strang-fix theory for approximation orderdownloads.hindawi.com/journals/jfs/2006/607497.pdf ·...

JOURNAL OF c© 2006, Scientific Horizon

FUNCTION SPACES AND APPLICATIONS http://www.jfsa.net

Volume 4, Number 1 (2006), 7-24

Strang-Fix theory for approximation order

in weighted Lp-spaces and Herz spaces

Naohito Tomita

(Communicated by Hans G. Feichtinger)

2000 Mathematics Subject Classification. 41A25, 42B25.

Keywords and phrases. Strang-Fix theory, multiresolution approximation,

Ap -weight, weighted Sobolev space, weighted Herz-Sobolev space.

Abstract. In this paper, we study the Strang-Fix theory for approximation

order in the weighted Lp -spaces and Herz spaces.

1. Introduction

In [10], Strang and Fix considered the relation between approximationorder in L2(Rn) of a given function and properties of its Fourier transform.In order to describe their results, we make the following definitions. Letϕ ∈ Cc(Rn), where Cc(Rn) consists of all continuous functions on R

n withcompact support. For a sequence c on Z

n , the semi-discrete convolutionproduct ϕ ∗′ c is defined by

ϕ ∗′ c =∑

ν∈Zn

ϕ(· − ν)c(ν).

8 Strang-Fix theory for approximation order

For h > 0, σh is the scaling operator defined by

σhf(x) = f(hx) (x ∈ Rn).

For a positive integer k , we denote the (unweighted) Sobolev space byLp

k(Rn) (see section 2) and define the semi-norm by

|f |Lpk(Rn) =

∑|α|=k

‖∂αf‖Lp(Rn) (f ∈ Lpk(Rn)),

where α = (α1, · · · , αn) ∈ Zn+ and ∂α = ∂α1

1 · · ·∂αnn . We say that

the collection Φ = {ϕ1, · · · , ϕN} of Cc(Rn) provides controlled Lp -approximation of order k if for each f ∈ Lp

k(Rn) there exist weightschj (h > 0, j = 1, · · · , N) such that

(i) ‖f − σ1/h

(N∑

j=1

ϕj ∗′ chj

)h−n/p‖Lp(Rn) ≤ C1h

k|f |Lpk(Rn)

and

(ii)N∑

j=1

‖chj ‖�p(Zn) ≤ C2‖f‖Lp(Rn)

hold for some C1 and C2 independent of f and h . The Fourier transformf of a function f ∈ L1(Rn) is defined by

f(ξ) =∫

Rn

e−iξ·xf(x) dx.

The collection Φ = {ϕ1, · · · , ϕN} of Cc(Rn) is said to satisfy the Strang-Fix condition of order k if there exist finitely supported sequences bj (j =

1, · · · , N) on Zn such that the function ϕ =

N∑j=1

ϕj ∗′ bj satisfies

ϕ(0) �= 0 and (∂αϕ)(2πν) = 0 (|α| < k, ν ∈ Zn \ {0}).

We note that the B-spline of degree k satisfies the Strang-Fix conditionof order k (for the definition of the B-spline, see the proof of Theorem 1below). Strang and Fix proved that, when Φ consists of only one function,Φ provides controlled L2 -approximation of order k if and only if Φ satisfiesthe Strang-Fix condition of order k ([10]). Moreover, they conjectured thatthis equivalence holds when Φ = {ϕ1, · · · , ϕN} . However, a counterexamplewas found by Jia in [6].

We say that Φ = {ϕ1, · · · , ϕN} provides local Lp -approximation of orderk if for each f ∈ Lp

k(Rn) there exist weights chj (h > 0, j = 1, · · · , N) such

that

N. Tomita 9

(iii) ‖f − σ1/h

(N∑

j=1

ϕj ∗′ chj

)‖Lp(Rn) ≤ Chk|f |Lp

k(Rn)

and(iv) ch

j (ν) = 0 whenever dist(hν, supp f) > r

hold for some C and r independent of f and h , where dist(hν, supp f) =infx∈supp f |hν − x| . Note that in the definition of local Lp -approximation,the weights ch

j depend on h . De Boor and Jia proved that Φ provideslocal Lp -approximation of order k if and only if Φ satisfies the Strang-Fixcondition of order k ([2]).

