Hyperbolic Wavelet Approximation - Texas A&M …rdevore/publications/85.pdfAbstract. We study the...

Constr. Approx. (1998) 14: 1–26CONSTRUCTIVEAPPROXIMATION© 1998 Springer-Verlag New York Inc.

Hyperbolic Wavelet Approximation

R. A. DeVore, S. V. Konyagin, and V. N. Temlyakov

Abstract. We study the multivariate approximation by certain partial sums (hyper-bolic wavelet sums) of wavelet bases formed by tensor products of univariate wavelets.We characterize spaces of functions which have a prescribed approximation error byhyperbolic wavelet sums in terms of aK -functional and interpolation spaces. The resultsparallel those for hyperbolic trigonometric cross approximation of periodic functions[DPT].

1. Introduction

Let ϕ be a univariate function that satisfies multiresolution analysis (see, e.g., [Da] fora description of multiresolution analysis). We denote byS := S(ϕ) the shift-invariantspace which is defined as theL2(R)-closure of finite linear combinations of the shiftsϕ(· − j ), j ∈ Z, of ϕ. By dilation, we obtain the univariate spaces

Sk := Sk(ϕ) := {S(2k·) : S∈ S}, k ∈ Z.

We obtain univariate waveletsψ by considering projectorsPk from L2(R) ontoSk. Thewavelet spaceWk−1 is defined to be the image ofQk := Pk − Pk−1. Wavelet functionsψ are generators of the shift-invariant spaceW := W0, i.e., W = S(ψ). We have inmind here the usual orthogonal wavelets in the case thePk are orthogonal projectorsand the biorthogonal wavelets (see [CDF]) obtained when considering certain obliqueprojectorsPk.

From the univariate waveletψ , we can construct efficient bases forL2(R) and otherfunction spaces by dilation and shifts. For example, the functions

ψj,k := 2k/2ψ(2k · − j ), j, k ∈ Z,

form a stable basis (orthogonal basis in the case of an orthogonal waveletψ) for L2(R).It is convenient to use a different indexing for the functionsψj,k. LetD(R) denote the

set of dyadic intervals. Each such intervalI is of the formI = [ j 2−k, ( j + 1)2−k]. Wedefine

ψI := ψj,k, I = [ j 2−k, ( j + 1)2−k].(1.1)

Thus the basis{ψj,k}j,k∈Z is the same as{ψI }I∈D(R).

Date received: October 16, 1995. Date revised: August 28, 1996. Communicated by Carl de Boor.AMS classification: 41A63, 46C99.Key words and phrases: Hyperbolic wavelets, Multivariate wavelets, Interpolation spaces.

1

2 R. A. DeVore, S. V. Konyagin, and V. N. Temlyakov

Multivariate wavelets are usually obtained from multiresolution analysis on the tensorproduct spacesSk ⊗ · · · ⊗ Sk. For example, in the case of bivariate approximation, thisleads to the bivariate wavelet basis consisting of all functions

ϕj,k(x)ψj ′,k(y) ψj,k(x)ϕj ′,k(y) ψj,k(x)ψj ′,k(y),(1.2)

with j, j ′, k ∈ Z. The approximation properties of these wavelets are now well under-stood (see, e.g., [M] or for nonlinear approximation [DJP]).

A second way to construct multivariate wavelet bases is to simply take tensor productsof the univariate basis functionsψj,k. If ψ is a univariate wavelet andd ≥ 1, then thefunctions

ψj1,k1(x1) · · ·ψjd,kd(xd), j = ( j1, . . . , jd) ∈ Zd, k = (k1, . . . , kd) ∈ Zd,(1.3)

are a basis forL2(Rd).There is quite a distinction between these two wavelet bases. The functions in (1.2)

have roughly the same support in each coordinate direction while the tensor products of(1.3) have support which is scaled independently in the different coordinate directions.As we shall see, this is also reflected in the approximation properties of the two sets ofwavelets.

Again, it is more convenient to use another indexing for the basis functions (1.3). WeletD(Rd) denote the set of all dyadic rectangles inRd. Any I ∈ D(Rd) is of the formI = I1× · · · × Id with I1, . . . , Id ∈ D(R). We define

ψI (x1, . . . , xd) := ψI1(x1) · · ·ψId(xd), I ∈ D(Rd).(1.4)

Therefore, the wavelet basis (1.3) is the same as the set of functions{ψI }I∈D(Rd).We are interested in the approximation properties of the functions (1.4). Forn =

0, 1, . . . and 0< p ≤ ∞, let

Hn := Hn(L p(Rd)) := span{ψI : |I | > 2−n}denote the closed linear span of the finite linear combinations of the functionsψI ,|I | > 2−n, with the closure taken with respect to theL p(Rd)-(quasi-)norm.

We call the approximation byHn hyperbolic wavelet approximation in analogy withthe approximation by trigonometric polynomials with frequencies from the hyperboliccross (see Temlyakov [T] for a discussion of this type of approximation).

We can also describeHn in terms of the scaling functionϕ. Namely,Hn is the closedlinear span of the functions

ϕI , |I | ≥ 2−n.

In most wavelet applications, approximation occurs over a compact subsetÄ of Rd andthe approximation takes place from a finite-dimensional linear subspaceHn ofHn.

The present paper is concerned with the approximation efficiency of the spacesHn. Tomeasure this efficiency, we introduce the following approximation error. For 0< p ≤ ∞and f ∈ L p(Rd), we define

En( f )p := E( f,Hn)p := infg∈Hn

‖ f − g‖p

with ‖ · ‖p here and later theL p(Rd)-norm.

Hyperbolic Wavelet Approximation 3

We are interested in characterizing functions which have a given order of approxi-mation. Forα > 0 and 0< p,q ≤ ∞, we defineAαq(L p(Rd)) as the collection of allfunctions f ∈ L p(Rd) such that

| f |Aαq (L p(Rd)) :=

{(∑k≥0[2kαEk( f )p]q

)1/q, 0< q <∞,

supk≥0 2kαEk( f )p, q = ∞,(1.5)

is finite. We define the norm on the approximation spaceAαq(L p(Rd)) by

‖ f ‖Aαq (L p(Rd)) := ‖ f ‖p + | f |Aα

q (L p(Rd)).

The characterization of the approximation classesAαq(L p(Rd)) is tantamount to prov-ing two inequalities (called Jackson and Bernstein inequalities) for the approximationprocess (see [DL, Chap. 7]). In the case of hyperbolic wavelet approximation, theseinequalities involve mixed derivatives.

If r is a positive integer, we define the differential operator

Dr := ∂r

∂xr1

· · · ∂r

∂xrd

.

For 1 ≤ p ≤ ∞, we letWr (L p(Rd)) be the set of all functionsf in L p(Rd) whosedistributional derivativeDr f is in L p(Rd) and define the seminorm onWr (L p(Rd)) by

| f |Wr (L p(Rd)) := ‖Dr f ‖p.

We will show in Section 3, that under certain conditions on the functionψ and for1< p <∞, we have the following two inequalities:

En( f )p ≤ C2−nr | f |Wr (L p(Rd)), n = 0, 1, . . . , f ∈Wr (L p(Rd)),(J)

with C independent ofn and f and

|g|Wr (L p(Rd)) ≤ C2nr‖g‖p, g ∈ Hn, n = 0, 1, . . . ,(B)

with C independent ofn andg.From these Jackson and Bernstein inequalities for hyperbolic wavelet approximation

it follows that we can characterize the spacesAαq(L p(Rd)) in terms of theK -functional

K ( f, t) := K ( f, t, Lp(Rd),Wr (L p(Rd))).(1.6)

Namely, for 0< q <∞, a function f ∈ L p(Rd) is inAαq(L p(Rd)) if and only if∫ ∞0

[t−αK ( f, tr )]q dt

t< ∞.(1.7)

A similar result holds forq = ∞ with the integral replaced by a sup. In the caseof periodic functions, we have shown in [DPT] thatK ( f, t) is equivalent to a certainmodulus of smoothnessÄr ( f, t)p based on mixed differences. In principle, this resultshould carry over to the case of approximation onRd; however, we have not yet carriedout the details of this equivalence.


Our proof of the Jackson and Bernstein inequalities for hyperbolic wavelet approxi-mation rests on the characterization of functionsf in L p(Rd) by a hyperbolic waveletseries

f =∑

I∈D(Rd)

cI ( f )ψI

and the calculation of its norm by the followingsquare function∥∥∥∥∥∑I∈D

cIψI

∥∥∥∥∥p

≈∥∥∥∥∥(∑

I∈D[cIψI ]

2

)1/2∥∥∥∥∥p

.(1.8)

Characterizations (1.8) are at the heart of what is called the Littlewood–Paley theory forwavelets.

