ≤ ∞,

(4.1) |f |Bα ≤ Cnα/d‖f‖p for all f ∈ Σn

where Bα = Bασ (Lσ) with σ = (α/d + 1/p)−1, C is a constant depending only on Φ and

p when p is small, and Σn is the set of all functions f of the form (1.7) with at most n ofthe coefficients aφ,I 6= 0.

Proof. Write S for S(Φ). Recall that Sk = g(2k·) : g ∈ S for k ∈ ZZ. A functionf ∈ Σn has a representation

(4.2) f =∑k∈K

fk

where K is a finite subset of ZZ, and fk ∈ Sk for each k ∈ K. Let Γk be the collection ofthe support cubes of fk in Dk (k ∈ K) and let Γ = ∪k∈KΓk. Denote by #Γ the numberof elements in Γ. Since the functions in Φ are compactly supported, #Γ ≤ Cn, where Cis a constant depending only on Φ.

We shall use Theorem 3.1 to estimate |f |Bα . Consider the case 0 < p < ∞ first. Notethat |I| = 2−kd for each I ∈ Γk. Applying Holder’s inequality to the integral

∫I|fk|σ dx,

we obtain∫I

|fk|σ dx ≤(∫

I

dx)1−σ/p(∫

I

|fk|p dx)σ/p = 2−kd(1−σ/p)(∫

I

|fk|p dx)σ/p

.

Since d(1− σ/p) = ασ, by (3.1) and the above inequality we have

|f |σBα ≤ Cσ(∑k∈K

2kασ‖fk‖σσ

)≤ Cσ

(∑k∈K

∑I∈Γk

(∫I

|fk|p dx)σ/p

).

Applying Holder’s inequality to the above double sum, we get

∑k∈K

∑I∈Γk

(∫I

|fk|p dx)σ/p ≤

(∑k∈K

∑I∈Γk

1)1−σ/p(∑

k∈K

∑I∈Γk

∫I

|fk|p dx)σ/p

.

Since #Γ ≤ Cn, we obtain from the above estimates that

(4.3) |f |Bα ≤ Cnα/d(∑k∈K

∑I∈Γk

∫I

|fk|p dx)1/p for 0 < p < ∞.

11

When p = ∞ we have d = ασ, hence for each I ∈ Γk,

2kασ‖fk‖σσ(I) = 2kd

∫I

|fk|σ dx ≤ ‖fk‖σ∞.

Therefore it follows from (3.1) and the above estimate that

(4.4) |f |Bα ≤ C(∑k∈K

2kασ‖fk‖σσ

)1/σ ≤ Cnα/d maxk∈K

‖fk‖∞,

where we have used the fact that #Γ ≤ Cn and 1/σ = α/d.Now let

Fk :=∑

i∈K,i≤k

fi for k ∈ K.

Given I ∈ Γk, we denote by FI the function FkχI with χI being the characteristic functionof I. If I is not a maximal cube in Γ, we let I∗ be the smallest cube in Γ that contains Istrictly . Suppose I∗ ∈ Γi for some i < k. Then by the very definition of I∗, fj vanisheson I for each j, i < j < k, and hence

fkχI = (Fk − Fi)χI = (FI − FI∗)χI .

When I is a maximal cube in Γ, we set FI∗ = 0. Thus, for 0 < p < ∞, we have∫I

|fk|p dx ≤ 2p(∫

I

|FI |p dx +∫

I

|FI∗ |p dx).

It follows that∑k∈K

∑I∈Γk

∫I

|fk|p dx ≤ 2p∑

k∈K

∑I∈Γk

(∫I

|FI |p dx +∫

I

|FI∗ |p dx)

= 2p∑

k∈K

∑I∈Γk

(∫I

|FI |p dx +∑

J∈Γ,J∗=I

∫J

|FI |p dx)

.

If J1 and J2 are two cubes in Γ such that J∗1 = J∗2 and J1 6= J2, then J1 and J2 areessentially disjoint. Hence

∑J∈Γ,J∗=I

∫J

|FI |p dx ≤∫

I

|FI |p dx.

The above estimates together with (4.3) yield

(4.5) |f |Bα ≤ Cnα/d(∑k∈K

∑I∈Γk

∫I

|Fk|p dx)1/p

, 0 < p < ∞.