Given a weight function w on Rn , we denote the weighted Lp -space and

the weighted Sobolev space by Lp(w) and Lpk(w) (see section 2). We define

the semi-norm on Lpk(w) by

|f |Lpk(w) =

∑|α|=k

‖∂αf‖Lp(w)

and say that Φ = {ϕ1, · · · , ϕN} provides local Lp(w)-approximation oforder k if for each f ∈ Lp

k(w) there exist weights chj (h > 0, j = 1, · · · , N)

such that (iv) and

(v) ‖f − σ1/h

(N∑

j=1

ϕj ∗′ chj

)‖Lp(w) ≤ Chk|f |Lp

k(w)

hold for some C and r independent of f and h .We denote the (homogeneous) weighted Herz space and the (homoge-

neous) weighted Herz-Sobolev space by Kα,pq (w1, w2) and Kα,p

q,k (w1, w2) (seesection 2). Similarly, we define the semi-norm by

|f |Kα,pq,k (w1,w2)

=∑|α|=k

‖∂αf‖Kα,pq (w1,w2)

and say that Φ = {ϕ1, · · · , ϕN} provides local Kα,pq (w1, w2)-approximation

of order k if for each f ∈ Kα,pq,k (w1, w2) there exist weights ch

j (h > 0, j =1, · · · , N) such that (iv) and

(vi) ‖f − σ1/h

(N∑

j=1

ϕj ∗′ chj

)‖Kα,p

q (w1,w2)≤ Chk|f |Kα,p

q,k(w1,w2)

hold for some C and r independent of f and h .Using the notion of multiresolution approximation, we prove the following

extension of [2] to weighted Lp -spaces. In the following two theorems, wedenote Muckenhoupt’s Ap -class by Ap (see section 2).

Theorem 1. Suppose that Φ = {ϕ1, · · · , ϕN} is a finite collection ofCc(Rn) . Then the following are equivalent.


(i) ′ Φ satisfies the Strang-Fix condition of order k .(ii) ′ For all p ∈ [1,∞] and w ∈ Ap , Φ provides local Lp(w)-

approximation of order k .(iii) ′ For some p ∈ [1,∞] and w ∈ Ap , Φ provides local Lp(w)-

approximation of order k .

We also consider the result of [2] in weighted Herz spaces.

Theorem 2. Suppose that Φ = {ϕ1, · · · , ϕN} is a finite collection ofCc(Rn) . Let w1 ∈ Aqw1

, w2 ∈ Aqw2, α ∈ R , 0 < p < ∞ and 1 < q < ∞ ,

where w1 and w2 satisfy the following(a) 1 ≤ qw1 ≤ q and −nqw1/q < αqw1 < n(1 − qw1/q) when w1 = w2 ,(b) 1 ≤ qw1 < ∞, 1 ≤ qw2 ≤ q and 0 < αqw1 < n(1 − qw2/q) when

w1 �= w2 .Then the following statements are equivalent.

(i) ′′ Φ satisfies the Strang-Fix condition of order k .(ii) ′′ Φ provides local Kα,p

q (w1, w2)-approximation of order k .

Lastly, we point out that the Strang-Fix theory for functions having non-compact support is given by, for example, Jia and Lei ([7]).

2. Preliminaries

This section is based on [3] and [7] (see also [5]). Given an appropriatefunction ϕ on R

n , we define the multiresolution approximation {Phf}h>0

of a function f on Rn with respect to ϕ by

Phf(x) =∑

ν∈Zn

〈f, ϕh,ν〉ϕh,ν(x),

whereϕh,ν(x) = h−n/2ϕ(x/h − ν)

and〈f, ϕh,ν〉 =

∫Rn

f(x)ϕh,ν(x) dx.

Let B(0, r) be the Euclidean ball of radius r centered at the origin. TheHardy-Littlewood maximal function Mf of a locally integrable function f

on Rn is defined by

Mf(x) = supr>0

1|B(0, r)|

∫B(0,r)

|f(x − y)| dy,

where |B(0, r)| denotes the Lebesgue measure of B(0, r).

N. Tomita 11

A weight w ≥ 0 on Rn is said to belong to Ap for 1 < p < ∞ if

Ap(w) = supB

(1|B|

∫B

w(x) dx

)(1|B|

∫B

w(x)1−p′dx

)p−1

< ∞,

where the supremum is taken over all balls B in Rn and p′ is the conjugate

exponent of p (1/p + 1/p′ = 1). Ap(w) is called the Ap -constant of w .The class A1 is defined by

A1(w) = supB

(1|B|

∫B

w(x) dx

)‖w−1‖L∞(B, dx) < ∞,

where ‖w−1‖L∞(B, dx) = ess supx∈Bw(x)−1 . The class A∞ is the union ofAp, 1 ≤ p < ∞ . If w ∈ A∞ , then w > 0 a.e. and w is locally integrable([3, p. 134]). These classes were introduced by Muckenhoupt ([8]).

C∞c (Rn) consists of all infinitely differentiable functions on R

n withcompact support.

Let w be a weight function on Rn and 1 ≤ p ≤ ∞ . Then the weighted

Lp -space Lp(w) consists of all f on Rn such that

‖f‖Lp(w) =(∫

Rn

|f(x)|p w(x) dx

)1/p

< ∞,

with the usual modification when p = ∞ .We give the definition of the weighted Sobolev space Lp

k(w), where1 ≤ p ≤ ∞ , k is a non-negative integer and w is a weight function onR

n ([9]). A function f on Rn belongs to Lp

k(w) if f ∈ Lp(w) and thepartial derivatives ∂αf , taken in the sense of distributions, belong to Lp(w),whenever 0 ≤ |α| ≤ k . The norm on Lp

k(w) is given by

‖f‖Lpk(w) =

∑|α|≤k

‖∂αf‖Lp(w).