Littlewood–Paley theory has a long and important history in harmonic analysis. For themost part, we will utilize known aspects of this theory adapted to the case of hyperbolicwavelet decompositions. We describe the results from Littlewood–Paley theory that wewill need and give their adaptation to hyperbolic wavelets in Section 2. There are sev-eral results which establish sufficient conditions for the family of functions{ψI }I∈D(Rd)

to satisfy (1.8). However, these conditions are not always applicable in wavelet theorysince they require smoothness ofψ not met by wavelets or their derivatives. We there-fore establish (in Section 4) sufficient conditions which allow only piecewise Lipschitzcontinuity on the functionψ for the Littlewood–Paley characterization to hold.

Finally, in Section 5 we give an application of our results to the Daubechies wavelets.Other wavelets can be handled in a similar manner. As mentioned earlier, we will restrictour development in this paper to approximation onRd. We could also give a similardevelopment for the case of approximation on a compact set inRd or on the torusTd. Inthis way, our approach could be applied to other wavelet-like bases such as the Franklinsystem and its generalizations. While preparing the present paper, we were sent a preprintby A. Kamont [K] that proves Jackson and Bernstein inequalities for the Franklin systemby utilizing Littlewood–Paley theory.

2. The Elements of Littlewood–Paley Theory

Littlewood–Paley theory gives a way of characterizing norms of linear combinations ofcertain basis functions. Its roots lie in the Littlewood–Paley theorems for Fourier seriesin which case the basis functions are the complex exponentials

ek(x) := eik·x = ei (k1x1+···+kdxd), x ∈ Rd.

However, the theory applies to many other orthogonal and nonorthogonal expansions(see, e.g., [FJ], [FJM], or [M]).

For us, Littlewood–Paley theory will provide a vehicle to prove our results on mul-tivariate wavelet approximation. We begin in this section by introducing various formsof the Littlewood–Paley theory for systems of functions. While the discussion we giveis for the most part known, it will enable us to set the framework for this paper andintroduce several important results which will be employed later in the paper.


LetD = D(Rd) denote the collection of all dyadic rectangles inRd. Thus, a rectangleI ⊂ Rd is inD if and only if I = I1 × · · · × Id with I1, . . . , Id dyadic intervals inR.We will consider in this section systems of functions{η(I , ·)}I∈D.

In the univariate case, one particular way to obtain such systems is by shifts and dilatesof a univariate wavelet. For a univariate functionψ , we use the notationψI of (1.1) todenote itsL2(R)-normalized, shifted dilates. The functionψ is an orthogonal waveletif the collection of functionsψI , I ∈ D(R), forms a complete orthonormal systemfor L2(R). Other cases of interest in wavelet theory are the prewavelets [BDR], splinewavelets [CW], and biorthogonal wavelets [CDF]. In the latter cases, the orthogonalityof the familyψI , I ∈ D(R), is replaced byL2-stability (Riesz basis).

Given a univariate functionψ , we can obtain a multivariate family of functions bytaking tensor products. For rectanglesI ∈ D(Rd), we define

ψI (x1, . . . , xd) := ψI1(x1) · · ·ψId(xd), I = I1× · · · × Id.(2.1)

We will use the notationψI to denote the family of functions obtained by tensor productsof shifted dilates of a univariate functionψ and will use the notationη(I , ·) to denotefamilies of functions indexed onI ∈ D(Rd) that are not necessarilly obtained by shifteddilates of one function. This notation will distinguish between the space dimension bythe Euclidean dimension of the index rectangles.

A particularly important example occurs when we take forψ the univariate Haarwavelet

H(x) :={+1, 0≤ x ≤ 1

2,

−1, 12 < x ≤ 1.

The Haar functionH is the simplest example of a univariate orthogonal wavelet.If 1 < p <∞, we say that a family of real-valued functionsη(I , ·), I ∈ D, satisfies the

strong Littlewood–Paley propertyfor p, if for any finite sequence(cI ) of real numbers,we have ∥∥∥∥∥∑

I∈DcI η(I , ·)

∥∥∥∥∥p

≈∥∥∥∥∥(∑

I∈D[cI η(I , ·)]2

)1/2∥∥∥∥∥p

(2.2)

with constants of equivalency depending at most onp andd. Here and later we use thenotationA ≈ B to mean that there are two constantsC1,C2 > 0 such that

C1A ≤ B ≤ C2A.

We will indicate what the constants depend on (in the case of (2.2) they may depend onp andd).

Here is another useful remark concerning (2.2). From the validity of (2.2) for finitesequences, we can deduce its validity for infinite sequences by a limiting argument. Forexample, if(cI )I∈D is an infinite sequence for which the sum on the left side of (2.2)converges inL p(Rd) with respect to some ordering of theI ∈ D, then the right sideof (2.2) will converge with respect to the same ordering and the right side of (2.2) willbe less than a multiple of the left. Likewise, we can reverse the roles of the left- andright-hand sides. Similar remarks hold for other statements like (2.2) made in this paper.

The termstrong Littlewood–Paley inequalityis used to differentiate (2.2) from otherpossible forms of Littlewood–Paley inequalities. For example, the Littlewood–Paley


inequalities for the complex exponentials take a different form (see [Z, Chap. XV]).Another form of interest in our considerations is the following:∥∥∥∥∥∑

I∈DcI η(I , ·)

∥∥∥∥∥p

≈∥∥∥∥∥(∑

I∈D[cI χI

]2

)1/2∥∥∥∥∥p

.(2.3)

We use the notationχ for the characteristic function of [0, 1] andχI

for its L2(Rd)-normalized, shifted dilates given by (2.1) (withψ = χ).

The two forms (2.2) and (2.3) are equivalent under very mild conditions on the func-tionsη(I , ·). To see this, we will use the Hardy–Littlewood maximal operator, which isdefined for a locally integrable functiong onRd by

Mg(x) := supJ3x

1

|J|∫

J|g(y)| dy

with the sup taken over all cubesJ that containx. It is well known thatM is a boundedoperator onL p(Rd) for all 1 < p ≤ ∞. The Fefferman–Stein inequality [FS] boundsthe mappingM on sequences of functions. We shall only need the following special caseof this inequality which says that for any functionsη(I , ·) and constantscI , I ∈ D, wehave for 1< p ≤ ∞,∥∥∥∥∥

(∑I∈D

(cI Mη(I , ·))2)1/2∥∥∥∥∥

p

≤ A

∥∥∥∥∥(∑

I∈D(cI η(I , ·))2

)1/2∥∥∥∥∥p

(2.4)

with A a constant depending only on the space dimensiond.Consider now as an example, the equivalence of (2.2) in the univariate case. If the

univariate functionsη(I , ·), I ∈ D, satisfy

|η(I , x)| ≤ C MχI(x), χ

I(x) ≤ C Mη(I , x), a.e. x ∈ R,(2.5)

then using (2.4), we see that (2.2) holds if and only if (2.3) holds. The left inequality in(2.5) is a decay condition onη(I , ·). For example, ifη(I , ·) is given by the normalized,shifted-dilates of the functionψ , η(I , ·) = ψI , then the left inequality in (2.5) holdswhenever

|ψ(x)| ≤ C[max(1, |x|)]−λ, a.e. x ∈ R,

with λ ≥ 1. The right condition in (2.5) is extremely mild. For example, it is alwayssatisfied in the case that the familyη(I , ·) is generated by the shifted dilates of a nonzerofunctionψ .

The Littlewood–Paley inequalitites are intimately connected with unconditional bases.Given a family of functions{η(I , ·)}I∈D from L p(Rd), we define its spanX in L p(Rd)

as theL p(Rd)-closure of the linear space spanned by its finite linear combinations.The ordered family{η(I , ·)}I∈D is a basis forX if each elementf ∈ X has a uniquerepresentation

f =∑I∈D

cI η(I , ·).(2.6)

In describing the convergence of the series (2.6), we should specify the ordering (i.e.,the partial sums). We will only consider unconditional bases (described in a moment) in


which order is not important. However, for completeness of the definition of a basis, wewill take the partial sumsSn of the series (2.6) to consist of the sum over all rectanglesI = I1× · · · × Id such that 2−n ≤ |I j | ≤ 2n, j = 1, . . . ,d.

We recall that a basisη(I , ·), I ∈ D(Rd), is said to be unconditional forL p(Rd) iffor each assignmentεI := ±1, I ∈ D(Rd), and each finite sequencecI , I ∈ D(Rd), wehave ∥∥∥∥∥∑

I∈DεI cI η(I , ·)

∥∥∥∥∥p

≈∥∥∥∥∥∑

I∈DcI η(I , ·)

∥∥∥∥∥p

(2.7)

with constants of equivalency independent of the sequences(cI )I∈D and(εI )I∈D.If a basis is unconditional, then the upper estimate in (2.7) also holds for any sequence

(εI ) taking the values 0, 1. From this, it follows easily that the series (2.6) convergesindependently of the ordering.

