2 A REMARK ON MULTIRESOLUTION ANALYSIS OF Lp(Rd) BY QIYU S U N (HANGZHOU) A condition on a scaling function which generates a multiresolution anal- ysis of Lp(Rd) is given

VOL. LXVI 1993 FASC. 2

A REMARK ON MULTIRESOLUTION ANALYSIS OF L^p(R^d)

BY

QIYU S U N (HANGZHOU)

A condition on a scaling function which generates a multiresolution anal- ysis of L^p(R^d) is given.

1. Introduction and results. A family of closed subspaces {Vj}_j∈Z of L^p(R^d) is called a multiresolution analysis of L^p(R^d) if

(i) Vj ⊂ V_j+1 and f (x) ∈ Vj if and only if f (2^−jx) ∈ V0; (ii)T

j∈ZVj = {0} andS

j∈ZVj = L^p(R^d);

(iii) there exists a scaling function φ with l^pstable integer translates such that

V0=n X

k∈Z^d

C(k)φ(· − k) : X

k∈Z^d

|C(k)|^p < ∞ o

.

We say that a function φ has l^p stable integer translates if there exist 0 < A ≤ B < ∞ such that

(1) A X

k∈Z^d

|C(k)|^p1/p

≤

X

k∈Z^d

C(k)φ(· − k) p

≤ B X

k∈Z^d

|C(k)|^p1/p

for every sequence {C(k)} ∈ l^p. Hereafter we assume 1 < p < ∞ and write L^p = L^p(R^d) = {f : kf kp = (R

R^d|f (x)|^pdx)^1/p < ∞} and l^p = l^p(Z^d) = {{C(k)} :P

k∈Z^d|C(k)|^p < ∞}. For simplicity we use P without index to replace the sum over Z^d. We say that a function φ generates a multiresolution analysis of L^p if φ has l^p stable integer translates and the family of closed subspaces {Vj}_j∈Z defined by

Vj =n X

C(k)φ(2^j · −k) :X

|C(k)|^p< ∞o (2)

= L^p closure of n X

C(k)φ(2^j· −k) : {C(k)} has finite lengtho is a multiresolution analysis of L^p. Hereafter we say that a sequence {C(k)}

has finite length if C(k) 6= 0 except for finitely many k.

1991 Mathematics Subject Classification: Primary 42C15.

The multiresolution analysis of L² was introduced by Mallat ([3]) and Meyer ([4]), and is well examined since we can use the Fourier transform ([2]). It becomes an important and almost unique scheme for construction of orthonormal bases of wavelets of L² which are unconditional bases of L^p (1 < p < ∞) under some conditions. The multiresolution analysis of L^p (p 6= 2) is still meaningful since the construction of bases of wavelets from a multiresolution analysis of L² is still a difficult problem in general, and a function φ with some decay at infinity which generates a multiresolution analysis of L²generates one of L^p also. Define

L^p_∗=n f : R

[0,1]^d

 X|f (x + k)|p

dx < ∞o .

Jia and Micchelli ([1]) proved that φ generates a multiresolution analysis of L^p if φ ∈ L^p∗ has l^p stable integer translates and satisfies the refinement equation

(3) φ(x) =X

a(k)φ(2x − k)

with the mask {a(k)} ∈ l¹. Let φ be a distribution having a continuous Fourier transform. We say that the integer translates of φ are globally lin- early independent for tempered sequences if bφ(ξ + 2kπ) is not identically zero on Z^d for every ξ ∈ R^d (cf. [6]).

In this paper we will use Fourier analysis to prove

Theorem 1. Suppose the integer translates of φ are l^pstable and globally linearly independent for tempered sequences. If φ satisfies the refinement equation (3) withP a(k) = 2 and P |a(k)|²(1 + |k|)^2l< ∞ for some integer l > d/2, then φ generates a multiresolution analysis of L^p.

In particular, if φ satisfies the refinement equation (3) with {a(k)} ∈ l¹, then the spaces Vj defined by (2) satisfy

(i) Vj ⊂ V_j+1 and f (x) ∈ Vj if and only if f (2^−jx) ∈ V0, sinceP C(k)φ(x − k) = P(P C(l)a(k − 2l))φ(2x − k) and

X  

XC(l)a(k − 2l)   

p

≤ X

|a(k)|pX

|C(l)|^p.

Let 1 < q < ∞. We say that a measurable function m is a local L^q multiplier if for every compact set K there exists a constant CKindependent of f such that

k(m bf )^∨k_q ≤ C_Kkf k_q

for every f ∈ L^q with supp bf ⊂ K, where bf and f^∨ denote the Fourier transform and inverse Fourier transform respectively.