In particular, we denote Lpk(w) by Lp

k(Rn) when w ≡ 1.Let Bk = B(0, 2k) and Rk = Bk \ Bk−1 , where k ∈ Z . For a weight

function w and a measurable set E , we write w(E) =∫

Ew(y) dy . Let

α ∈ R, 0 < p, q < ∞, w1 and w2 be weight functions on Rn . The

(homogeneous) weighted Herz space Kα,pq (w1, w2) consists of all f on R

n

such that

‖f‖Kα,pq (w1,w2)

=

(∑k∈Z

w1(Bk)αp/n‖fχRk‖p

Lq(w2)

)1/p

< ∞.


Similarly, the weighted Herz-Sobolev space Kα,pq,k (w1, w2) is defined by using

Kα,pq (w1, w2) instead of Lp(w) ([9]).Lastly, we give the necessary lemmas.

Lemma 2.1 ([3, Proposition 2.7]). Let ϕ be a function on Rn which is

non-negative, radial, decreasing (as a function on (0,∞)) and integrable.Then

supα>0

|(ϕα ∗ f)(x)| ≤ ‖ϕ‖L1(Rn)Mf(x),

where ϕα(x) = α−nϕ(x/α) .

Lemma 2.2 ([3, Theorem 7.3]). If 1 < p < ∞ , then the Hardy-Littlewoodmaximal operator M is bounded on Lp(w) if and only if w ∈ Ap .

Lemma 2.3. Let 1 ≤ p < ∞ and w ∈ Ap . Then the following statementshold.

(a) ([3, p. 134]) There exists a constant C > 0 such that

w(B)w(E)

≤ C

( |B||E|)p

for all balls B and measurable sets E such that E ⊂ B .(b) ([3, Corollary 7.6]) There exist constants C and δ > 0 such that

w(E)w(B)

≤ C

( |E||B|)δ

for all balls B and measurable sets E such that E ⊂ B .

Lemma 2.4 ([9, Theorem 1.1]). Let 1 ≤ p < ∞, w ∈ Ap and k be anon-negative integer. Then C∞

c (Rn) is dense in the weighted Sobolev spaceLp

k(w) .

Lemma 2.5 ([9, Corollary 3.3]). Let w1 ∈ Aqw1, w2 ∈ Aqw2

, α ∈ R ,0 < p < ∞ , 1 < q < ∞ and k be a non-negative integer, where w1 and w2

satisfy either Theorem 2 (a) or Theorem 2 (b). Then C∞c (Rn) is dense in

the weighted Herz-Sobolev space Kα,pq,k (w1, w2) .

Lemma 2.6 ([9, p. 126]). Let w1 ∈ Aqw1, w2 ∈ Aqw2

, α ∈ R , 0 < p < ∞and 1 < q < ∞ , where w1 and w2 satisfy either Theorem 2 (a) orTheorem 2 (b). Then the Hardy-Littlewood maximal operator M is boundedon Kα,p

q (w1, w2) .

Corollary 2.3, Corollary 2.4 and Lemma 2.6 in [7] give the followinglemma.

N. Tomita 13

Lemma 2.7. Let Φ = {ϕ1, · · · , ϕN} be a finite collection in Cc(Rn) . IfΦ satisfies the Strang-Fix condition of order k , then there exist finitelysupported sequences bj (j = 1, · · · , N) on Z

n such that the function

ϕ =N∑

j=1


ϕ(0) = 1, ϕ(2πν) = 0 (ν ∈ Zn \ {0})

and

(∂αϕ)(2πν) = 0 (0 < |α| < k, ν ∈ Zn).

Lemma 2.8 ([7, Lemma 5.2]). Let k be a positive integer and Φ be afinite collection in Cc(Rn) . For each h > 0 , let Sh(Φ) denote the linearspace spanned by ϕ(·/h− ν) , where ϕ ∈ Φ and ν ∈ Z

n . Suppose that thereis a family {uh}0<h<1 of functions satisfying the following conditions

(a) uh ∈ Sh(Φ) (0 < h < 1) ,(b) lim

h→0uh(0) = 1 ,

(c) limh→0

(∂αuh)(ξ/h)/hk−1 = 0 (ξ ∈ Rn \ {0}, |α| < k) .

Then Φ satisfies the Strang-Fix condition of order k .

3. Lemmas

Lemma 3.1 (a) ′ appears as [4, Chapter 5, Proposition 3.14].