12

With the help of (4.4), a similar argument gives the following estimate for the case p = ∞:

(4.6) |f |Bα ≤ Cnα/d maxk∈K

‖Fk‖∞.

The rest of our proof will be based on the estimates (4.5) and (4.6).Comparing (4.5) and (4.6) with (4.1) we find that (4.1) will be proved if we can show

(4.7)(∑k∈K

∑I∈Γk

∫I

|Fk|p dx)1/p ≤ C‖f‖p, 0 < p ≤ ∞.

Our goal is to find a “good” representation of f such that (4.7) is valid.We observe that if g is a compactly supported function in Sj for some j ∈ ZZ, then g

is also an element of Sj+1, because S = S(Φ) is refinable by assumption. Moreover, anysupport cube of g in Dj+1 must be contained in some support cube of g in Dj . This showsthat the number of the support cubes of g in Dj+1 is at most 2d times the number of thesupport cubes of g in Dj . Now suppose f ∈ Σn has a representation of the form (4.2).Let m be a positive integer. Let λ be the least integer such that λm ≥ minK and µ theleast integer such that µm ≥ maxK. For λ ≤ j ≤ µ, rewrite∑

k∈K,(j−1)m<k≤jm

fk

as a function fjm in Sjm. Then f has a new representation: f =∑µ

j=λ fjm. From theabove remark we see that the total number of the support cubes of fjm in Djm (λ ≤ j ≤ µ)is at most 2md times that of fk in Dk (k ∈ K). The integer m will be chosen later in sucha way that it depends only on Φ and p when p is small. Thus, we may assume from thebeginning that f has a representation of the form (4.2) with

(4.8) K = jm : j ∈ ZZ, λ ≤ j ≤ µ

for some integers λ and µ. Again we denote by Γk the collection of the support cubes offk in Dk for each k ∈ K and let Γ = ∪k∈KΓk. It has already been shown that #Γ ≤ Cnfor some constant C depending only on Φ and p when p is small.

Let I and J be two dyadic cubes in ∪k∈KDk. If J ⊂ I, then we say that J is adescendant of I. Here we caution the reader that in this paper J ⊂ I means that J is aproper subset of I, i.e., J ⊆ I but J 6= I. If, in addition, there is no cube L ∈ ∪k∈KDk

such that J ⊂ L ⊂ I, then we say that J is a child of I and I is the parent of J . Notethat our definition of children depends on the choice of the index set K. A dyadic cubeI ∈ ∪k∈KDk is said to be good with respect to Γ, if all the descendants of I in Γ arecontained in one child of I; otherwise, I is said to be bad with respect to Γ. If Γ is clearfrom context, the reference to Γ will be omitted.

We shall show that the number of bad cubes does not exceed #Γ. This will be provedby induction on #K. (Our proof is motivated by [10, Lemma 4.1].) If K contains only oneelement, then every cube in ∪k∈KDk is good, hence there is nothing to prove. Suppose

13

#K > 1 and our statement has been proved for any finite subset Γ′ ⊂ ∪k∈K′Dk with#K ′ < #K. Let Γb denote the collection of all bad cubes (with respect to Γ). Suppose jis the largest number of K. Let K ′ := K \ j. Then #K ′ < #K. Let i be the largestnumber of K ′. Let P1 be the collection of those dyadic cubes in Di which have exactly onechild in Γj , and let P2 be the collection of those dyadic cubes in Di which have at leasttwo children in Γj . Let

Γ′ :=(∪k∈K′Γk

)∪ P1 ∪ P2.

Denote by Γ′b the collection of all bad cubes with respect to Γ′. Then we have

Γb ⊆ Γ′b ∪ P2.

Indeed, if I ∈ Γb, then I has two children, say J1 and J2, each of which contains a cube inΓ. If I ∈ Di, then I ∈ P2; otherwise, I ∈ Dk for some k < i. In the latter case if J1 containsa cube in Γj = Γ \ Γ′, then J1 must also contain its parent, which is in P1 ∪ P2 ⊆ Γ′. Thesame is true for J2. This shows I ∈ Γ′b. Thus Γb ⊆ Γ′b ∪ P2. By induction hypothesis,#Γ′b ≤ #Γ′. It follows that

#Γb ≤ #Γ′b + #P2 ≤ #Γ′ + #P2

≤ #(∪k∈K′Γk) + #P1 + 2(#P2) ≤ #(∪k∈K′Γk) + #Γj = #Γ.

