CONSTRUCTIVE APPROXIMATION

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1 Constr. Arox. (1998) 14: 1 26 CONSTRUCTIVE APPROXIMATION 1998 Sringer-Verlag New York Inc. Hyerbolic Wavelet Aroximation R. A. DeVore, S. V. Konyagin, and V. N. Temlyakov Abstract. We study the multivariate aroximation by certain artial sums (hyerbolic wavelet sums) of wavelet bases formed by tensor roducts of univariate wavelets. We characterize saces of functions which have a rescribed aroximation error by hyerbolic wavelet sums in terms of a K -functional and interolation saces. The results arallel those for hyerbolic trigonometric cross aroximation of eriodic functions [DPT]. 1. Introduction Let ϕ be a univariate function that satisfies multiresolution analysis (see, e.g., [Da] for a descrition of multiresolution analysis). We denote by S := S(ϕ) the shift-invariant sace which is defined as the L 2 (R)-closure of finite linear combinations of the shifts ϕ( j), j Z,ofϕ. By dilation, we obtain the univariate saces S k := S k (ϕ) := {S(2 k ):S S}, k Z. We obtain univariate wavelets ψ by considering rojectors P k from L 2 (R) onto S k. The wavelet sace W k 1 is defined to be the image of Q k := P k P k 1. Wavelet functions ψ are generators of the shift-invariant sace W := W 0, i.e., W = S(ψ). Wehavein mind here the usual orthogonal wavelets in the case the P k are orthogonal rojectors and the biorthogonal wavelets (see [CDF]) obtained when considering certain oblique rojectors P k. From the univariate wavelet ψ, we can construct efficient bases for L 2 (R) and other function saces by dilation and shifts. For examle, the functions ψ j,k := 2 k/2 ψ(2 k j), j, k Z, form a stable basis (orthogonal basis in the case of an orthogonal wavelet ψ) for L 2 (R). It is convenient to use a different indexing for the functions ψ j,k. Let D(R) denote the set of dyadic intervals. Each such interval I is of the form I = [ j2 k,(j +1)2 k ]. We define (1.1) ψ I := ψ j,k, I = [ j2 k,(j +1)2 k ]. Thus the basis {ψ j,k } j,k Z is the same as {ψ I } (R). Date received: October 16, Date revised: August 28, Communicated by Carl de Boor. AMS classification: 41A63, 46C99. Key words and hrases: Hyerbolic wavelets, Multivariate wavelets, Interolation saces. 1

2 2 R. A. DeVore, S. V. Konyagin, and V. N. Temlyakov Multivariate wavelets are usually obtained from multiresolution analysis on the tensor roduct saces S k S k. For examle, in the case of bivariate aroximation, this leads to the bivariate wavelet basis consisting of all functions (1.2) ϕ j,k (x)ψ j,k(y) ψ j,k (x)ϕ j,k(y) ψ j,k (x)ψ j,k(y), with j, j, k Z. The aroximation roerties of these wavelets are now well understood (see, e.g., [M] or for nonlinear aroximation [DJP]). A second way to construct multivariate wavelet bases is to simly take tensor roducts of the univariate basis functions ψ j,k.ifψis a univariate wavelet and d 1, then the functions (1.3) ψ j1,k 1 (x 1 ) ψ jd,k d (x d ), j = ( j 1,..., j d ) Z d, k =(k 1,...,k d ) Z d, are a basis for L 2 (R d ). There is quite a distinction between these two wavelet bases. The functions in (1.2) have roughly the same suort in each coordinate direction while the tensor roducts of (1.3) have suort which is scaled indeendently in the different coordinate directions. As we shall see, this is also reflected in the aroximation roerties of the two sets of wavelets. Again, it is more convenient to use another indexing for the basis functions (1.3). We let D(R d ) denote the set of all dyadic rectangles in R d.any(r d )is of the form I = I 1 I d with I 1,...,I d D(R). We define (1.4) ψ I (x 1,...,x d ):=ψ I1 (x 1 ) ψ Id (x d ), I D(R d ). Therefore, the wavelet basis (1.3) is the same as the set of functions {ψ I } (R d ). We are interested in the aroximation roerties of the functions (1.4). For n = 0, 1,...and 0 <, let H n := H n (L (R d )) := san{ψ I : I > 2 n } denote the closed linear san of the finite linear combinations of the functions ψ I, I > 2 n, with the closure taken with resect to the L (R d )-(quasi-)norm. We call the aroximation by H n hyerbolic wavelet aroximation in analogy with the aroximation by trigonometric olynomials with frequencies from the hyerbolic cross (see Temlyakov [T] for a discussion of this tye of aroximation). We can also describe H n in terms of the scaling function ϕ. Namely, H n is the closed linear san of the functions ϕ I, I 2 n. In most wavelet alications, aroximation occurs over a comact subset of R d and the aroximation takes lace from a finite-dimensional linear subsace H n of H n. The resent aer is concerned with the aroximation efficiency of the saces H n.to measure this efficiency, we introduce the following aroximation error. For 0 < and f L (R d ), we define E n ( f ) := E( f, H n ) := inf g H n f g with here and later the L (R d )-norm.