Lemma 3.1. Let k be a positive integer and ε > 0 . Suppose that ϕ is afunction on R

n such that

(a) |ϕ(x)| ≤ C

(1 + |x|)k−1+n+ε(x ∈ R

n),

(b) ϕ(0) = 1, ϕ(2πν) = 0 (ν ∈ Zn \ {0}),

(c) (∂αϕ)(2πν) = 0 (0 < |α| < k, ν ∈ Zn),

where C is independent of x ∈ Rn . Then for h > 0 we have that

(a)′∫

Rn

h−n∑

ν∈Zn

ϕ(x/h − ν)ϕ(y/h − ν) dy = 1 a.e.

(b)′∫

Rn

(y − x)α∑

ν∈Zn

ϕ(x/h − ν)ϕ(y/h − ν) dy = 0 a.e. (0 < |α| < k).


Proof. We prove only (b)’. We note that ϕ is in Ck−1(Rn) and thereexists a constant C such that

(3.1)∑

ν∈Zn

|ϕ(x − ν)ϕ(y − ν)| ≤ C

(1 + |x − y|)k−1+n+ε(x, y ∈ R

n).

Let α = (α1, · · · , αn) ∈ Zn+ be such that 0 < |α| < k . Condition (c) gives

(3.2)∫

Rn

yβϕ(y) dy = 0 (0 < |β| < k).

By (3.2) and ϕ(0) = 1, we have that∫Rn

(y − x)α∑

ν∈Zn

ϕ(x/h − ν)ϕ(y/h − ν) dy

=∑

ν∈Zn

ϕ(x/h − ν)∫

Rn

(y − x)αϕ(y/h − ν) dy

= hn∑

ν∈Zn

ϕ(x/h − ν)∫

Rn

(h(y + ν) − x)αϕ(y) dy

= hn∑

ν∈Zn

ϕ(x/h − ν)∫

Rn

(hν − x)αϕ(y) dy

= (−1)|α|h|α|+n∑

ν∈Zn

(x/h − ν)αϕ(x/h − ν).

Hence, it suffices to prove that∑ν∈Zn

(x − ν)αϕ(x − ν) = 0 a.e. x ∈ Rn.

Since xαϕ(x) is in L1(Rn), if we set A(x) =∑

k∈Zn

(x − k)αϕ(x − k), then

A(x/2π) is a 2πZn -periodic function which is integrable on [−π, π)n . We

show that A(x) = 0 a.e. To do this, it is enough to prove that all the Fouriercoefficients are zero. By (c), we see that

1(2π)n

∫[−π,π)n

A(x/2π) e−i�·x dx

=∫

[− 12 , 1

2 )n

(∑ν∈Zn

(x − ν)αϕ(x − ν)

)e−2πi�·x dx

=∫

Rn

xαϕ(x) e−2πi�·x dx = i|α|(∂αϕ)(2π ) = 0 for all ∈ Zn.

N. Tomita 15

The proof is complete. �

When we consider local L1(w)-approximation, the following lemma playsan important role.

Lemma 3.2. Let 1 ≤ p < ∞ and w ∈ Ap . If ϕ is a function on Rn

which is non-negative, radial, decreasing and integrable, then there exists aconstant C such that∫

Rn

(∫Rn

|f(x − αy)|ϕ(y) dy

)p

w(x) dx ≤ C

∫Rn

|f(x)|p w(x) dx

for all f ∈ Lp(w) and α ∈ R .

Proof. Since ϕ is radial, by a change of variable, we may assume α > 0.We first consider the case p = 1. We note that w ∈ A1 if and only ifthere exists a constant C such that Mw(x) ≤ Cw(x) a.e. ([3, p.134]). Bya change of variable, Fubini’s theorem and ϕ(y) = ϕ(−y), we have that∫

Rn

∫Rn

|f(x − αy)|ϕ(y) dy w(x) dx

=∫

Rn

∫Rn

|f(y′)|ϕ(

x − y′

α

)(dy′

αn

)w(x) dx

=∫

Rn

|f(y)|(∫

Rn

1αn

ϕ

(y − x

α

)w(x) dx

)dy

=∫

Rn

|f(y)|(ϕα ∗ w)(y) dy.

From Lemma 2.1, we see that∫Rn

|f(y)|(ϕα ∗ w)(y) dy ≤∫

Rn

|f(y)| supα>0

(ϕα ∗ w)(y) dy

≤∫

Rn

|f(y)|‖ϕ‖L1(Rn)Mw(y) dy.

Hence, since w ∈ A1 , we get that∫Rn

∫Rn

|f(x − αy)|ϕ(y) dy w(x) dx ≤ ‖ϕ‖L1(Rn)

∫Rn

|f(y)|Mw(y) dy

≤ C‖ϕ‖L1(Rn)

∫Rn

|f(y)|w(y) dy.