This completes the induction procedure.Recall that Fk :=

∑i∈K,i≤k fi for k ∈ K. Each Fk has an expansion as follows:

(4.9) Fk =∑

φ∈Φ,I∈Dk

aφ,IφI

where the coefficients aφ,I are uniquely determined by Fk, because the shifts of the func-tions in Φ are locally linearly independent. We say that a representation of f ∈ Σn of theform (4.2) is good , if #Γ ≤ Cn, and if for each k ∈ K every φI in (4.9) whose coefficientaφ,I 6= 0 has at least one good support cube. The initial representation (4.2) of f mightnot be good. But, as we shall demonstrate, (4.2) can be modified so that the resultingrepresentation of f is good. To this end, for each k ∈ K, we let gk be the sum of thoseterms aφ,IφI in (4.9) for which all the support cubes of φI are bad. Put

f?k := gk′ + fk − gk, k ∈ K,

where k′ denotes the largest integer in K less than k. When k is the least number of K,set gk′ = 0. Since S is refinable, we have f?

k ∈ Sk. By our previous assumption (4.8) aboutK, k′ = k−m. Thus, when gk′ is rewritten as a function in Sk, the number of its supportcubes in Dk is at most 2md times the number of its support cubes in Dk′ . Let Γ?

k be thecollection of the support cubes of f?

k in Dk, k ∈ K, and let Γ? = ∪k∈KΓ?k. Since all the

support cubes of gk in Dk (k ∈ K) are bad, and since the number of bad cubes does notexceed #Γ, we have

#Γ? ≤ (2md + 2)(#Γ) ≤ Cn.

14

Let F ?k :=

∑j∈K,j≤k f?

j for each k ∈ K. Then F ?k = Fk − gk. From the definition of gk

we find that F ?k is the sum of those terms aφ,IφI in (4.9) for which aφ,I 6= 0 and φI has at

least one good support cube (with respect to Γ). If we can show that any good cube (withrespect to Γ) is also good with respect to Γ?, then we can conclude that the representation

f =∑k∈K

f?k

is good. For this purpose, let I be a good cube (with respect to Γ). Then I has a child Jsuch that J contains all the descendants of I in Γ. Let A be a descendant of I in Γ?

k \ Γk

for some k ∈ K. We wish to show A ⊆ J . Observe that A is either a support cube of gk inDk, or a child of a support cube of gk′ in Dk′ . In the former case, A itself is a bad cube,while in the latter case the parent of A, denoted by A∗, is a bad cube. Since I is good, A∗

cannot be I, so in the latter case A∗ is also a descendant of I. Now either A or A∗ has adescendant in Γ which must be contained in J . Hence A itself is contained in J . Thus, Iis also good with respect to Γ?.

To summarize, we have proved that f has a good representation of the form (4.2)with K as given in (4.8). We are now in a position to prove that (4.7) is true for a goodrepresentation of f . To this end, for each k ∈ K, we denote by Λk the collection of thesupport cubes of those φI in (4.9) for which the corresponding coefficient aφ,I is nonzeroand φI has both good and bad support cubes. Let Γk and Λk be the collection of thegood cubes in Γk and Λk, respectively. We denote by Gk the sum of those terms aφ,IφI

in (4.9) for which φI has a support cube in Λk. Then Gk agrees with Fk on every cubeJ ∈ Λk. Assume 0 < p < ∞ for the moment. Since the shifts of the functions in Φ arelocally linearly independent, by Lemma 2.3 we have

(4.10)∫

IRd

|Gk|p dx ≤ Cp( ∑J∈Λk

∫J

|Gk|p dx).