We take for granted the known fact (see, e.g., [KS]) that for each 1< p < ∞, theunivariate Haar familyHI , I ∈ D(R), satisfies the strong Littlewood–Paley property.These functions also form an unconditional basis forL p(R), for all 1 < p < ∞; thiscan be found in [KS] and also follows from Lemma 2.2 below. We want next to concludefrom this that the multivariate Haar systemHI , I ∈ D(Rd), also satisfies the strongLittlewood–Paley property.

Let r j (t) := sign(sin 2j+1π t), t ∈ [0, 1], j = 0, 1, . . ., be the univariate Rademacherfunctions. We take any one-to-one correspondence of the natural numbers with therectangles ofD(R). This gives an indexingr (I , ·), I ∈ D(R), of the Rademacherfunctions. InRd, we let

r (I , (x1, . . . , xd)) := r (I1, x1) · · · r (Id, xd), I = I1× · · · × Id,

be the tensor products of the Rademacher functions. We recall Khinchine’s inequality(see [KS]) which says that for 1≤ p < ∞ and for any finite sequencecI , I ∈ D(Rd),we have ∥∥∥∥∥∑

I∈DcI r (I , ·)

∥∥∥∥∥L p([0,1]d)

≈(∑

I∈D|cI |2

)1/2

.(2.8)

Lemma 2.1. Let1< p <∞and letψ be a univariate function such that the univariatefamilyψI , I ∈ D(R), is an unconditional basis for Lp(R). Then the multivariate familyψI , I ∈ D(Rd), satisfies the Littlewood–Paley property(2.2) for this value of p.

Proof. For notational simplicity, we give the proof only in the cased = 2. Let cI ,I ∈ D(R2), be a sequence with finitely many nonzero terms, and letI = I1× I2. Then,using the unconditionality of the univariate basis, for eacht1, t2 ∈ [0, 1] which are notendpoints of dyadic intervals, we have with constants of equivalency depending at moston p andψ ,∫

R

∫R

∣∣∣∣∣ ∑I∈D(R2)

cI r (I , (t1, t2))ψI (x1, x2)

∣∣∣∣∣p

dx1 dx2(2.9)

=∫

R

∫R

∣∣∣∣∣ ∑I1∈D(R)

r (I1, t1)

[ ∑I2∈D(R)

cI r (I2, t2)ψI2(x2)

]ψI1(x1)

∣∣∣∣∣p

dx1 dx2


≈∫

R

∫R

∣∣∣∣∣ ∑I1∈D(R)

[ ∑I2∈D(R)

cI r (I2, t2)ψI2(x2)

]ψI1(x1)

∣∣∣∣∣p

dx1 dx2

=∫

R

∫R

∣∣∣∣∣ ∑I2∈D(R)

r (I2, t2)

[ ∑I1∈D(R)

cIψI1(x1)

]ψI2(x2)

∣∣∣∣∣p

dx1 dx2

≈∫

R

∫R

∣∣∣∣∣ ∑I2∈D(R)

[ ∑I1∈D(R)

cIψI1(x1)

]ψI2(x2)

∣∣∣∣∣p

dx1 dx2

=∫

R

∫R

∣∣∣∣∣ ∑I∈D(R2)

cIψI (x1, x2))

∣∣∣∣∣p

dx1 dx2.

We now integrate (2.9) with respect tot1, t2 ∈ [0, 1] and interchange the order ofintegration in the first term. By Khinchine’s inequalities (2.8) with respect to the normin t1, t2, the first term of (2.9) is equivalent to

∫R

∫R

( ∑I∈D(R2)

|cIψI (x1, x2)|2)p/2

dx1 dx2.

Comparing this with the last term in (2.9), we see that we have proved the lemma.

It follows, in particular from Lemma 2.1, that the multivariate Haar functionsHI ,I ∈ D(Rd), satisfy the strong Littlewood–Paley properties (2.2) and (2.3) (note that|HI | = χI ).

Lemma 2.2. Let 1 < p < ∞ and letη(I , ·), I ∈ D(Rd), be any collection of multi-variate functions. Concerning the following statements:

(i) (η(I , ·))I∈D satisfies the Littlewood–Paley condition(2.2) for this value of p;(ii) (η(I , ·))I∈D satisfies the Littlewood–Paley condition(2.3) for this value of p;

(iii) (η(I , ·))I∈D ≈ (HI )I∈D; and(iv) (η(I , ·))I∈D is an unconditional basis for Lp(Rd);

we have that(i) and(iv) are equivalent, (ii) and(iii) are equivalent, and(ii) implies(iv).Moreover, if (2.5)holds, then all these statements are equivalent.

Proof. We leave the proof of this lemma to the reader.

3. Approximation by Hyperbolic Wavelets

In this section, we will discuss approximation inL p(Rd), 1 < p < ∞, from thehyperbolic wavelet spacesHn := Hn(L p(Rd)). Letψ be a univariate function and letψI , I ∈ D(Rd), be defined as in (2.1) and letHn be the closed linear span of the finitelinear combinations of theψI with |I | > 2−n. For n = 0, 1, . . ., and f ∈ L p(Rd), we


define

En( f )p := E( f,Hn)p := infg∈Hn

‖ f − g‖p

with ‖ · ‖p here and later theL p(Rd) norm.Our main interest in this paper is the characterization of the approximation spacesAαq(L p(Rd)) defined by (1.5). We will characterize the spacesAαq(L p(Rd)) by prov-ing Bernstein and Jackson inequalities forL p(Rd) andWr (L p(Rd)) (recall the spacesWr (L p(Rd)) and the operatorDr of the Introduction). We will give two approaches toproving companion Jackson and Bernstein inequalities in this section. The first approachassumes certain conditions onψ that allow us to characterizeWr (L p(Rd)), 1< p <∞,in terms of expansions in the basisψI , I ∈ D(Rd). Using this characterization, we willthen easily prove the Jackson and Bernstein inequalities. Our second approach will as-sume weaker (and more easily verifiable conditions onψ) that still allow the proof ofthe Jackson and Bernstein inequalities. We begin with the first approach.

If ψI , I ∈ D(Rd), is a Schauder basis, then associated to this basis we have its dualbasis. In the case that 1< p <∞, the dual basis is given by linear functionalscI with

cI ( f ) =∫

Rd

f (x)λ(I , x) dx

and the functionsλ(I , ·) are inL p′(Rd) with 1/p+ 1/p′ = 1.If λ(x) := λ([0, 1], x), then it is easy to see (by using shifts and dilations) that we

can takeλ(I , ·) = λI , I ∈ D(Rd), with theλI defined as in (2.1). We note that in thecase thatψ is suitably differentiable, we have(Drψ)I = |I |r DrψI , I ∈ D(Rd). We willmake for our first approach the following assumptions about the multivariate basisψI ,I ∈ D(Rd), and its dual basisλI , I ∈ D(Rd):

(A1) ψI , I ∈ D(Rd), spanL p(Rd), 1 < p < ∞, and satisfy the Littlewood–Paleyinequalities (2.3);

(A2) (Drψ)I , I ∈ D(Rd), spanL p(Rd), 1 < p < ∞, and satisfy the Littlewood–Paley inequalities (2.3);

(A3)∫

R x jλ(x) dx = 0, j = 0, . . . , r ; and(A4) |λ(x)| ≤ C max(1, |x|−r−1−ε), for someε > 0.

Because of (A3) and (A4), we can integrate the univariate functionλ, r times tofind a functionµ ∈ L p′(R) which satisfies(−1)rµ(r ) = λ. It follows that DrµI =(−1)rd |I |−rλI , I ∈ D(Rd). Integration by parts then shows that∫

Rd

(Drψ)IµJ dx = |I |r |J|−r∫

Rd

ψI λJ dx = δ(I , J), I , J ∈ D(Rd),

with δ the Kronecker delta. Hence,µI , I ∈ D(Rd), is the dual basis for(Drψ)I ,I ∈ D(Rd).

Theorem 3.1. Let r be a positive integer, 1 < p < ∞, and letψ be a univariatefunction which satisfies assumptions(A1)–(A4). Then a function f∈ Lp(Rd) is inWr (L p(Rd)) if and only if

f =∑

I∈D(Rd)

cI ( f )ψI(3.1)


with ∥∥∥∥∥( ∑

I∈D(Rd)

[|I |−r |cI ( f )|χI]2

)1/2∥∥∥∥∥p

< ∞.(3.2)

Furthermore, the left side of(3.2) is equivalent to| f |Wr (L p(Rd)).

Proof. Suppose first thatf ∈Wr (L p(Rd)). Assumption (A1) gives that the functionsψI , I ∈ D(Rd), satisfy the strong Littlewood–Paley inequalities. Hence, these functionsare an unconditional basis forL p(Rd) and

f =∑

I∈D(Rd)

cI ( f )ψI

with

cI ( f ) =∫

Rd

f λI dx.