Since∫

Rn |f(x−αy)|ϕ(y) dy = ϕα∗|f |(x), by Lemma 2.1 and Lemma 2.2,we can prove Lemma 3.2 when p > 1. The proof is complete. �


4. Strang-Fix theory in weighted Lp -spaces

In this section, we prove Theorem 1. The boundedness of Ph on Lp(w)is given in, for example, [1]. Since the proof of the boundedness of Ph onL1(w) is omitted in [1], we give the proof.

Theorem 4.1. Let 1 ≤ p < ∞, w ∈ Ap, ε > 0 and k be a positiveinteger. Suppose that ϕ is a function on R

n such that

(a) |ϕ(x)| ≤ C

(1 + |x|)k+n+ε(x ∈ R

n),

(b) ϕ(0) = 1, ϕ(2πν) = 0 (ν ∈ Zn \ {0}),

(c) (∂αϕ)(2πν) = 0 (0 < |α| < k, ν ∈ Zn),

where C is independent of x ∈ Rn . Then there exists a constant C such

that‖Phf − f‖Lp(w) ≤ Chk

∑|α|=k

‖∂αf‖Lp(w)

for all f ∈ Lpk(w) and h > 0 .

Proof. We first show that, for h > 0, Ph is bounded on Lp(w). Letf ∈ Lp(w). From (3.1), we have that

|Phf(x)| =

∣∣∣∣∣∑

ν∈Zn

〈f, ϕh,ν〉ϕh,ν(x)

∣∣∣∣∣=

∣∣∣∣∣∫

Rn

h−nf(y)∑

ν∈Zn


∣∣∣∣∣≤ C

∫Rn

h−n|f(y)| 1(1 + h−1|x − y|)k+n+ε

dy

= C

∫Rn

1(1 + |y′|)k+n+ε

|f(x − hy′)| dy′.(4.1)

Thus, from Lemma 3.2, we can see that

‖Phf‖Lp(w) ≤ C

(∫Rn

|f(x)|p w(x) dx

)1/p

.

Hence, by the boundedness of Ph on Lp(w) and Lemma 2.4, it suffices toprove that

‖Phf − f‖Lp(w) ≤ Chk∑|α|=k

‖∂αf‖Lp(w) (f ∈ C∞c (Rn)).

N. Tomita 17

Let f ∈ C∞c (Rn). By Lemma 3.1, for almost all x ∈ R

n we have that

Phf(x) − f(x)

=∫

Rn

h−nf(y)∑

ν∈Zn


− f(x)∫

Rn

h−n∑

ν∈Zn


=∫

Rn

h−nf(y)∑

ν∈Zn


− f(x)∫

Rn

h−n∑

ν∈Zn


−∑

0<|α|<k

(∂αf)(x)α!

∫Rn

h−n(y − x)α∑

ν∈Zn


=∫

Rn

h−n

⎛⎝f(y) −

∑|α|<k

(∂αf)(x)α!

(y − x)α

⎞⎠∑

ν∈Zn

ϕ(x/h − ν)ϕ(y/h − ν) dy.

By Taylor’s theorem, (3.1) and a change of variable, we see that

|Phf(x) − f(x)|

≤∫

Rn

h−n C

(1 + h−1|y − x|)k+n+ε

×∣∣∣∣∣∣k∑|α|=k

(y − x)α

α!

∫ 1

0

(1 − t)k−1(∂αf)(x + t(y − x)) dt

∣∣∣∣∣∣ dy

≤ C∑|α|=k

∫Rn

h−n|y − x||α|

× 1(1 + h−1|y − x|)k+n+ε

∫ 1

0

|(∂αf)(x + t(y − x))| dt dy

= C∑|α|=k

∫ 1

0

∫Rn

h−n|y − x|k

× 1(1 + h−1|y − x|)k+n+ε

|(∂αf)(x + t(y − x))| dy dt

= Chk∑|α|=k

∫ 1

0

∫Rn

|y′|k 1(1 + |y′|)k+n+ε

|(∂αf)(x + thy′)| dy′ dt


≤ Chk∑|α|=k

∫ 1

0

∫Rn

1(1 + |y|)n+ε

|(∂αf)(x − thy)| dy dt.(4.2)

Thus, by Minkowski’s inequality for integrals and Lemma 3.2, we get that

‖Phf − f‖Lp(w)

≤ Chk∑|α|=k

∫ 1

0

{∫Rn

(∫Rn

1(1 + |y|)n+ε

|(∂αf)(x − thy)| dy

)p

w(x) dx

}1/p

dt

≤ Chk∑|α|=k

∫ 1

0

‖∂αf‖Lp(w) dt = Chk∑|α|=k

‖∂αf‖Lp(w)

The proof is complete. �

We are now ready to prove Theorem 1. If w > 0 a.e. , then ‖ · ‖L∞(w) =‖ · ‖L∞(Rn) . On the other hand, if w ∈ A∞ then w > 0 a.e. (see section2). Thus, since L∞

k (w) = L∞k (Rn) when w ∈ A∞ , Theorem 1 for p = ∞

was proved by de Boor and Jia ([2]). Hence, for the proof of Theorem 1, weconsider only the case 1 ≤ p < ∞ .