3 Hyerbolic Wavelet Aroximation 3 We are interested in characterizing functions which have a given order of aroximation. For α>0and 0 <, q, we define Aq α(l (R d )) as the collection of all functions f L (R d ) such that {( k 0 f A α q (L (R d )) := [2kα E k ( f ) ] q) 1/q, 0 < q <, (1.5) su k 0 2 kα E k ( f ), q =, is finite. We define the norm on the aroximation sace A α q (L (R d )) by f A α q (L (R d )) := f + f A α q (L (R d )). The characterization of the aroximation classes A α q (L (R d )) is tantamount to roving two inequalities (called Jackson and Bernstein inequalities) for the aroximation rocess (see [DL, Cha. 7]). In the case of hyerbolic wavelet aroximation, these inequalities involve mixed derivatives. If r is a ositive integer, we define the differential oerator D r := r x1 r xd r. For 1,weletW r (L (R d )) be the set of all functions f in L (R d ) whose distributional derivative D r f is in L (R d ) and define the seminorm on W r (L (R d )) by r f W r (L (R d )) := D r f. We will show in Section 3, that under certain conditions on the function ψ and for 1 < <, we have the following two inequalities: (J) E n ( f ) C2 nr f W r (L (R d )), n = 0, 1,..., f W r (L (R d )), with C indeendent of n and f and (B) g W r (L (R d )) C2 nr g, g H n, n = 0, 1,..., with C indeendent of n and g. From these Jackson and Bernstein inequalities for hyerbolic wavelet aroximation it follows that we can characterize the saces A α q (L (R d )) in terms of the K -functional (1.6) K ( f, t) := K ( f, t, L (R d ), W r (L (R d ))). Namely, for 0 < q <, a function f L (R d ) is in A α q (L (R d )) if and only if (1.7) 0 [t α K ( f, t r )] q dt t <. A similar result holds for q = with the integral relaced by a su. In the case of eriodic functions, we have shown in [DPT] that K ( f, t) is equivalent to a certain modulus of smoothness r ( f, t) based on mixed differences. In rincile, this result should carry over to the case of aroximation on R d ; however, we have not yet carried out the details of this equivalence.