Likewise, the functions(Drψ)I , I ∈ D(Rd), are also a basis forL p(Rd), and we have

Dr f =∑

I∈D(Rd)

dI ( f )(Drψ)I

with

(3.3)

dI ( f ) =∫

Rd

Dr f µI dx = (−1)rd∫

Rd

f DrµI dx = |I |−r∫

Rd

f λI dx = |I |−r cI ( f ).

We can compute‖Dr f ‖p from the Littlewood–Paley condition for the basis(Drψ)I ,I ∈ D(Rd). This gives that the left side of (3.2) is equivalent to‖Dr f ‖p.

Conversely, assume thatf ∈ L p(Rd) is such that (3.2) is finite. BecauseψI , I ∈D(Rd), is an unconditional basis forL p(Rd), we have

f =∑

I∈D(Rd)

cI ( f )ψI

in the sense ofL p(Rd)-convergence. From (3.2) and the fact that(Drψ)I , I ∈ D(Rd),satisfies the Littlewood–Paley inequalities, we find that there is a functiong ∈ L p(Rd)

with

g =∑

I∈D(Rd)

|I |−r cI ( f )(Drψ)I

again in the sense ofL p(Rd)-convergence. We compute the coefficients ofg with respectto the basis(Drψ)I ), I ∈ D(Rd), and find∫

Rd

gµI = |I |−r cI ( f ) = |I |−r∫

Rd

f λI = (−1)rd∫

Rd

f DrµI .

This shows that g is the distributional derivativeDr f and therefore f ∈Wr (L p(Rd)).


The following theorem gives the Jackson and Bernstein inequalities for approximationby the spacesHn.

Theorem 3.2. Let r be a positive integer, 1 < p < ∞, and letψ be a univariatefunction which satisfies assumptions(A1)–(A4). If f ∈Wr (L p(Rd)), then

En( f )p ≤ C2−nr | f |Wr (L p(Rd)), n = 0, 1, . . . ,(3.4)

with C independent of n and f. If g ∈ Hn(L p(Rd)), then

|g|Wr (L p(Rd)) ≤ C2nr‖g‖p, n = 0, 1, . . . ,(3.5)

with C independent of n and g.

Proof. First, let f ∈Wr (L p(Rd)). Then

f =∑

I∈D(Rd)

cI ( f )ψI

in the sense ofL p(Rd)-convergence and (3.2) is satisfied. We letg :=∑|I |>2−n cI ( f )ψI

which is a function inHn(L p(Rd)). The remainderf − g is given by

f − g =∑|I |≤2−n

cI ( f )ψI .

We can estimate‖ f − g‖p by using the Littlewood–Paley inequalities:

‖ f − g‖p ≤ C

∥∥∥∥∥( ∑|I |≤2−n

[|cI ( f )|χI]2

)1/2∥∥∥∥∥p

≤ C2−nr

∥∥∥∥∥( ∑|I |≤2−n

[|I |−r |cI ( f )|χI]2

)1/2∥∥∥∥∥p

≤ C2−nr

∥∥∥∥∥( ∑

I∈D(Rd)

[|I |−r |cI ( f )|χI]2

)1/2∥∥∥∥∥p

≤ C2−nr‖Dr f ‖p.

This proves (3.4).Suppose now thatg ∈ Hn(L p(Rd)). Then,

g =∑|I |>2−n

cI (g)ψI ,

and from Theorem 3.1, we have

‖Dr g‖p ≤ C

∥∥∥∥∥( ∑|I |>2−n

[|I |−r |cI (g)|χI]2

)1/2∥∥∥∥∥p

≤ C2nr

∥∥∥∥∥( ∑|I |>2−n

[cI (g)|χI]2

)1/2∥∥∥∥∥p

≤ C2nr‖g‖p

becauseψI , I ∈ D(Rd), satisfies the Littlewood–Paley inequalities (2.3). This proves(3.5).


As we have mentioned earlier in this section, the Jackson and Bernstein inequalities(3.4) and (3.5) allow the characterization of the approximation spacesAαq(L p(Rd)). Forthis, we will use theK -functional

K ( f, t) := K ( f, t, L p(Rd),Wr (L p(Rd)))(3.6)

:= infg∈Wr (L p(Rd))

‖ f − g‖p + t |g|Wr (L p(Rd)).

We recall the interpolation spaces(L p(Rd),Wr (L p(Rd)))θ,q defined by the real methodof interpolation (see Chapter 6 of [DL]).

Corollary 3.3. Let r be a positive integer, 1 < p < ∞, 0 < q ≤ ∞, and 0 <

α < r . Let ψ be a univariate function which satisfies assumptions(A1)–(A4). Then,f ∈ Aαq(L p(Rd)) if and only if f ∈ (L p(Rd),Wr (L p(Rd)))α/r,q with equivalent norms.

Proof. This corollary follows from Theorem 3.2 and general facts about approximationandK -functionals that can be found in Chapter 7 of [DL].

While the above approach is simple and direct, the assumption (A2) is too severe forsome applications. It is also uncomfortable to make such an assumption for a Jacksoninequality since it is unclear why the Jackson inequality should depend on the smoothnessof ψ . Recall that in the univariate case of wavelet approximation, Jackson inequalitiesdepend only on conditions (A1), (A3), and (A4). We shall therefore now give a secondapproach to proving the Jackson and Bernstein inequalities which separates the proofof these inequalities. This approach allows us to prove the Jackson inequality under amuch weaker assumption than (A2).

We first consider the Jackson inequality. Suppose that we have in hand two multi-variate familiesη(I , ·), µ(I , ·), I ∈ D(Rd). We will use the notation{η(I , ·)}I∈D ≺{µ(I , ·)}I∈D, if there is a constantC > 0 such that∥∥∥∥∥∑

I∈DcI η(I , ·)

∥∥∥∥∥p

≤ C

∥∥∥∥∥∑I∈D

cIµ(I , ·)∥∥∥∥∥

p

(3.7)

holds for all finite sequences(cI )I∈DwithC independent of the sequence. If{η(I , ·)}I∈D ≺{µ(I , ·)}I∈D and{µ(I , ·)}I∈D ≺ {η(I , ·)}I∈D, then we write{η(I , ·)}I∈D ≈ {µ(I , ·)}I∈D.

Given two multidimensional familiesη(I , ·), µ(I , ·), I ∈ D(Rd), we define the op-eratorT which mapsµ(I , ·) into η(I , ·) for all I ∈ D and we extendT to finite linearcombinations of theµ(I , ·) by linearity. Then (3.7) holds if and only ifT is a boundedoperator with respect to theL p-norm and{µ(I , ·)}I∈D ≺ {η(I , ·)}I∈D holds if and onlyif T has a bounded inverse with respect to theL p-norm.

We recall the Haar basisHI , I ∈ D(Rd). In place of (A2), we will assume

(A2′) {µI }I∈D(Rd) ≺ {HI }I∈D(Rd),

where as beforeµ satisfiesµ(r ) = (−1)rλ. It follows from (A2′) that the operatorTdefined by

T f :=∑

I∈D(Rd)

〈 f, HI 〉µI


is bounded onL p(Rd)) for each 1< p <∞. Hence, by duality, its adjoint

T∗ f :=∑

I∈D(Rd)

〈 f, µI 〉HI

is also bounded onL p(Rd)), for each 1< p <∞.

Theorem 3.4. Assume that(A1), (A2′), (A3), and (A4) hold. If 1 < p < ∞, r is apositive integer and f∈Wr (L p(Rd)), then

En( f )p ≤ C2−nr | f |Wr (L p(Rd)), n = 0, 1, . . . ,(3.8)

with C independent of n and f.

Proof. Let f ∈Wr (L p(Rd)). From assumption (A1), we have (see (3.3))

f =∑

I∈D(Rd)

cI ( f )ψI , cI ( f ) = 〈 f, λI 〉 = |I |r 〈Dr f, µI 〉,

in the sense ofL p(Rd)-convergence. We letg :=∑|I |>2−n cI ( f )ψI which is a functioninHn(L p(Rd)). We can estimate the remainderf − g by

‖ f − g‖p =∥∥∥∥∥ ∑|I |≤2−n

cI ( f )ψI

∥∥∥∥∥p

≤ C

∥∥∥∥∥ ∑|I |≤2−n

cI ( f )HI

∥∥∥∥∥p

= C

∥∥∥∥∥ ∑|I |≤2−n

|I |r 〈Dr f, µI 〉HI

∥∥∥∥∥p

≤ C

∥∥∥∥∥∥( ∑|I |≤2−n

[|I |r 〈Dr f, µI 〉χI]2

)1/2∥∥∥∥∥∥

p

≤ C2−nr

∥∥∥∥∥∥( ∑|I |≤2−n

[〈Dr f, µI 〉χI]2

)1/2∥∥∥∥∥∥

p

≤ C2−nr

∥∥∥∥∥ ∑I∈D(Rd)

〈Dr f, µI 〉HI

∥∥∥∥∥p

≤ C2−nr‖Dr f ‖p,

where the last inequality uses the boundedness of the adjoint operatorT∗ (which followsfrom (A2′)).