Proof of Theorem 1. We first prove (i)’ ⇒ (ii)’. Suppose that 1 ≤p < ∞, w ∈ Ap and Φ = {ϕ1, · · · , ϕN} ⊂ Cc(Rn) satisfies the Strang-Fix condition of order k . By Lemma 2.7, there exist finitely supported

sequences bj (j = 1, · · · , N) such that the function ϕ =N∑

j=1


ϕ(0) = 1, ϕ(2πν) = 0 (ν ∈ Zn \ {0})

and(∂αϕ)(2πν) = 0 (0 < |α| < k, ν ∈ Z

n).

Since ϕ ∈ Cc(Rn), by Theorem 4.1, we have that

(4.3) ‖Phf − f‖Lp(w) ≤ Chk|f |Lpk(w) (f ∈ Lp

k(w), h > 0),

where Phf =∑

�∈Zn

〈f, ϕh,�〉ϕh,� . Since ϕh,� =N∑

j=1

(ϕj ∗′ bj)h,� , we have that

Phf =∑�∈Zn

〈f, ϕh,�〉⎛⎝ N∑

j=1

(ϕj ∗′ bj)h,�

⎞⎠

=N∑

j=1

∑�∈Zn

〈f, ϕh,�〉(

h−n/2∑

ν∈Zn

ϕj(·/h − − ν)bj(ν)

)

N. Tomita 19

=N∑

j=1

∑�∈Zn

〈f, ϕh,�〉(

h−n/2∑

ν∈Zn

ϕj(·/h − ν)bj(ν − )

)

=N∑

j=1

∑ν∈Zn

(∑�∈Zn

h−n/2〈f, ϕh,�〉bj(ν − )

)ϕj(·/h − ν).

If chj (ν) =

∑�∈Zn

h−n/2〈f, ϕh,�〉bj(ν − ), then we can write that

(4.4) Phf = σ1/h

⎛⎝ N∑

j=1

ϕj ∗′ chj

⎞⎠ .

Hence, by (4.3) and (4.4), it suffices to prove that there exists a constantr > 0 such that ch

j (ν) = 0 whenever dist(hν, supp f) > r (that is,(iv)). Let R > 0 and M > 0 be such that suppϕ ⊂ B(0, R) andbj(ν) = 0 (j = 1, · · · , N, |ν| > M). We also suppose 0 < h < 1. Since

chj (ν) =

∑�∈Zn

h−n/2〈f, ϕh,ν−�〉bj( ) =∑

|�|≤M

h−n/2〈f, ϕh,ν−�〉bj( ),

if we take r = R + M , then chj (ν) = 0 whenever dist(hν, supp f) > r . This

shows (i)’ ⇒ (ii)’.(ii) ′ ⇒ (iii) ′ is clear. We next prove (iii) ′ ⇒ (i) ′ . The proof is similar

to one of [7, Theorem 5.1]. Let 1 ≤ p < ∞, w ∈ Ap and 0 < h < 1.For a positive integer m , Bm denotes the B-spline of degree m defined byB1 = χ[−1/2,1/2) and for m = 2, 3, · · ·

Bm(x) = Bm−1 ∗ B1(x) (x ∈ R),

where χ[−1/2,1/2) is the characteristic function of [−1/2, 1/2). Weapproximate the following tensor product

u(x) =n∏

j=1

Bk+1(xj) (x = (x1, · · · , xn) ∈ Rn).

Since u ∈ Lpk(w), we can find weights ch

j (0 < h < 1, j = 1, · · · , N) so

that conditions (iv) and (v) are satisfied. Let uh = σ1/h

(N∑

j=1

ϕj ∗′ chj

)

and gh = u − uh . Then conditions (iv) and (v) imply chj (ν) = 0 whenever

dist(hν, supp f) > r and

(4.5) ‖gh‖Lp(w) ≤ Chk (0 < h < 1).


Since supp u ⊂ [−k, k]n , if R = r +√

nk + 1, then chj (ν) = 0 (|hν| > R).

Hence, we have that

(4.6) uh =N∑

j=1

∑ν∈Zn

ϕj(·/h − ν)chj (ν) =

N∑j=1

∑|hν|≤R

ϕj(·/h − ν)chj (ν).

By (4.6) and ϕj ∈ Cc(Rn) (j = 1, · · · , N), there exists a constant R′ suchthat supp gh ⊂ B(0, R′) (0 < h < 1). Thus, by Holder’s inequality and(4.5), we see that

|∂αgh(ξ)| =∣∣∣∣∫

Rn

eix·ξxαgh(x)dx

∣∣∣∣≤∫

B(0,R′)|x||α||gh(x)|w(x)1/pw(x)−1/pdx

≤ CR′‖gh‖Lp(w)

(w1−p′

(BR′))1/p′

≤ Cαhk (0 < h < 1).