4 4 R. A. DeVore, S. V. Konyagin, and V. N. Temlyakov Our roof of the Jackson and Bernstein inequalities for hyerbolic wavelet aroximation rests on the characterization of functions f in L (R d ) by a hyerbolic wavelet series f = c I ( f )ψ I (R d ) and the calculation of its norm by the following square function ( ) 1/2 c I ψ I [c I ψ I ] 2 (1.8). Characterizations (1.8) are at the heart of what is called the Littlewood Paley theory for wavelets. Littlewood Paley theory has a long and imortant history in harmonic analysis. For the most art, we will utilize known asects of this theory adated to the case of hyerbolic wavelet decomositions. We describe the results from Littlewood Paley theory that we will need and give their adatation to hyerbolic wavelets in Section 2. There are several results which establish sufficient conditions for the family of functions {ψ I } (R d ) to satisfy (1.8). However, these conditions are not always alicable in wavelet theory since they require smoothness of ψ not met by wavelets or their derivatives. We therefore establish (in Section 4) sufficient conditions which allow only iecewise Lischitz continuity on the function ψ for the Littlewood Paley characterization to hold. Finally, in Section 5 we give an alication of our results to the Daubechies wavelets. Other wavelets can be handled in a similar manner. As mentioned earlier, we will restrict our develoment in this aer to aroximation on R d. We could also give a similar develoment for the case of aroximation on a comact set in R d or on the torus T d.in this way, our aroach could be alied to other wavelet-like bases such as the Franklin system and its generalizations. While rearing the resent aer, we were sent a rerint by A. Kamont [K] that roves Jackson and Bernstein inequalities for the Franklin system by utilizing Littlewood Paley theory. 2. The Elements of Littlewood Paley Theory Littlewood Paley theory gives a way of characterizing norms of linear combinations of certain basis functions. Its roots lie in the Littlewood Paley theorems for Fourier series in which case the basis functions are the comlex exonentials e k (x) := e ik x = e i(k 1x 1 + +k d x d ), x R d. However, the theory alies to many other orthogonal and nonorthogonal exansions (see, e.g., [FJ], [FJM], or [M]). For us, Littlewood Paley theory will rovide a vehicle to rove our results on multivariate wavelet aroximation. We begin in this section by introducing various forms of the Littlewood Paley theory for systems of functions. While the discussion we give is for the most art known, it will enable us to set the framework for this aer and introduce several imortant results which will be emloyed later in the aer.

5 Hyerbolic Wavelet Aroximation 5 Let D = D(R d ) denote the collection of all dyadic rectangles in R d. Thus, a rectangle I R d is in D if and only if I = I 1 I d with I 1,...,I d dyadic intervals in R. We will consider in this section systems of functions {η(i, )} I D. In the univariate case, one articular way to obtain such systems is by shifts and dilates of a univariate wavelet. For a univariate function ψ, we use the notation ψ I of (1.1) to denote its L 2 (R)-normalized, shifted dilates. The function ψ is an orthogonal wavelet if the collection of functions ψ I, I D(R), forms a comlete orthonormal system for L 2 (R). Other cases of interest in wavelet theory are the rewavelets [BDR], sline wavelets [CW], and biorthogonal wavelets [CDF]. In the latter cases, the orthogonality of the family ψ I, I D(R), is relaced by L 2 -stability (Riesz basis). Given a univariate function ψ, we can obtain a multivariate family of functions by taking tensor roducts. For rectangles I D(R d ), we define (2.1) ψ I (x 1,...,x d ):=ψ I1 (x 1 ) ψ Id (x d ), I = I 1 I d. We will use the notation ψ I to denote the family of functions obtained by tensor roducts of shifted dilates of a univariate function ψ and will use the notation η(i, ) to denote families of functions indexed on I D(R d ) that are not necessarilly obtained by shifted dilates of one function. This notation will distinguish between the sace dimension by the Euclidean dimension of the index rectangles. A articularly imortant examle occurs when we take for ψ the univariate Haar wavelet { +1, 0 x 1 H(x) := 1 1, 2 < x 1. The Haar function H is the simlest examle of a univariate orthogonal wavelet. If 1 < <, we say that a family of real-valued functions η(i, ), I D, satisfies the strong Littlewood Paley roerty for, if for any finite sequence (c I ) of real numbers, we have ( ) 1/2 c I η(i, ) [c I η(i, )] 2 (2.2) with constants of equivalency deending at most on and d. Here and later we use the notation A B to mean that there are two constants C 1, C 2 > 0 such that I D C 1 A B C 2 A. We will indicate what the constants deend on (in the case of (2.2) they may deend on and d). Here is another useful remark concerning (2.2). From the validity of (2.2) for finite sequences, we can deduce its validity for infinite sequences by a limiting argument. For examle, if (c I ) is an infinite sequence for which the sum on the left side of (2.2) converges in L (R d ) with resect to some ordering of the I D, then the right side of (2.2) will converge with resect to the same ordering and the right side of (2.2) will be less than a multile of the left. Likewise, we can reverse the roles of the left- and right-hand sides. Similar remarks hold for other statements like (2.2) made in this aer. The term strong Littlewood Paley inequality is used to differentiate (2.2) from other ossible forms of Littlewood Paley inequalities. For examle, the Littlewood Paley 2,