We next consider the Bernstein inequality. In place of (A2), we will assume that

(A2′′) {(Drψ)I }I∈D(Rd) ≺ {HI }I∈D(Rd).

Theorem 3.5. Assume that(A1), (A2′′), (A3), and(A4) hold. Then, for each positiveinteger r and each g∈ Hn(L p(Rd)), 1< p <∞, we have

‖Dr g‖p ≤ C2nr‖g‖p, n = 0, 1, . . . ,(3.9)

with C independent of n and g.


Proof. Because of (A1), we can write

g =∑|I |>2−n

cI (g)ψI .(3.10)

In the proof of (3.9), it is enough to consider functionsg for which the sum in (3.10) hasonly a finite number of terms; the general case follows by a limiting argument. In thiscase, we have

‖Dr g‖p =∥∥∥∥∥ ∑|I |>2−n

cI (g)DrψI

∥∥∥∥∥p

=∥∥∥∥∥ ∑|I |>2−n

|I |−r cI (g)(Drψ)I

∥∥∥∥∥p

≤ C

∥∥∥∥∥ ∑|I |>2−n

|I |−r cI (g)HI

∥∥∥∥∥p

≤ C

∥∥∥∥∥ ∑|I |>2−n

|I |−r cI (g)ψI

∥∥∥∥∥p

≤ C2nr

∥∥∥∥∥ ∑|I |>2−n

cI (g)ψI

∥∥∥∥∥p

= C2nr‖g‖p.

4. Sufficient Conditions for the Littlewood–Paley Inequalities

In the previous section, we have characterized the approximation spaces for hyperbolicwavelet approximation under certain assumptions on the univariate functionsψ andψ(r )

relating to Littlewood–Paley theory. For many functionsψ that occur in wavelet theory,it is possible to utilize the existing Littlewood–Paley theory to verify these assumptions.However, in some instances (e.g., for spline wavelets), the application of the existingtheory will not give the largest possible value ofr because this theory requires globalsmoothness ofψ (respectivelyψ(r )). The purpose of the present section is to prove aLittlewood–Paley theorem which does not require global smoothness (rather it is enoughto have certain piecewise continuity). We shall also address some related questionsassociated with Littlewood–Paley theory.

It is possible to formulate our theorems without assuming that the family of functionsunder consideration are all shifted-dilates of a single function. We shall therefore revertback to our notationη(I , ·), I ∈ D := D(R1), to denote an arbitrary family of functionsindexed on dyadic intervals.

The strong Littlewood–Paley inequalities (2.3) are the same as the equivalence{η(I , ·)}≈ {HI }. We begin this section by discussing sufficient conditions in order that{η(I , ·)} ≺{HI }. Let ξI , I ∈ D, denote the center of the dyadic intervalI . We will assume in thissection thatη(I , ·), I ∈ D, is a family of univariate functions that satisfy the followingassumptions:

(A5) There is anε > 0, and a constantC1 such that for allt ∈ R and allJ ∈ D, wehave

|η(J, ξJ + t |J|)| ≤ C1|J|−1/2(1+ |t |)−1−ε.

(A6) There is anε > 0 and a constantC2 and a partition of [− 12,

12] into intervals

J1, . . . , Jm that are dyadic with respect to [− 12,

12], such that for anyJ ∈ D, any

j ∈ Z, and anyt1, t2 in the interior of the same intervalJk, k = 1, . . . ,m, we


have

|η(J, ξJ+ j |J|+t1|J|)−η(J, ξJ+ j |J|+t2|J|)| ≤ C2|J|−1/2(1+| j |)−1−ε|t2−t1|ε,(A7) For anyJ ∈ D, we have ∫

Rη(J, x) dx = 0.

In the case thatη(J, ·) = ψJ for a functionψ , it is enough to check these assumptionsfor J = [0, 1], i.e., for the functionψ alone. They follow for all other dyadic intervalsJ by dilation and translation.

The condition (A5) is a standard decay assumption and (A7) is the zero momentcondition. The condition (A6) requires that the functionsη(I , ·) be piecewise in Lipε.

Let T be the linear operator which satisfies

T

(∑I∈D

cI HI

)=∑I∈D

cI η(I , ·)(4.1)

for each finite linear combination∑

I∈D cI HI of the HI . We wish to show that∥∥∥∥∥T

(∑I∈D

cI HI

)∥∥∥∥∥p

≤ C

∥∥∥∥∥∑I∈D

cI HI

∥∥∥∥∥p

for each such sum. From this it would follow thatT extends (by continuity) to a boundedoperator on all ofL p(R) and therefore{η(I , ·)} ≺ {HI }.

We can expandη(J, ·) into its Haar decomposition. Let

λ(I , J) :=∫

Rη(J, x)HI (x) dx,(4.2)

so that

η(J, ·) =∑I∈D

λ(I , J)HI .

It follows that

T

(∑J∈D

cJ HJ

)=∑I∈D

∑J∈D

λ(I , J)cJ HI .(4.3)

We see that the mappingT is tied to the bi-infinite matrix3 := (λ(I , J))I ,J∈D whichmaps the sequencec := (cJ) into the sequence

(c′I ) := 3c.

One approach to proving Littlewood–Paley inequalities is to show that the matrix3

decays sufficiently fast away from the diagonal (see [FJ, §3]). Following [FJ], we saythat a matrixA = (a(I , J))I ,J∈D is almost diagonalif for someε > 0, we have

|a(I , J)| ≤ Cω(I , J)(4.4)

with

ω(I , J) :=(

1+ |ξI − ξJ |max(|I |, |J|)

)−1−ε (min

( |I ||J| ,|J||I |))(1+ε)/2

.(4.5)

We will use the following special case of a theorem of Frazier and Jawerth [FJ,Theorem 3.3] concerning almost diagonal operators.


Theorem 4.1. If (a(I , J))I ,J∈D is an almost diagonal matrix, then the operator Adefined by

A

(∑J∈D

cJ HJ

):=∑I∈D

∑J∈D

a(I , J)cJ HI(4.6)

is bounded on Lp(R) for each1< p <∞.

Proof. For the completeness of the present paper, we give the following proof of thistheorem. LetW be the operator defined as in (4.6) with(ω(I , J))I ,J∈D in place of(a(I , J))I ,J∈D. From the Littlewood–Paley inequalities for the Haar functions, we have∥∥∥∥∥A

(∑J∈D

cJ HJ

)∥∥∥∥∥p

≤ C

∥∥∥∥∥∑I∈D

∑J∈D|a(I , J)||cJ |HI

∥∥∥∥∥p

≤ C

∥∥∥∥∥∑I∈D

∑J∈D

ω(I , J)|cJ |HI

∥∥∥∥∥p

≤ C‖W‖∥∥∥∥∥∑

J∈D|cJ |HJ

∥∥∥∥∥p

≤ C‖W‖∥∥∥∥∥∑

J∈DcJ HJ

∥∥∥∥∥p

with ‖W‖ the norm ofW as an operator fromL p(R) into itself and with the constantsC depending only onp. Thus, it is sufficient to show that‖W‖ is finite.

We write

ω(I , J) = ω+(I , J)+ ω−(I , J),

with

ω+(I , J) :={ω(I , J), |J| ≥ |I |,0, |J| < |I |,

and we letW+ andW− be defined as in (4.6) for these two sequences. We shall showthatW+ andW− are bounded onL p(R) which will complete the proof of the theorem.Since the proof of boundedness in these two cases is similar, we shall only considerW+.

We will employ the following inequality for nonnegative sequences(b`):

∞∑`=1

b``−τ ≤ Cτ max

m≥1

1

m

m∑`=1

b`, τ > 1,

which is easily proved by summation by parts.To boundW+, it is enough to consider its action on

∑J∈D cJ HJ where thecJ are

nonnegative and only a finite number of them are nonzero. We let

c′I :=∑J∈D

ω+(I , J)cJ .

From the above inequality, it follows that for each intervalI with |I | = 2ν andµ ≥ ν,we have ∑

|J|=2µ

(1+ |ξI − ξJ |

max(|I |, |J|))−1−ε

cJ ≤ C2µ/2M

( ∑|J|=2µ

cJχJ, x

)= C2µ/2M( fµ, x), x ∈ I ,


with M the Hardy–Littlewood maximal operator, and with

fµ :=∑|J|=2µ

cJχJ=( ∑|J|=2µ

c2Jχ

2

J

)1/2

,

and withC here and later in this proof depending onε. Hence, from (4.5)

|I |−1/2|c′I | ≤ C∑µ≥ν

2−µε/22νε/2M( fµ, x), x ∈ I .