Therefore, we get that

(4.7) ‖∂αgh‖L∞(Rn) ≤ Chk (|α| < k, 0 < h < 1).

On the other hand, since

u(ξ) =n∏

j=1

{sin(ξj/2)

ξj/2

}k+1

(ξ = (ξ1, · · · , ξn) ∈ Rn),

we have that

(4.8) u(0) = 1 and limh→0

(∂αu)(ξ/h)hk−1

= 0 (ξ ∈ Rn \ {0}, |α| < k).

Hence, by (4.7) and (4.8), we get that(4.9)

limh→0

uh(0) = 1 and limh→0

(∂αuh)(ξ/h)hk−1

= 0 (ξ ∈ Rn \ {0}, |α| < k).

From (4.6) and (4.9), we see that {uh}0<h<1 satisfies the hypothesis ofLemma 2.8. Thus, by Lemma 2.8, Φ satisfies the Strang-Fix condition oforder k . The proof is complete.

N. Tomita 21

5. Strang-Fix theory in weighted Herz spaces

In this section, we prove Theorem 2. We first prove the weighted Herzspaces version of Theorem 4.1.

Theorem 5.1. Let w1 ∈ Aqw1, w2 ∈ Aqw2

, α ∈ R , 0 < p < ∞ ,1 < q < ∞ , ε > 0 and k be a positive integer, where w1 and w2 satisfyeither Theorem 2 (a) or Theorem 2 (b). Suppose that a function ϕ on R

n

satisfies the hypothesis of Theorem 4.1. Then there exists a constant C suchthat

‖Phf − f‖Kα,pq (w1,w2)

≤ Chk∑|α|=k


for all f ∈ Kα,pq,k (w1, w2) and h > 0 .

Proof. By (4.1) and Lemma 2.1, we see that

|Phf(x)| ≤ C

∫Rn

1(1 + |y|)k+n+ε

|f(x − hy)| dy ≤ CMf(x).

Hence, from Lemma 2.6, we have that Ph is bounded on Kα,pq . By the

boundedness of Ph on Kα,pq (w1, w2) and Lemma 2.5, it is enough to prove

that

(5.1) ‖Phf − f‖Kα,pq (w1,w2)

≤ Chk∑|α|=k


(f ∈ C∞c (Rn)).

Let f ∈ C∞c (Rn). From (4.2) and Lemma 2.1, we have that

|Phf(x) − f(x)| ≤ Chk∑|α|=k

∫ 1

0

∫Rn

1(1 + |y|)n+ε

|(∂αf)(x − thy)| dy dt

≤ Chk∑|α|=k

M [∂αf ](x).

Hence, by Lemma 2.6, we see that

‖Phf − f‖Kα,pq (w1,w2)

≤ Chk∑|α|=k

‖M [∂αf ]‖Kα,pq (w1,w2)

≤ Chk∑|α|=k


.

Hence, we get (5.1). The proof is complete. �

We are now ready to prove Theorem 2.


Proof of Theorem 2. Since we have proved Theorem 5.1, the proof of (i)”⇒ (ii)” is similar to one of Theorem 1 (i)’ ⇒ (ii)’. We prove only (ii)” ⇒(i)”. From the proof of Theorem 1 (iii)’ ⇒ (i)’, to prove (ii)” ⇒ (i)”, it isenough to show that ⊗n

j=1Bk+1 ∈ Kα,pq,k (w1, w2) and for each R > 0 there

exists a constant CR > 0 such that

(5.2)∫

B(0,R)

|f(y)|dy ≤ CR‖f‖Kα,pq (w1,w2)

(f ∈ Kα,pq (w1, w2)),

where ⊗nj=1Bk+1 denotes the tensor product of B-splines of degree k + 1.

Since ∂α(⊗n

j=1Bk+1

) ∈ L∞c (Rn) (|α| ≤ k), to prove ⊗n

j=1Bk+1 ∈Kα,p

q,k (w1, w2), it is enough to show L∞c (Rn) ⊂ Kα,p

q (w1, w2), where L∞c (Rn)

consists of all L∞(Rn)-functions with compact support.We first prove L∞

c (Rn) ⊂ Kα,pq (w1, w2). Let f ∈ L∞

c (Rn). If we takek0 ∈ Z such that supp f ⊂ Bk0 = B(0, 2k0), then


=

(k0∑

k=−∞w1(Bk)αp/n‖fχRk

‖pLq(w2)

)1/p

≤ ‖f‖L∞(Rn)

(k0∑

k=−∞w1(Bk)αp/nw2(Bk)p/q

)1/p

= Ck0,w1,w2‖f‖L∞(Rn)

(k0∑

k=−∞

[w1(Bk)w1(Bk0 )

]αp/n [w2(Bk)w2(Bk0 )

]p/q)1/p

.