Let J ∈ Γk \ Γk. We claim that Fk and Gk agree on J . Indeed, J is a bad cube; if J isa support cube for some φI with aφ,I 6= 0, then φI must also have a good support cube,because the representation (4.2) of f is good. Then φI has a support cube in Λk, andhence the term aφ,IφI was included in Gk. Thus, it follows from (4.10) that∑

J∈Γk\Γk

∫J

|Fk|p dx ≤ Cp( ∑J∈Λk

∫J

|Fk|p dx).

Consequently, we have ∑J∈Γk

∫J

|Fk|p dx ≤ Cp( ∑J∈Γk∪Λk

∫J

|Fk|p dx).

Let Γ := ∪k∈K Γk and Λ := ∪k∈KΛk. Then it follows that

(4.11)∑k∈K

∑J∈Γk

∫J

|Fk|p dx ≤ Cp( ∑J∈Γ∪Λ

∫J

|FJ |p dx),

15

where FJ := FkχJ for each J ∈ Γk. Let J ∈ ∪k∈KDk and consider

J ′ :=⋃

I∈Γ∪Λ,I⊂J

I,

where Λ := ∪k∈KΛk. If J ∈ Dj for some j ∈ K, then for every k > j, k ∈ K, fk issupported on ∪I∈Γk

I; hence fk(x) = 0 for x ∈ J \ J ′. This shows that f and FJ agreeon J \ J ′. Now let ε be the positive constant in Lemma 2.1 such that the estimate (2.1)holds. We claim that for a sufficiently large m,

(4.12) |J ′| ≤ ε|J | for every good cube J ∈ ∪k∈KDk.

This will be proved later. For the time being we assume that (4.12) is valid. Thus, byLemma 2.1 we find that for each cube J ∈ Γ ∪ Λ,

(4.13)∫

J

|FJ |p dx ≤ Cp

∫J\J′

|FJ |p dx = Cp

∫J\J′

|f |p dx.

Evidently, the sets J \ J ′ (J ∈ Γ ∪ Λ) are mutually essentially disjoint. Thus, it followsfrom (4.13) that∑

J∈Γ∪Λ

∫J

|FJ |p dx ≤ Cp∑

J∈Γ∪Λ

∫J\J′

|f |p dx ≤ Cp

∫IRd

|f |p dx.

This together with (4.11) and (4.5) proves the desired estimate (4.1) for the case 0 < p < ∞.By using an argument similar to the above we obtain

(4.14) maxk∈K

maxJ∈Γk

‖Fk‖∞(J) ≤ C maxk∈K

maxJ∈Γk∪Λk

‖FJ‖∞.

By Lemma 2.1, (4.12) implies that for J ∈ Γk ∪ Λk,

‖FJ‖∞ ≤ C‖FJ‖∞(J \ J ′) ≤ C‖f‖∞,

where we have used the fact that f and FJ agree on J \ J ′. This together with (4.14) and(4.6) yields the desired estimate for p = ∞:

|f |Bα ≤ Cnα/d‖f‖∞ for all f ∈ Σn.

Now let us justify our claim that (4.12) is true as long as m is sufficiently large. Sincethe functions in Φ are compactly supported, there exists a positive integer N such that[−N,N ]d contains the support of every φ ∈ Φ. Choose m in such a way that 2m > 2N .Let J be a good cube in ∪k∈KDk. If J has descendants in Γ, we let J1 be the only childof J that contains all the descendants of J in Γ; otherwise, set J1 = ∅. By our assumption(4.8) about K, there exist some j ∈ ZZ and ν ∈ 2−jmZZd such that

J = ν + 2−jm[0, 1]d.

16

It follows that J1 ∈ D(j+1)m. Consider the set

(4.15) J ′′ :=(ν + 2−(j+1)m[2N, 2m − 2N ]d

)\

(J1 + 2−(j+1)m[−2N, 2N ]d

).