Using the Cauchy–Schwartz inequality, we obtain

|c′I |2χ2

I(x) ≤ C

∑µ≥ν

2(ν−µ)ε/2(M( fµ, x))2.

Since the functionsχ2I

have disjoint supports, we have∑|I |=2ν|c′I |2χ2

I≤ C

∑µ≥ν

2(ν−µ)ε/2(M( fµ))2.

Summing overν ∈ Z gives∑I∈D|c′I |2χ2

I≤ C

∑ν∈Z

∑µ≥ν

2(ν−µ)ε/2(M( fµ))2.

We now apply the Littlewood–Paley inequalities for the Haar system and the Fefferman–Stein inequality (2.4) to find∥∥∥∥∥W+

(∑I∈D

cI HI

)∥∥∥∥∥p

≤ C

∥∥∥∥∥(∑

I∈D|c′I |2χ2

I

)1/2∥∥∥∥∥p

≤ C

∥∥∥∥∥(∑ν∈Z

∑µ≥ν

2(ν−µ)ε/2 f 2µ

)1/2∥∥∥∥∥p

≤ C

∥∥∥∥∥(∑µ∈Z

f 2µ

)1/2∥∥∥∥∥p

≤ C

∥∥∥∥∥∑I∈D

cI HI

∥∥∥∥∥p

,

where the last inequality follows from the Littlewood–Paley inequalities (2.3) for theHaar functions and the definition of the functionsfµ.

Theorem 4.2. If η(I , ·), I ∈ D, satisfy assumptions(A5)–(A7), then the operator Tdefined by(4.1) is bounded from Lp(R) into itself for each1< p <∞.

Proof. For an intervalI ∈ D, let I± denote the dyadic intervals with|I±| = |I |, whichare immediately to the right and left ofI , respectively. We defineI ∗ := I− ∪ I ∪ I+. Fortheλ(I , J) of (4.2), we define

λ′(I , J) :={λ(I , J), |I | ≤ |J|,0, else,

λ′′(I , J) :={λ(I , J), |I | > |J|, J * I ∗,0, else,

λ′′′(I , J) :={λ(I , J), |I | > |J|, J ⊂ I ∗,0, else.


Then,

λ(I , J) = λ′(I , J)+ λ′′(I , J)+ λ′′′(I , J).

We let T1, T2, T3 be the operators defined as in (4.3) withλ replaced byλ′, λ′′, λ′′′,respectively. We will show that each of these operators is bounded fromL p(R) intoitself.

We first show thatT1 satisfies the conditions of Theorem 4.1. Let 2−α be the lengthof the smallest dyadic interval appearing in the statement of condition (A6). We firstconsider intervalsI such that|I | ≤ 2−α|J|. Then from property (A6), there is a constanta such that

|η(J, x)− a| ≤ C2|J|−1/2

(1+ |ξI − ξJ |

|J|)−1−ε ( |I |

|J|)ε, x ∈ int (I ).

Indeed, for one of the intervalsJk of assumption (A6) and an appropriately chosenj ,anyx ∈ I can be written asx = ξJ + j |J| + t |J| with t ∈ Jk. Using the last inequality,we obtain

|λ′(I , J)| ≤∣∣∣∣∫

IHI (x)(η(J, x)− a) dx

∣∣∣∣≤ C2|I |1/2|J|−1/2

(1+ |ξI − ξJ |

|J|)−1−ε ( |I |

|J|)ε.

By using (A5), we see that this last inequality is also valid if|J| ≥ |I | > 2−α|J|because the term(|I |/|J|)ε can be replaced by 1. This shows that(λ′(I , J))I ,J∈D isalmost diagonal and hence the boundedness ofT1 on L p(R) follows from Theorem 4.1.

A similar calculation shows that(λ′′(I , J))I ,J∈D is almost diagonal and henceT2 isalso bounded onL p(R) because of Theorem 4.1.

The proof of the boundedness ofT3 will require more care since this operator is notnecessarily almost diagonal. We can decompose this operator into a sum of four operatorscorresponding to whetherJ is contained in the left- or right-half ofI , or J is containedin I+ or I−. We can show that each of these operators is bounded onL p(R) in a similarway. We consider in detail the operatorA0 corresponding to the left-half ofI and showthat it is bounded onL p, 1< p <∞. Later we shall note the modifications that need tobe made to handle the remaining three cases.

We denote the left-half of intervalsI ∈ D by I ′. ThenA0 has the associated matrix

λ0(I , J) :

{λ(I , J), J ⊂ I ′,0, else.

We can write

A0 =∞∑

m=1

Am,

whereAm has the associated matrix

λm(I , J) :={λ(I , J), |J| = 2−m|I |, J ⊂ I ′,0, else.


We will show that for a certainδ > 0 (depending onp),

‖Am‖ ≤ C2−mδ,(4.7)

with ‖ · ‖ denoting the norm of the operatorAm from L p(R) into itself, and withCindependent ofm. This will then complete the proof of the theorem.

We recall that an operator onL p(R) has the same norm as its adjoint onL p′ , 1/p+1/p′ = 1. It is therefore enough to show that the adjoint operatorsA∗m satisfy (4.7) foreach 1< p <∞. The operatorA∗m is defined by

A∗m

(∑J∈D

cJ HJ

)=∑J∈D

∑|I |=2m|J|

I ′⊃J

λm(I , J)cI HJ .

We first estimateλm(I , J), |I | = 2m|J|, J ⊂ I ′. Let C(I ′) denote the complement ofI ′. Using assumptions (A7) and (A5), we have

|λm(I , J)| =∣∣∣∣∫

Rη(J, x)HI (x) dx

∣∣∣∣(4.8)

≤ |I |−1/2

(∣∣∣∣∫I ′η(J, x) dx

∣∣∣∣+ ∣∣∣∣∫I \I ′

η(J, x) dx

∣∣∣∣)≤ 2|I |−1/2

∫C(I ′)|η(J, x)| dx

≤ 2C1|I |−1/2|J|−1/2∫C(I ′)

(1+ |x − ξJ |

|J|)−1−ε

dx

≤ C|J|1/2|I |1/2 k(I , J)−ε,

wherek(I , J) ≥ 1 is the largest integerk such that

dist(ξJ, C(I ′)) ≥ k|J|/2.

Let c′J be theHJ-coefficient ofA∗m(∑

cI HI ). Then, there is at most one intervalI ′

with J ⊂ I ′ and|I | = 2m|J|, and for that intervalI , we have from (4.8),

|c′J | ≤ |λm(I , J)||cI | ≤ Ck(I , J)−ε2−m/2|cI |.(4.9)

This give

∑J⊂I ′|c′J |2 ≤ C2−m

2m∑k=1

k−2ε|cI |2 ≤ C2−2εm|cI |2.

We now sum over allI ∈ D to find∑J∈D|c′J |2 ≤ C2−2εm

∑I∈D|cI |2.


Since(HI )I∈D is an orthonormal system, it follows that∥∥∥∥∥A∗m

(∑J∈D

cJ HJ

)∥∥∥∥∥2

≤ C2−εm∥∥∥∥∥∑

J∈DcJ HJ

∥∥∥∥∥2

.(4.10)

In other words, the operatorA∗m has norm at mostC2−εm when acting onL2(R).We now consider 1< q < ∞ and boundA∗m on the spaceLq(R). We use (4.9) and

the fact thatk(I , J) ≥ 1 to find that∑J⊂I

|c′J |2χ2

J≤ C2−m|cI |2

∑J⊂I

χ2

J= C|cI |2χ2

I.

Therefore, summing overI ∈ D, we find∑J∈D|c′J |2χ2

J≤ C

∑I∈D|cI |2χ2

I.(I)

From this, and the strong Littlewood–Paley inequalities for the Haar functions, weobtain ∥∥∥∥∥A∗m

(∑J∈D

cJ HJ

)∥∥∥∥∥q

≤ C

∥∥∥∥∥∑J∈D

cJ HJ

∥∥∥∥∥q

.(4.11)

In other words, the operatorA∗m has norm at mostC when acting onLq(R).Suppose that 1< p ≤ 2. We choose a value ofq with 1 < q < p. If 0 < θ < 1

satisfies 1/p = (1− θ)/q+ θ/2, then (4.10), (4.11), and the Riesz–Thorin interpolationtheorem for linear operators gives

‖A∗m‖L p→L p ≤ ‖A∗m‖1−θLq→Lq‖A∗m‖θL2→L2

≤ C2−δm, δ := θε.A similar argument applies when 2< p < ∞. Thus, we have shown that the operatorA0 is bounded fromL p to L p for all 1< p <∞.