Observe that P |a(k)|²(1 + |k|)^2l < ∞ for some l > d/2 implies {a(k)}

∈ l¹. Therefore the matter reduces to

Theorem 2. Suppose the integer translates of φ are l^p stable and glob- ally linearly independent for tempered sequences. Assume bφ is a continuous local L^q multiplier for some ∞ > q > max(p, p/(p − 1)). If φ satisfies the refinement equation (3) with {a(k)} ∈ l¹ and bφ(0) 6= 0, then T

j∈ZVj = {0}

and S

j∈ZVj = L^p.

Theorem 3. If φ satisfies the refinement equation (3) with P a(k) = 2 andP |a(k)|²(1+|k|)^2l< ∞ for some integer l > d/2, then bφ is a continuous local L^q multiplier for 1 < q < ∞.

Therefore conditions (i) and (iii) imply (ii) provided φ ∈ L¹and the inte- ger translates of φ are globally linearly independent for tempered sequences, since k( bφ bf )^∨kq = kR φ(· − y)f (y) dykq ≤ kφk1kf kq for 1 < q < ∞. Observe that L¹⊃ L^p_∗ for any 1 ≤ p ≤ ∞ and the integer translates of φ ∈ L^p∗ which has l^p stable integer translates must be globally linearly independent for tempered sequences (see also Section 3 below). Hence Theorem 2 improves the result of Jia and Micchelli ([1]). In particular, Theorem 1 is new even when p = 2.

The author would like to thank the referee for his (her) useful sugges- tions.

2. Proofs. The proof of Theorem 2 depends on the following two technical lemmas.

Lemma 1. Let m be a continuous local L^q multiplier for some q > 2.

Then for every x0 such that m(x0) 6= 0 there exist a compact set K and a constant C independent of f such that x0 is an inner point of K and k(m⁻¹f )b ^∨k_p ≤ Ckf k_p for every f ∈ L^p with supp bf ⊂ K where q/(q − 1) <

p < q.

P r o o f. Without loss of generality we assume x0= 0. Observe that k(m bef )^∨k₂≤ C sup

B(r)

|m(x)| kf ke 2

for every f ∈ L² with supp bf ⊂ B(r) = {x : |x| ≤ r}, where m(x) =e m(x) − m(0) and r > 0. Recall that

k(m bef )^∨k_p≤ k(m bf )^∨k_p+ |m(0)| kf kp≤ Ckf k_p for every f ∈ L^q with supp bf ⊂ B(r). Therefore we get

k(m bef )^∨k_p≤ C(sup

B(r)

|m(x)|)e ^θkf k_p

for every f ∈ L^pwith supp bf ⊂ B(r) by the Marcinkiewicz real interpolation between 2 and q or q/(q − 1), where q/(q − 1) < p < q and θ = θ(p, q) > 0

([5], p. 21). Furthermore,

k(m bef )^∨k_p≤ 1

2|m(0)| kf k_p

holds for every f ∈ L^p with supp bf ⊂ B(r0) when r0 > 0 is chosen small enough by the continuity of m(x). Observe that

m⁻¹(x) = m(0)⁻¹

 1 +

∞

X

k=1

 m(0) − m(x) m(0)

k . Therefore

k(m⁻¹f )b ^∨kp≤ |m(0)|⁻¹



kf kp+

∞

X

k=1

 m(x) − m(0) m(0)

k

fb

∨ p



≤ 2|m(0)|⁻¹kf kp

for every f ∈ L^p with supp bf ⊂ B(r0) and Lemma 1 is proved.

Lemma 2. Let Vj defined by (2) for some φ ∈ L^p satisfy Vj ⊂ V_j+1 and ψ be any Schwartz function. Then for every f ∈ V0there exists gj ∈ Vj such that kψ ∗ f − gjk_p→ 0 as j → ∞.

P r o o f. Let gj(x) = 2^−jdP

kψ(2^−jk)f (x − 2^−jk) ∈ Vj. Then kg_j − ψ ∗ f k_p≤ X

k

R

[0,2^−j]^d

|ψ(y + 2^−jk) − ψ(2^−jk)| dy kf kp

+X

k

2^−jd|ψ(2^−jk)|ωp(f, 2^−j)

≤ C2^−jkf kp+ Cωp(f, 2^−j) → 0 as j → ∞, where ωp(f, t) = sup_|y|≤tkf (· − y) − f (·)k_p, and Lemma 2 is proved.