6 6 R. A. DeVore, S. V. Konyagin, and V. N. Temlyakov inequalities for the comlex exonentials take a different form (see [Z, Cha. XV]). Another form of interest in our considerations is the following: ( ) 1/2 c I η(i, ) [c I χ I ] 2 (2.3). We use the notation χ for the characteristic function of [0, 1] and χ I for its L 2 (R d )- normalized, shifted dilates given by (2.1) (with ψ = χ). The two forms (2.2) and (2.3) are equivalent under very mild conditions on the functions η(i, ). To see this, we will use the Hardy Littlewood maximal oerator, which is defined for a locally integrable function g on R d by 1 Mg(x) := su g(y) dy J x J J with the su taken over all cubes J that contain x. It is well known that M is a bounded oerator on L (R d ) for all 1 <. The Fefferman Stein inequality [FS] bounds the maing M on sequences of functions. We shall only need the following secial case of this inequality which says that for any functions η(i, ) and constants c I, I D, we have for 1 <, ( ) 1/2 ( (c I Mη(I, )) 2 2)1/2 (2.4) A I η(i, )) I D (c with A a constant deending only on the sace dimension d. Consider now as an examle, the equivalence of (2.2) in the univariate case. If the univariate functions η(i, ), I D, satisfy (2.5) I D η(i, x) CMχ I (x), χ I (x) CMη(I, x), a.e. x R, 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 on η(i, ). For examle, if η(i, ) is given by the normalized, shifted-dilates of the function ψ, η(i, ) = ψ I, then the left inequality in (2.5) holds whenever ψ(x) C[max(1, x )] λ, a.e. x R, with λ 1. The right condition in (2.5) is extremely mild. For examle, it is always satisfied in the case that the family η(i, ) is generated by the shifted dilates of a nonzero function ψ. The Littlewood Paley inequalitites are intimately connected with unconditional bases. Given a family of functions {η(i, )} I D from L (R d ), we define its san X in L (R d ) as the L (R d )-closure of the linear sace sanned by its finite linear combinations. The ordered family {η(i, )} I D is a basis for X if each element f X has a unique reresentation (2.6) f = c I η(i, ). In describing the convergence of the series (2.6), we should secify the ordering (i.e., the artial sums). We will only consider unconditional bases (described in a moment) in

7 Hyerbolic Wavelet Aroximation 7 which order is not imortant. However, for comleteness of the definition of a basis, we will take the artial sums S n of the series (2.6) to consist of the sum over all rectangles I = I 1 I d such that 2 n I j 2 n, j=1,...,d. We recall that a basis η(i, ), I D(R d ), is said to be unconditional for L (R d ) if for each assignment ε I :=±1, I D(R d ), and each finite sequence c I, I D(R d ),we have (2.7) ε I c I η(i, ) c I η(i, ) with constants of equivalency indeendent of the sequences (c I ) and (ε I ). If a basis is unconditional, then the uer estimate in (2.7) also holds for any sequence (ε I ) taking the values 0, 1. From this, it follows easily that the series (2.6) converges indeendently of the ordering. We take for granted the known fact (see, e.g., [KS]) that for each 1 < <, the univariate Haar family H I, I D(R), satisfies the strong Littlewood Paley roerty. These functions also form an unconditional basis for L (R), for all 1 < < ; this can be found in [KS] and also follows from Lemma 2.2 below. We want next to conclude from this that the multivariate Haar system H I, I D(R d ), also satisfies the strong Littlewood Paley roerty. Let r j (t) := sign(sin 2 j+1 πt), t [0, 1], j = 0, 1,..., be the univariate Rademacher functions. We take any one-to-one corresondence of the natural numbers with the rectangles of D(R). This gives an indexing r(i, ), I D(R), of the Rademacher functions. In R d,welet r(i,(x 1,...,x d )) := r(i 1, x 1 ) r(i d,x d ), I = I 1 I d, be the tensor roducts of the Rademacher functions. We recall Khinchine s inequality (see [KS]) which says that for 1 < and for any finite sequence c I, I D(R d ), we have ( ) 1/2 (2.8) c I r(i, ) c I 2. L ([0,1] d ) Lemma 2.1. Let 1 < < and let ψ be a univariate function such that the univariate family ψ I, I D(R), is an unconditional basis for L (R). Then the multivariate family ψ I, I D(R d ), satisfies the Littlewood Paley roerty (2.2) for this value of. Proof. For notational simlicity, we give the roof only in the case d = 2. Let c I, I D(R 2 ), be a sequence with finitely many nonzero terms, and let I = I 1 I 2. Then, using the unconditionality of the univariate basis, for each t 1, t 2 [0, 1] which are not endoints of dyadic intervals, we have with constants of equivalency deending at most on and ψ, (2.9) c I r(i,(t 1,t 2 ))ψ I (x 1, x 2 ) dx 1 dx 2 R R (R 2 ) [ ] = r(i 1, t 1 ) c I r(i 2, t 2 )ψ I2 (x 2 ) ψ I1 (x 1 ) dx 1 dx 2 R R I 1 D(R) I 2 D(R) I D