In the same way, we can show that the operatorA0 corresponding to the caseJ iscontained in the right-half ofI is bounded onL p, 1< p <∞.

To show the boundedness of the operatorA0 corresponding to the caseJ ⊂ I± wewill need a slight modification of the above proof. The same argument as above givesthe inequality

λm(I , J) ≤ |I |−1/2∫

I|η(J, x)| dx ≤ C

|J|1/2|I |1/2 k(I , J)−1/2

with k(I , J) the largest integer such that

dist(ξJ, I ) ≥ k|J|/2.As in the previous case, we obtain (4.10) and in place of (I) we obtain∑

J∈D|c′J |2χ2

J≤ C

∑I∈D|cI |2χ2

I±.

SinceχI±≤ C M(χ

I) with M the Hardy–Littlewood maximal function, we can use the

Fefferman–Stein inequality together with the strong Littlewood–Paley inequalities forHaar functions to arrive at (4.11). The remainder of the proof of the boundedness ofA0

in this case is the same as above. This completes the proof of the theorem.


Corollary 4.3. If η(I , ·), I ∈ D,satsify the assumptions(A5)–(A7),then{η(I , ·}I∈D ≺{HI }I∈D.

We can use a duality argument to give sufficient conditions that the operatorT of (4.1)is boundedly invertible. For this, we assume thatη(I , ·), I ∈ D, is a family of functionsfor which there is a dual familyλ(I , ·), I ∈ D, that satisfies

〈η(I , ·), λ(J, ·)〉 = δ(I , J), I , J ∈ D.

Theorem 4.4. If the functionsλ(I , ·), I ∈ D, satisfy the assumptions(A5)–(A7), then{HI }I∈D ≺ {η(I , ·}I∈D.

Proof. We need to show that for each 1< p < ∞ and each sequence(cI )I∈D, wehave ∥∥∥∥∥∑

I∈DcI HI

∥∥∥∥∥p

≤ C

∥∥∥∥∥∑I∈D

cI η(I , ·)∥∥∥∥∥

p

(4.12)

with a constantC independent of the sequence(cI )I∈D. We can assume that the sequence(cI )I∈D has at most a finite number of nonzero entries. We have∥∥∥∥∥∑

I∈DcI HI

∥∥∥∥∥p

= sup(dI )

⟨∑I∈D

cI HI ,∑I∈D

dI HI

⟩= sup

(dI )

⟨∑I∈D

cI η(I , ·),∑I∈D

dI λ(I , ·)⟩

with the sup taken over all sequences(dI )I∈D with at most a finite number of nonzeroentries which satisfy‖∑I∈D dI HI ‖p′ = 1. From Holder’s inequality, we have∥∥∥∥∥∑

I∈DcI HI

∥∥∥∥∥p

≤∥∥∥∥∥∑

I∈DcI η(I , ·)

∥∥∥∥∥p

∥∥∥∥∥∑I∈D

dI λ(I , ·)∥∥∥∥∥

p′

≤ C

∥∥∥∥∥∑I∈D

cI η(I , ·)∥∥∥∥∥

p

∥∥∥∥∥∑I∈D

dI HI

∥∥∥∥∥p′= C

∥∥∥∥∥∑I∈D

cI η(I , ·)∥∥∥∥∥

p

,

where in the last inequality we used Theorem 4.2 for the sequence(λ(I , ·))I∈D.

We now apply Theorems 4.2 and 4.4 to the setting of Section 3. Letψ be a univariatefunction and let(ψI )I∈D be its univariate shifted-dilates. We also suppose that(ψI ) hasa dual basis(λI )I∈D satisfying∫

RψI (x)λI (x) dx = δ(I , J).

Corollary 4.5. If the functionsψ andλ satisfy conditions(A5)–(A7) in the case J=[0, 1], then(ψI )I∈D satisfies the strong Littlewood–Paley property(2.3).

Proof. If follows from Theorem 4.2 that(ψI )I∈D ≺ (HI )I∈D and from Theorem 4.4that (HI )I∈D ≺ (ψI )I∈D. Thus,(ψI )I∈D ≈ (HI )I∈D and the theorem follows fromLemma 2.2.


We shall use the remainder of this section to give an example which shows that, in acertain sense, the assumption of piecewise Lipschitzε continuity of theη(I , ·) in (A6)cannot be removed. Namely, we will show that there is a continuous functionψ supportedon [0, 1] with mean value 0 for which the Littlewood–Paley inequalities (2.2) and (2.3)do not hold.

Let X denote the set of all functions inC[0, 1] which vanish at the endpoints (f (0) =f (1) = 0) and have mean value zero (

∫ 10 f (x) dx = 0). This is a closed subspace of

C[0, 1]. For the formulas that follow, we considerf (x) := 0 for x ∈ R\[0, 1]. Let usconsider the operatorR defined for any function supported on [0, 1] by

R f(x) := f (2x)+ f (2x − 1), x ∈ [0, 1].(4.13)

Then,R has norm one onL2[0, 1]. We will use the following lemma:

Lemma 4.6. For eachε > 0, there is a function f= fε in X such that‖ f ‖L2[0,1] ≥1− ε/4 and

‖ f − R f‖L2[0,1] < ε.(4.14)

Proof. We choose an integerN > 1 such that

2−N

(2N + 1

N

)1/2

< ε/4.(4.15)

Each point inx ∈ (0, 1) has a dyadic expansion with digitsx1, x2, . . .. We require thatinfinitely many of thexi are zero; then these digits are unique. We define

f (x) :={

1, if∑2N+1

i=1 xi ≤ N,

−1, else.

Then f is piecewise constant taking the values±1 on dyadic intervals of length 2−2N−1.We have

R f (x) ={

1,∑2N+2

i=2 xi ≤ N,

−1, else.

It follows that R f (x) = f (x) except for the setE of pointsx for which∑2N+1

i=1 xi =N, N + 1. SinceE has measure

2−2N−1

((2N + 1

N

)+(

2N + 1

N + 1

))= 2−2N

(2N + 1

N

).

We have (using (4.15))

‖ f − R f ‖L2[0,1] ≤ 2−N+1

(2N + 1

N

)1/2

< ε/2.

The function f has a finite number of discontinuities which occur at pointsj 2−2N−1,j = 1, . . . ,22N+1. We can adjustf near its points of discontinuity to obtain a functionf ∈ X with ‖ f ‖C[0,1] = 1 and

‖ f − f ‖L2[0,1] < ε/4.


Then,‖ f ‖L2[0,1] ≥ ‖ f ‖L2[0,1] − ε/4= 1− ε/4.

Since the operatorR has norm one onL2[0, 1], it follows that

‖ f − R f‖L2[0,1] ≤ 2‖ f − f ‖L2[0,1] + ‖ f − R f ‖L2[0,1] < 2ε/4+ ε/2= ε.

Theorem 4.7. There is a continuous functionψ supported on[0, 1] with mean valuezero such that the Littlewood–Paley inequalities(2.2) and (2.3) do not hold forψI ,I ∈ D.

Proof. Consider again the operatorR defined by (4.13) and define for eachn =1, 2, . . . , the operator

Sn f := 1√n

n−1∑k=0

Rk f, f ∈ X.

LetDn be the set of dyadic intervals in [0, 1] of length≥ 2−n−1, it follows that

Sn f = 1√n

∑I∈Dn

|I |1/2 f I .

For the Haar functionH , we have by orthogonality that‖Sn H‖L2[0,1] = 1. It is thereforeenough to show that there is a functionψ ∈ X such that

supn≥1‖Snψ‖L2[0,1] = ∞.

By the Banach–Steinhaus theorem, we need only show that the operatorsSn are un-bounded as mappings fromX into L2[0, 1]. To this end, letε > 0 and let f = fε be thefunction in X of Lemma 4.6. SinceR has norm one onL2[0, 1], we have

‖Rk f − f ‖L2[0,1] ≤k∑`=1

‖R` f − R`−1 f ‖L2[0,1] ≤k∑`=1

‖ f − R f‖L2[0,1] ≤ kε.

Therefore,

‖Sn f −√n f ‖L2[0,1] = 1√n

∥∥∥∥∥n−1∑k=0

(Rk f − f )

∥∥∥∥∥L2[0,1]

≤ 1√n

n−1∑k=0

‖Rk f − f ‖L2[0,1] ≤√

n(n− 1)ε

2.

It follows that

‖Sn f ‖L2[0,1] ≥√

n‖ f ‖L2[0,1] − ‖Sn f −√n f ‖L2[0,1]

≥ √n‖ f ‖L2[0,1] −√

n(n− 1)ε

2.

Since‖ f ‖L2[0,1] ≥ 1− ε/4, by lettingε→ 0, we see that the norm ofSn is≥ √n.