Now we start to prove Theorem 2. First, Y = T

j∈ZVj = {0}. Let K0

be a compact set such that for every ξ ∈ R^d there exists η ∈ K0 such that φ(η) 6= 0 and (ξ − η)/(2π) ∈ Zb ^dsince the integer translates of φ are globally linearly independent for tempered sequences. Then for every ξ06∈ K0 there exists a Schwartz function ψ such that supp bψ ∩ K0 = ∅ and bψ(ξ0) = 1.

Let f be any function in Y . Therefore ψ ∗ f ∈ Y ⊂ V0 by Lemma 2 and ψ(ξ) bb f (ξ) = τ (ξ) bφ(ξ), where τ (ξ) =P C(k)e^ikξ is a 2π-periodic distribution and {C(k)} ∈ l^p. Let η0 be some point in K0 such that bφ(η0) 6= 0 and (ξ0 − η₀)/(2π) ∈ Z^d. Therefore τ (ξ) = 0 on some neighborhood of η0

and furthermore on some neighborhood of η0+ 2πZ^d since bψ bf = 0 in some neighborhood of η0, bφ(η0) 6= 0 and τ is 2π-periodic. Hence bf (ξ) = 0 on some neighborhood of ξ0 and supp bf ⊂ K0 for every f ∈ Y . Observe that f ∈ Y if and only if f (2^j·) ∈ Y for j ∈ Z and any function f with supp bf = {0}

is a nonzero polynomial. Recall that K0 is bounded. Therefore supp bf = ∅, f = 0 andT

j∈ZVj = {0}.

Second,S

j∈ZVj = L^p(R^d). Define

X⁰= {f ∈ L^p: supp bf is compact in R^d} and X = S

j∈ZVj. Therefore the matter reduces to X⁰ ⊂ X since X⁰ is dense in L^p and X is a closed subspace of L^p. Let f be any function in X⁰. Recall that bφ(0) 6= 0. Therefore there exists a positive integer j1 such that |2^−j1y| ≤ π/2 and | bφ(2^−j1y)| ≥ ¹₂| bφ(0)| for any y ∈ supp bf . Since bφ is a local L^q multiplier for some q > max(p, p/(p − 1)), by Lemma 1 we get τ (ξ)^∨= ( bφ(2^−j1ξ)⁻¹f (ξ))b ^∨ ∈ L^p, where we set τ (ξ) = bφ(2^−j1ξ)⁻¹f (ξ).b Furthermore,P |τ^∨(2^−j1k)|^p < ∞ by the Shannon sampling theorem which says that the L^pnorm of a function f whose Fourier transform is supported in [−π/2, π/2]^d is equivalent to the l^p norm of the sampling values of f at the integer lattice points. Let

g(x) = 2^j1dX

k

τ^∨(2^−j1k)φ(2^j1x − k) ∈ Vj1 ⊂ X

and ψ be some Schwartz function such that bψ = 1 on supp bf and supp bψ ⊂ {|y| ≤ 2^j1−1π}. Then

(ψ ∗ g)(ξ) = bd ψ(ξ)bg(ξ) = bψ(ξ) X

k

τ^∨(2^−j1k)e^i2−j1ξk

φ(2b ^−j1ξ) = bf (ξ) , f = ψ ∗ g ∈ X by Lemma 2 and Theorem 2 holds true.

The proof of Theorem 3 depends on the following lemma.

Lemma 3. If D^αm ∈ L²_loc for all |α| ≤ l and some integer l > d/2, then m is a continuous local L^q multiplier for all 1 < q < ∞, where L^q_loc = {f : R

K|f (x)|^q < ∞ for every compact set K}, and α = (α1, . . . , αd) and

|α| =Pd i=1|α_i|.

The proof of Lemma 3 follows from the Marcinkiewicz multiplier theorem ([5], p. 96).

Now we start to prove Theorem 3. Let φ satisfy the refinement equation (3) with the mask {a(k)}. Define H(ξ) = ¹₂P

ka(k)e^ikξ. Observe that φ(ξ) =b Q∞

j=1H(ξ/2^j) bφ(0) and D^αφ(ξ) =b X

α1+...+αs=α αi∈Z^d,αi6=0

Cα,α1,...,αs

X

j1,...,js

s

Y

m=1

D^αmH

 ξ 2^jm



× 2^−jm|αm| Y

j6=j1,...,js

H ξ 2^j

 .