Since α/n+1/q > 0 when w1 = w2 or α > 0 when w1 �= w2 , from Lemma2.3 (b), we have that

[w1(Bk)w1(Bk0)

]αp/n [w2(Bk)w2(Bk0)

]p/q

≤ Cw1,w2

[ |Bk||Bk0 |

]αpδw1/n [ |Bk||Bk0 |

]pδw2/q

= Cn,w1,w22np(k−k0)(αδw1/n+δw2/q).

Thus, since αδw1/n + δw2/q > 0, we get that


≤ Ck0,w1,w2‖f‖L∞(Rn)

(k0∑

k=−∞2np(k−k0)(αδw1/n+δw2/q)

)1/p

< ∞.

N. Tomita 23

We next prove (5.2). We take k0 ∈ Z such that B(0, R) ⊂ Bk0 =B(0, 2k0). By w ∈ Aq and the definition of Ap -weights, we see that

∫B(0,R)

|f(y)| dy ≤∫

Bk0

|f(y)| dy =k0∑

k=−∞

∫Bk\Bk−1

|f(y)| dy

≤k0∑

k=−∞‖fχRk

‖Lq(w2)

(w1−q′

2 (Bk))1/q′

≤ Aq(w2)1/qk0∑

k=−∞‖fχRk

‖Lq(w2)|Bk|w2(Bk)−1/q

= Ck0,w1,w2

k0∑k=−∞

w1(Bk)α/n‖fχRk‖Lq(w2)

× |Bk|[w1(Bk0)w1(Bk)

]α/n [w2(Bk0)w2(Bk)

]1/q

.

Since α/n+1/q > 0 when w1 = w2 or α > 0 when w1 �= w2 , from Lemma2.3 (a), we have that

[w1(Bk0)w1(Bk)

]α/n [w2(Bk0 )w2(Bk)

]1/q

≤ Cw1,w2

[ |Bk0 ||Bk|

]αqw1/n [ |Bk0 ||Bk|

]qw2/q

= Cn,w1,w22n(k0−k)(αqw1/n+qw2/q).

Thus, we see that

∫B(0,R)

|f(y)|dy≤C

k0∑k=−∞

w1(Bk)α/n‖fχRk‖Lq(w2)2

n(k0−k)(αqw1/n+qw2/q−1),

where C depends only on R , w1 and w2 . Since αqw1/n + qw2/q − 1 < 0,we get (5.2). The proof is complete.

Acknowledgment

The author gratefully acknowledges helpful discussions with Prof. M. Na-gase, Prof. E. Nakai, Prof. T. Nishitani, Prof. K. Saka, Prof. D.F. Walnutand Prof. K. Yabuta. Moreover, he would like to express his deep gratitudeto the referee for many important comments. Nakai provided the simpleproof for Lemma 3.2 when 1 < p < ∞ . Then Yabuta proved the casep = 1.


References

[1] H. Aimar, A. Bernardis and F. J. Martın-Reyes, Multiresolutionapproximations and wavelet bases of weighted Lp -spaces, J. FourierAnal. and Appl., 9 (2003), 497–510.

[2] C. de Boor and R. Q. Jia, Controlled approximation and acharacterization of the local approximation order, Proc. Amer. Math.Soc., 95 (1985), 547–553.

[3] J. Duoandikoetxea, Fourier Analysis, Amer. Math. Soc. GraduateStudies in Mathematics, 29, American Mathematical Providence, RI,2001.

[4] E. Hernandez and G. Weiss, A First Course on Wavelets, CRC Press,1996.

[5] J. Garcıa-Cuerva and J. L. Rubio de Francia, Weighted NormInequalities and Related Topics, North-Holland, 1985.

[6] R. Q. Jia, A counterexample to a result concerning controlledapproximation, Proc. Amer. Math. Soc., 97 (1986), 647–654.

[7] R. Q. Jia and J. Lei, Approximation by multi-integer translates offunctions having global support, J. Approx. Theory, 72 (1993), 2–23.

[8] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximalfunction, Trans. Amer. Math. Soc., 165 (1972), 207–226.

[9] E. Nakai, N. Tomita and K. Yabuta, Density of the set of allinfinitely differentiable functions with compact support in weightedSobolev spaces, Sci. Math. Jpn., 60 (2004), 121–127.

[10] G. Strang and G. Fix, A Fourier analysis of the finite-elementvariational method, In : Constructive Aspects of Functional Analysis,(1973), 793–840.

Department of MathematicsOsaka UniversityMachikaneyama 1-16Toyonaka, Osaka 560-0043Japan(E-mail : [email protected])

(Received : September 2004 )

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