5. Further Remarks

The results we have developed in the previous sections can be applied to any of the knownwavelets. We shall consider in detail only the example of the Daubechies wavelets. Thiswill indicate how the results of the previous sections are applied. Other examples suchas the biorthogonal wavelets of Cohen, Daubechies, and Feauveau [CDF] can be treatedin a similar way.

Let ψ := Dk, k > 1, be thekth univariate, orthogonal wavelet of Daubechies (see§6.4 of [Da]). We define, as usual, the multivariate family{ψI }I∈D(Rd) by (1.4). We firstconsider how one verifies the Jackson inequality (J) of Section 1 for the spaceHn. We donot want to use Theorem 3.2 to establish the Jackson inequality because it requires toomuch smoothness forψ . Instead, we will use Theorem 3.4. This requires us to establishconditions (A1), (A2′), (A3), (A4) of Section 3 for the functionsη(I , ·) = ψI .

The usual theory of multiresolution tells us that theψI , I ∈ D(Rd), spanL p(Rd),1< p <∞. It is also well known (see, e.g., [M]) thatψI , I ∈ D(R), is an unconditionalbasis forL p(R), 1 < p < ∞ (this also follows from our Section 4). The results inSection 2 then tell us thatψI , I ∈ D(Rd), is an unconditional basis forL p(Rd) and theLittlewood–Paley relations (2.3) hold for this basis. Hence, condition (A1) is satisfied.

Sinceλ = ψ , the moment condition (A3) holds for anyr ≤ k. The decay condition(A4) obviously holds for anyr becauseψ has compact support.

We are left with showing (A2′). We can show that this condition holds forr = k− 1.We need to show that{µI }I∈D ≺ {HI }I∈D whereµ is ther th integral ofψ . For this, wecan use Corollary 4.3. SinceDk hask vanishing moments, the functionsµI , I ∈ D(R),have compact support and therefore satisfy (A5). The functionµ is at least Lipschitzcontinuous since it is the integral of a bounded function. Hence, condition (A6) is valid.SinceDk hask vanishing moments,µ has at least one vanishing moment and so (A7) issatisfied. Theorem 4.3 now implies that (A2′) is satisfied. We therefore have the Jacksoninequality for hyperbolic approximation with the Daubechies wavelets

En( f )p ≤ C2−nr | f |Wr (L p(Rd)), n = 0, 1, . . . , 1< p <∞,(5.1)

with r = k− 1 and withC independent ofn and f .It is important to contrast the difference between (5.1) in the univariate case and the

multivariate case. It is well known that (5.1) holds in the univariate case forr = k. Thereason for this is that in the univariate case one does not need to assumer momentsare zero as in (A3) but only thatr − 1 moments are zero. However, in the multivariatecase, we cannot make this less restrictive assumption as can be seen already for the HaarfunctionD1. For example, if we defineF(x) :=∏d

j=1 f (xj ), with f a univariate function

from W1(L2(R)) which has compact support and for whichf (t) = t , t ∈ ( 14,

34). Then

for any dyadic rectangleI ⊂ ( 14,

34)

d, we have

|〈 f, HI 〉| ≥ C|I |3/2

and therefore

En( f )22 ≥ C∑|I |<2−n

|〈 f, HI 〉|2 ≥ C∑

|I |<2−n,I⊂(1/4,3/4)d|I |3 ≥ C

(n(d−1)/22−n

)2.

This example shows, in particular, that the relation (5.1) is not correct for the case


r = 1 and approximation using the Haar system. However, we do not know if (5.1) iscorrect for the Haar system in the case 0< r < 1. Note that the classWr (L p(Rd)) canbe defined for 0< r < 1 using fractional derivatives.

We next discuss the Bernstein inequalities for Daubhechies wavelets. Let againψ =Dk, k > 1. We can use Theorem 3.5 to establish a Bernstein inequality. To apply thistheorem, we need to verify assumptions (A1), (A2′′), (A3), (A4). We have already notedthat (A1), (A3), (A4) hold for anyr ≤ k. Letρ = ρ(k) be the maximum of allα such thatDk has Holder smoothness of orderα. There are several papers dealing with the valuesof ρ. A discussion of this topic can be found, for example, in Chapter 7 of Daubechiesbook [Da]. It is known that

c0k ≤ ρ(k, p) ≤ c1k(5.2)

for constants 0< c0, c1 < 1. For example, it is known that for sufficiently largek, theconstantc0 = 0.20754 suffices.

We can again use Corollary 4.3 to show that( A2′′) holds for allr < ρ, i.e.,{DrψI }I∈D(Rd)

≺ {HI }I∈D(Rd). The decay ssumption (A5) and the moment condition (A7) of that corol-lary is satisfied becauseDrψ has compact support. The definition ofρ andr < ρ givesthatDrψ has Lipschitz smoothness and therefore (A6) follows. So all the conditions ofCorollary 4.3 are satisfied and property (A2′′) follows.

In summary, we have shown

Theorem 5.1. Letψ = Dk, k = 2, 3, . . . , be one of the Daubechies wavelets. Then:

(i) The Jackson inequality(3.8)holds for r= k− 1.(ii) If r ≤ c0k with c0 any constant for which(5.2) is valid, then the Bernstein

inequality(3.8)holds for this value of r.(iii) For the r in (ii) and for any0 < α < r , the approximation classAαq(L p(Rd)),

1< p <∞, of (1.5) is identical with the class of functions satisfying(1.2)withK ( f, t) the K-functional of(1.6).

Acknowledgments. The authors wish to thank Professor Pencho Petrushev for severalvaluable discussions concerning the material in this paper. The authors would also liketo thank the referee for many constructive comments about this paper. This research wassupported by the Office of Naval Research Contract N0014-91-J1343 and the NationalScience Foundation Grant EHR 9108772 and was completed while S. V. Konyagin wasa visiting scholar at the University of South Carolina. S. V. Konyagin was supported inpart by the Russian Fund of Fundamental Investigations Contract N 93-01-0024.

References

[BDR] C. DE BOOR, R. A. DEVORE, A. RON (1993):On the construction of multivariate(pre) wavelets.Const. Approx.,9:123–166.

[CW] C. K. CHUI, J. Z. WANG (1992):On compactly supported spline wavelets and a duality principle.Trans. Amer. Math. Soc.,330:903–915.

[CDF] A. COHEN, I. DAUBECHIES, J.-C. FEAUVEAU (1992): Biorthogonal bases of compactly supportedwavelets. Comm. Pure Appl. Math.,45:485–560.

[Da] I. DAUBECHIES(1992): Ten Lectures on Wavelets. CBMS–NSF Regional Conference Series in AppliedMathematics, vol. 61. Philadelphia, PA: SIAM.


[DJP] R. DEVORE, B. JAWERTH, V. POPOV(1992):Compression of wavelet decompositions. Amer. J. Math.,114:737–785.

[DL] R. DEVORE, G. LORENTZ(1993): Constructive Approximation. Heidelberg: Springer-Verlag.[DPT] R. DEVORE, P. PETRUSHEV, V. TEMLYAKOV (1994):Multivariate trigonometric approximation with

frequencies from the hyperpolic cross. Mat. Zametki,56(3):36–63; English translation in Math. Notes,56.

[Di] J. DIESTEL (1984): Sequences and Series in Banach Spaces. Graduate Series in Mathematics. NewYork: Springer-Verlag.

[FS] C. FEFFERMAN, E. STEIN (1972):H p spaces of several variables. Acta Math.,129:137–193.[FJ] M. FRAZIER, B. JAWERTH (1990):A discrete transform and decomposition of distribution spaces.

J. Funct. Anal.,93:34–170.[FJM] M. FRAZIER, B. JAWERTH, G. WEISS (1991): Littlewood–Paley Theory and the Study of Function

Spaces. CBMS Regional Conference Series in Mathematics, vol. 79. Providence, RI: AmericanMathematical Society.

[K] A. K AMONT (1996):On hyperbolic summation and hyperbolic moduli of smoothness. Const. Approx.,12:111–125.

[KS] B. S. KASHIN, A. A. SAAKYAN (1989): Orthogonal Series. Providence, RI: American MathematicalSociety.

[M] Y. M EYER (1990): Ondelettes et Op´erateurs, vols. 1 and 2. Paris: Hermann.[T] V. T EMLYAKOV (1989):Approximation of functions with bounded mixed derivative. Proc. Steklov

Inst. Math.,178:1–122.[Z] A. Z YGMUND (1977): Trigonometric Series, vols. I and II. London: Cambridge University Press.

R. A. DeVoreDepartment of MathematicsUniversity of South CarolinaColumbiaSC 29208USA

S. V. KonyaginDepartment OPU, Mech.-Math.Moscow State UniversityLeninskie GoryMoscow 117234Russia

V. N. TemlyakovDepartment of MathematicsUniversity of South CarolinaColumbiaSC 29208USA

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