We wish to show that J ′ is essentially disjoint from J ′′. For this purpose, let A be a cubein Γ ∪ Λ such that A ⊂ J . Then A ∈ Dkm for some integer k > j. If A ∈ Γ, then A ⊆ J1

by the choice of J1, and hence A is essentially disjoint from J ′′. Now let A ∈ Λ. By thedefinition of Λ, there exist some φ ∈ Φ and I ∈ Dkm such that A is a support cube of φI

and φI has both good and bad support cubes. In particular, the support of φI is containedin A1 := A + 2−(j+1)m[−2N, 2N ]d. If the set A ∩ J ′′ had positive measure, then A wouldbe contained in the closure of J ′′. By the definition of J ′′ we would have

A1 = A + 2−(j+1)m[−2N, 2N ]d ⊆ ν + 2−(j+1)m[0, 2m]d = J,

hence every support cube of φI would be contained in J . Moreover, the set A1 would beessentially disjoint from J1. This shows that every support cube of φI would be good,for otherwise J would have a descendant which is not contained in J1. This contradictionshows that A is essentially disjoint from J ′′. Thus, we have proved that J ′ is essentiallydisjoint from J ′′. Consequently, by (4.15) we obtain

|J ′| ≤ |J | − |J ′′| ≤ |J |[1− (1− 4N2−m)d + 2−md(4N + 1)d

].

From this we see that |J ′| ≤ ε|J | as long as m is sufficiently large. Note that m dependsonly on ε and Φ, while ε depends only on Φ and p when p is small, so m depends only onΦ and p when p is small.

5. Some Examples

Before giving any example, we make an apparent observation that if the Bernsteininequality (4.1) is valid for a shift invariant subspace T of W s

∞(IRd), then it is also truefor any shift invariant subspace S of T .

We consider first the univariate case. Let S be a shift invariant subspace of W s∞(IR).

If S is refinable, and if S is generated by a finite collection Φ of compactly supportedfunctions, then the Bernstein inequality (4.1) holds for each α, 0 < α < s, and each p,0 < p ≤ ∞. Note that the shifts of the functions in Φ were not assumed to be locally linearlyindependent. Indeed, it is known (see [3] and [15]) that there exists a finite collection Ψof compactly supported functions in S(Φ) such that S(Ψ) = S(Φ) and the shifts of thefunctions in Ψ are linearly independent over any interval [r, r +R], where r is an arbitraryinteger and R is a fixed positive integer depending only on Φ. Thus, Theorem 4.1 can beapplied to this case. Here we would like to mention that inequalities of Bernstein typewere established by Burchard in [5] for piecewise polynomials on optimal meshes. Morerecently, Petrushev in [16] studied free knot spline approximation in Lp[0, 1], p > 0, andproved Bernstein inequalities for this type of approximation. In fact, using his method, onecan prove that the Bernstein inequality (4.1) is valid for refinable shift invariant subspacesof W s

∞(IR) having local finite dimension.

17

Next, we discuss the multivariate case. Suppose Φi (i = 1, 2) are finite collectionsof compactly supported functions in W s

∞(IRdi). Given φi ∈ Φi (i = 1, 2), we denote byφ1 ⊗ φ2 the tensor product of φ1 and φ2, that is,

φ1 ⊗ φ2(x) := φ1(x1)φ2(x2) for x = (x1, x2) ∈ IRd1 × IRd2 .

LetΦ = Φ1 ⊗ Φ2 := φ1 ⊗ φ2 : φ1 ∈ Φ1, φ2 ∈ Φ2.

If the shifts of the functions in Φi (i=1,2) are locally linearly independent, then so are theshifts of the functions in Φ. Thus the Bernstein inequality (4.1) holds in this case.

Piecewise polynomials are important in wavelet decomposition. As an example, let usconsider a shift invariant subspace S of W s

∞(IR2) which consists of piecewise polynomialson the square mesh, that is, for every f ∈ S and every j ∈ ZZ2, f agrees with a polynomialof (total) degree ≤ k on the square j + [0, 1]2, where k is a fixed positive integer. In thiscase, one can find two finite sets Φ1 and Φ2 of compactly supported functions in W s

∞(IR)such that the shifts of the functions in Φi (i = 1, 2) are locally linearly independent andS ⊆ S(Φ1 ⊗ Φ2). Thus, by what has been proved before, the Bernstein inequality (4.1)applies to this case.

An interesting class of piecewise polynomials are the box splines introduced in [2]. LetV := vi : 1 ≤ i ≤ m be a family of integer vectors in IRd which span IRd. Some vectorsin V may be repeated several times. The box spline M = MV is the function defined bythe distributional equation∫

IRd

M(x)f(x) dx =∫

[0,1]m

( m∑i=1

yivi) dy, f ∈ C∞0 (IRd).