Therefore the matter reduces to proving P

j1,...,js

Qs

m=1|D^αmH(ξ/2^jm)|

× 2^−jm|αm| ∈ L²_loc, or there exist C and ε > 0 independent of j1, . . . , js for every R ≥ 1 and every α with αm6= 0 and |α| =Ps

m=1|α_m| ≤ l₀= [d/2]+1, such that

R

|x|≤R

 ^s Y

m=1

D^αmH

 ξ 2^jm

   

2^−jm|αm|

2

dx ≤ C2^{−ε(j1+...+js)},

where [x] denotes the integer part of x. Recall thatP

k|a(k)|²(1 + |k|)^2l<

∞. Therefore D^αH ∈ L²_loc for every |α| ≤ l0. By the Sobolev imbedding theorem ([5], p. 124), D^αH ∈ L^p_loc^α for every pα such that 1/pα> 1/2 − (l0−

|α|)/d. Since

s

X

m=1

 1

2 −l − |αm| d



< 1 2, there exist pαm such that D^αmH ∈ L^p_loc^αm, 1/r = Ps

m=11/pαm ≤ 1/2 and d/pαm < |αm|. Therefore

R

|x|≤R

 ^s Y

m=1

D^αmH

 x 2^jm

   

2^−jm|αm|

2

dx

≤ C



R

|x|≤R

 s

Y

m=1

D^αmH

 x 2^jm

   

2^−jm|αm|

r

dx

2/r

≤ C

s

Y

m=1



R

|x|≤R

D^αmH

 x 2^jm

   

p_αm

dx

2/p_αm

2^−2jm|αm|

≤ C

s

Y

m=1

2^{jm(−2|αm|+2d/pαm)}≤ C2^{−ε(j1+...+js)}, where ε is chosen as min(2|αm| − 2d/p_αm) and Theorem 3 is proved.

3. Remarks. If φ has compact support, then (4)

X

k

C(k)φ(· − k) p

≤ B X

k

|C(k)|^p1/p

holds if and only if φ ∈ L^p. Jia and Micchelli ([1]) proved that φ ∈ L^p∗, i.e.

R

[0,1]^d

 X

k

|φ(x + k)|p

dx < ∞ ,

is a sufficient condition for (4) to hold. Obviously φ ∈ L^p is a necessary

condition. By the inequalities for Rademacher functions ([5], pp. 104, 276), we know that (4) implies

R

R^d

 X

k

|C(k)|²|φ(x + k)|²p/2

dx ≤ CX

k

|C(k)|^p. Furthermore, we have

R

[0,1]^d

 X

k

|φ(x + k)|²p/2

dx

≤ C lim

k→∞2^−kd X

|s|≤2^k

R

[0,1]^d

 X

|j|≤2^k+1

|φ(x + j − s)|²p/2

dx

≤ C lim

k→∞2^−kd R

R^d

 X

|j|≤2^k+1

|φ(x + j)|²p/2

dx < ∞ . Therefore

R

[0,1]^d

 X

k

|φ(x + k)|²p/2

dx < ∞ ,

which is stronger than φ ∈ L^p when p > 2, is a necessary condition for (4) to hold.

For some functions {φs}^N_s=1, define Vj =

nX^N

s=1

X

k

Cs(k)φs(x − k) :

N

X

s=1

X

k

|C_s(k)|^p< ∞ o

.

Then the corresponding result of Theorem 2 holds (see [1], [6] for the defi- nition of l^p stable integer translates of {φs}^N_s=1).

REFERENCES

[1] R.-Q. J i a and C. A. M i c c h e l l i, Using the refinement equations for the construction of prewavelet II: power of two, in: Curves and Surfaces, P. J. Laurent, A. Le Mehaute and L. L. Schumaker (eds.), Academic Press, 1990, 1–36.

[2] W. R. M a d y c h, Some elementary properties of multiresolution analysis of L²(Rⁿ), in: Wavelets—A Tutorial in Theory and Applications, C. K. Chui (ed.), Academic Press, 1992, 259–294.

[3] S. M a l l a t, Multiresolution approximation and wavelet orthonormal bases of L²(Rⁿ), Trans. Amer. Math. Soc. 315 (1989), 69–88.

[4] Y. M e y e r, Ondelettes, fonctions spline et analyses gradu´ees, Rapport CEREMADE 8703, 1987.

[5] E. M. S t e i n, Singular Integrals and Differentiability Properties of Functions, Prince- ton Univ. Press, Princeton, N.J., 1970.

[6] Q. S u n, Sequences spaces and stability of integer translates, Z. Anal. Anwendungen 12 (1993), 567–584.

CENTER FOR MATHEMATICAL SCIENCES DEPARTMENT OF MATHEMATICS

ZHEJIANG UNIVERSITY HANGZHOU UNIVERSITY

HANGZHOU, ZHEJIANG 310027 HANGZHOU, ZHEJIANG 310028

P.R. CHINA P.R. CHINA

Re¸cu par la R´edaction le 20.10.1992;

en version modifi´ee le 27.1.1993 et 29.3.1993