8 8 R. A. DeVore, S. V. Konyagin, and V. N. Temlyakov = = R R R R R R R R I 1 D(R) I 2 D(R) I 2 D(R) I D(R 2 ) [ ] c I r(i 2, t 2 )ψ I2 (x 2 ) ψ I1 (x 1 ) dx 1 dx 2 I 2 D(R) [ ] r(i 2,t 2 ) c I ψ I1 (x 1 ) ψ I2 (x 2 ) dx 1 dx 2 I 1 D(R) [ ] c I ψ I1 (x 1 ) ψ I2 (x 2 ) dx 1 dx 2 I 1 D(R) c I ψ I (x 1,x 2 )) dx 1 dx 2. We now integrate (2.9) with resect to t 1, t 2 [0, 1] and interchange the order of integration in the first term. By Khinchine s inequalities (2.8) with resect to the norm in t 1, t 2, the first term of (2.9) is equivalent to ( ) /2 c I ψ I (x 1, x 2 ) 2 dx 1 dx 2. R R (R 2 ) Comaring this with the last term in (2.9), we see that we have roved the lemma. It follows, in articular from Lemma 2.1, that the multivariate Haar functions H I, I D(R d ), satisfy the strong Littlewood Paley roerties (2.2) and (2.3) (note that H I =χ I ). Lemma 2.2. Let 1 < < and let η(i, ), I D(R d ), be any collection of multivariate functions. Concerning the following statements: (i) (η(i, )) satisfies the Littlewood Paley condition (2.2) for this value of ; (ii) (η(i, )) satisfies the Littlewood Paley condition (2.3) for this value of ; (iii) (η(i, )) (H I ) ; and (iv) (η(i, )) is an unconditional basis for L (R d ); we have that (i) and (iv) are equivalent, (ii) and (iii) are equivalent, and (ii) imlies (iv). Moreover, if (2.5) holds, then all these statements are equivalent. Proof. We leave the roof of this lemma to the reader. 3. Aroximation by Hyerbolic Wavelets In this section, we will discuss aroximation in L (R d ),1 < <, from the hyerbolic wavelet saces H n := H n (L (R d )). Let ψ be a univariate function and let ψ I, I D(R d ), be defined as in (2.1) and let H n be the closed linear san of the finite linear combinations of the ψ I with I > 2 n.forn=0,1,..., and f L (R d ),we