Then M is a piecewise polynomial of compact support. Moreover, the box spline M is inW s∞(IRd), s := s0 − 1, where s0 is the smallest integer for which there are s0 vectors in V

whose removal results in a set of vectors which do not span IRd (see [4] for these facts). Thebox spline M is a refinable function (see [6]). Thus Theorem 4.1 applies to S(M) as longas the shifts of M are locally linear independent. Dahmen and Micchelli ([7]) and Jia ([14])have characterized when the shifts of M are locally linearly independent. In particular, ifV consists of vectors (1, 0), (0, 1) and (1, 1) in IR2 (these vectors may be repeated severaltimes), then the shifts of MV are locally linearly independent. However, if M = MV is theZwart element , where

V = (1, 0), (0, 1), (1, 1), (−1, 1),

then the shifts of M are not locally linearly independent. It is interesting to point out thatthe Bernstein inequality (4.1) is still valid for S(M) with M being the Zwart element.

To see this, let us consider shift invariant triangulations of IR2. A triangulation ∆ ofIR2 is said to be shift invariant, if it is invariant under integer translation. Typical examplesof shift invariant triangulations are the 3-direction mesh and the 4-direction mesh. The3-direction mesh is formed by the three lines x1 = 0, x2 = 0, x1 = x2 on the plane IR2

and their shifts. The 4-direction mesh is formed by the four lines x1 = 0, x2 = 0, x1 = x2,

18

x1 = −x2 on the plane IR2 and their shifts. If ∆ is a triangulation of IR2, we denote by Πk,∆

the space of all piecewise polynomials of degree k on ∆, and let Πsk,∆ := Πk,∆∩W s+1

∞ (IR2).It has been proved by de Boor in [1] that if k ≥ 3s + 2, then Πs

k,∆ has a locally linearlyindependent basis. In particular, if ∆ is shift invariant and k ≥ 3s + 2, then one can finda finite collection Φ of compactly supported functions in Πs

k,∆ such that Πsk,∆ = S(Φ)

and that the shifts of the functions in Φ are locally linearly independent. Moreover, ∆ issaid to be refinable if ∆/2 is a subdivision of ∆. Evidently, the 3 or 4-direction mesh isrefinable. It is easily seen that Πs

k,∆ is a refinable shift invariant space provided that ∆ isshift invariant and refinable. Thus, we have the following result.

Theorem 5.1. Let ∆ be a shift invariant and refinable triangulation of IR2. Suppose Sis a subspace of Πs

k,∆ for some positive integer k. Then for each α, 0 < α < s + 1, theBernstein inequality (4.1) is valid for S.

In particular, if M is the Zwart element, then S(M) = Π12,∆, where ∆ is the 4-direction

mesh. Hence the Bernstein inequality holds true for S(M).The above discussion motivates us to put forward the following conjecture.

Conjecture. If S is a shift invariant subspace of W s∞(IRd) which is refinable and has finite

local dimension, then there exists a finite collection Φ of compactly supported functions inW s∞(IRd) such that (i) S ⊆ S(Φ), (ii) S(Φ) is refinable, and (iii) the shifts of the functions

in Φ are locally linearly independent.

If this conjecture were true, then Theorem 4.1 would be valid with the condition oflocal linear independence removed.

19

References

[1] C. de Boor, A local basis for certain smooth bivariate pp spaces, in MultivariateApproximation Theory V , W. Schempp and K. Zeller eds., Birkhauser, Basel (1990),25–30.

[2] C. de Boor and R. DeVore, Approximation by smooth multivariate splines, Trans.Amer. Math. Soc. 276 (1983), 775–788.

[3] C. de Boor and R. DeVore, Partition of unity and approximation, Proc. Amer. Math.Soc. 93 (1985) 705–709.

[4] C. de Boor and K. Hollig, B-splines from parallelepipeds, J. d’Anal. Math. 42 (1982/3),99–115.

[5] H. G. Burchard, On the degree of convergence of piecewise polynomial approximationon optimal meshes, Trans. Amer. Math. Soc. 234 (1977), 531–559.

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