9 Hyerbolic Wavelet Aroximation 9 define E n ( f ) := E( f, H n ) := inf g H n f g with here and later the L (R d ) norm. Our main interest in this aer is the characterization of the aroximation saces Aq α(l (R d )) defined by (1.5). We will characterize the saces Aq α(l (R d )) by roving Bernstein and Jackson inequalities for L (R d ) and W r (L (R d )) (recall the saces W r (L (R d )) and the oerator D r of the Introduction). We will give two aroaches to roving comanion Jackson and Bernstein inequalities in this section. The first aroach assumes certain conditions on ψ that allow us to characterize W r (L (R d )),1< <, in terms of exansions in the basis ψ I, I D(R d ). Using this characterization, we will then easily rove the Jackson and Bernstein inequalities. Our second aroach will assume weaker (and more easily verifiable conditions on ψ) that still allow the roof of the Jackson and Bernstein inequalities. We begin with the first aroach. If ψ I, I D(R d ), is a Schauder basis, then associated to this basis we have its dual basis. In the case that 1 < <, the dual basis is given by linear functionals c I with c I ( f ) = f (x)λ(i, x) dx R d and the functions λ(i, ) are in L (R d ) with 1/ + 1/ = 1. If λ(x) := λ([0, 1], x), then it is easy to see (by using shifts and dilations) that we can take λ(i, ) = λ I, I D(R d ), with the λ I defined as in (2.1). We note that in the case that ψ is suitably differentiable, we have (D r ψ) I = I r D r ψ I,(R d ). We will make for our first aroach the following assumtions about the multivariate basis ψ I, I D(R d ), and its dual basis λ I, I D(R d ): (A1) ψ I, I D(R d ), san L (R d ),1< <, and satisfy the Littlewood Paley inequalities (2.3); (A2) (D r ψ) I, I D(R d ), san L (R d ),1< <, and satisfy the Littlewood Paley inequalities (2.3); (A3) R x j λ(x) dx = 0, j = 0,...,r; and (A4) λ(x) Cmax(1, x r 1 ε ), for some ε>0. Because of (A3) and (A4), we can integrate the univariate function λ, r times to find a function µ L (R) which satisfies ( 1) r µ (r) = λ. It follows that D r µ I = ( 1) rd I r λ I, I D(R d ). Integration by arts then shows that (D r ψ) I µ J dx = I r J r ψ I λ J dx = δ(i, J), I, J D(R d ), R d R d with δ the Kronecker delta. Hence, µ I, I D(R d ), is the dual basis for (D r ψ) I, I D(R d ). Theorem 3.1. Let r be a ositive integer, 1< <,and let ψ be a univariate function which satisfies assumtions (A1) (A4). Then a function f L (R d ) is in W r (L (R d )) if and only if (3.1) f = c I ( f )ψ I (R d )

10 10 R. A. DeVore, S. V. Konyagin, and V. N. Temlyakov with (3.2) ( (R d ) [ I r c I ( f ) χ I ] 2 ) 1/2 <. Furthermore, the left side of (3.2) is equivalent to f W r (L (R d )). Proof. Suose first that f W r (L (R d )). Assumtion (A1) gives that the functions ψ I, I D(R d ), satisfy the strong Littlewood Paley inequalities. Hence, these functions are an unconditional basis for L (R d ) and f = c I ( f )ψ I with (R d ) c I ( f ) = R d f λ I dx. Likewise, the functions (D r ψ) I, I D(R d ), are also a basis for L (R d ), and we have D r f = d I ( f )(D r ψ) I with (R d ) (3.3) d I ( f ) = D r f µ I dx = ( 1) rd fd r µ I dx = I r fλ I dx = I r c I (f). R d R d R d We can comute D r f from the Littlewood Paley condition for the basis (D r ψ) I, I D(R d ). This gives that the left side of (3.2) is equivalent to D r f. Conversely, assume that f L (R d ) is such that (3.2) is finite. Because ψ I, I D(R d ), is an unconditional basis for L (R d ),wehave f = c I (f)ψ I I D(R d ) in the sense of L (R d )-convergence. From (3.2) and the fact that (D r ψ) I, I D(R d ), satisfies the Littlewood Paley inequalities, we find that there is a function g L (R d ) with g = I r c I ( f )(D r ψ) I (R d ) again in the sense of L (R d )-convergence. We comute the coefficients of g with resect to the basis (D r ψ) I ), I D(R d ), and find gµ I = I r c I (f)= I r fλ I =( 1) rd R d R d R d fd r µ I. This shows that g is the distributional derivative D r f and therefore f W r (L (R d )).

11 Hyerbolic Wavelet Aroximation 11 The following theorem gives the Jackson and Bernstein inequalities for aroximation by the saces H n. Theorem 3.2. Let r be a ositive integer, 1< <,and let ψ be a univariate function which satisfies assumtions (A1) (A4). If f W r (L (R d )), then (3.4) E n ( f ) C2 nr f W r (L (R d )), n = 0, 1,..., with C indeendent of n and f. If g H n (L (R d )), then (3.5) g W r (L (R d )) C2 nr g, n = 0, 1,..., with C indeendent of n and g. Proof. First, let f W r (L (R d )). Then f = c I ( f )ψ I (R d ) in the sense of L (R d )-convergence and (3.2) is satisfied. We let g := I >2 c n I ( f )ψ I which is a function in H n (L (R d )). The remainder f g is given by f g = c I ( f )ψ I. I 2 n We can estimate f g by using the Littlewood Paley inequalities: ( ) 1/2 ( ) 1/2 f g C [ c I ( f ) χ I ] 2 C2 nr [ I r c I ( f ) χ I ] 2 I 2 n I 2 n C2 nr ( I D(R d ) [ I r c I ( f ) χ I ] 2 ) 1/2 C2 nr D r f. This roves (3.4). Suose now that g H n (L (R d )). Then, g = c I (g)ψ I, I >2 n and from Theorem 3.1, we have ( ) 1/2 D r g C [ I r c I (g) χ I ] 2 I >2 n ( ) 1/2 C2 nr [c I (g) χ I ] 2 C2 nr g I >2 n because ψ I, I D(R d ), satisfies the Littlewood Paley inequalities (2.3). This roves (3.5).

12 12 R. A. DeVore, S. V. Konyagin, and V. N. Temlyakov As we have mentioned earlier in this section, the Jackson and Bernstein inequalities (3.4) and (3.5) allow the characterization of the aroximation saces A α q (L (R d )).For this, we will use the K -functional (3.6) K ( f, t) := K ( f, t, L (R d ), W r (L (R d ))) := inf f g + t g W r (L (R d )). g W r (L (R d )) We recall the interolation saces (L (R d ), W r (L (R d ))) θ,q defined by the real method of interolation (see Chater 6 of [DL]). Corollary 3.3. Let r be a ositive integer, 1< <,0<q, and 0 < α < r. Let ψ be a univariate function which satisfies assumtions (A1) (A4). Then, f A α q (L (R d )) if and only if f (L (R d ), W r (L (R d ))) α/r,q with equivalent norms. Proof. This corollary follows from Theorem 3.2 and general facts about aroximation and K -functionals that can be found in Chater 7 of [DL]. While the above aroach is simle and direct, the assumtion (A2) is too severe for some alications. It is also uncomfortable to make such an assumtion for a Jackson inequality since it is unclear why the Jackson inequality should deend on the smoothness of ψ. Recall that in the univariate case of wavelet aroximation, Jackson inequalities deend only on conditions (A1), (A3), and (A4). We shall therefore now give a second aroach to roving the Jackson and Bernstein inequalities which searates the roof of these inequalities. This aroach allows us to rove the Jackson inequality under a much weaker assumtion than (A2). We first consider the Jackson inequality. Suose that we have in hand two multivariate families η(i, ), µ(i, ), I D(R d ). We will use the notation {η(i, )} I D {µ(i, )} I D, if there is a constant C > 0 such that (3.7) c I η(i, ) C c I µ(i, ) holds for all finite sequences (c I ) with C indeendent 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 families η(i, ), µ(i, ), I D(R d ), we define the oerator T which mas µ(i, ) into η(i, ) for all I D and we extend T to finite linear combinations of the µ(i, ) by linearity. Then (3.7) holds if and only if T is a bounded oerator with resect to the L -norm and {µ(i, )} I D {η(i, )} I D holds if and only if T has a bounded inverse with resect to the L -norm. We recall the Haar basis H I, I D(R d ). In lace of (A2), we will assume (A2 ) {µ I } (R d ) {H I } I D(R d ), where as before µ satisfies µ (r) = ( 1) r λ. It follows from (A2 ) that the oerator T defined by Tf := f,h I µ I I D(R d ) I D

13 Hyerbolic Wavelet Aroximation 13 is bounded on L (R d )) for each 1 < <. Hence, by duality, its adjoint T f := f,µ I H I (R d ) is also bounded on L (R d )), for each 1 < <. Theorem 3.4. Assume that (A1), (A2 ), (A3), and (A4) hold. If 1 < <, risa ositive integer and f W r (L (R d )), then (3.8) E n ( f ) C2 nr f W r (L (R d )), n = 0, 1,..., with C indeendent of n and f. Proof. Let f W r (L (R d )). From assumtion (A1), we have (see (3.3)) f = c I ( f )ψ I, c I ( f ) = f,λ I = I r D r f,µ I, (R d ) in the sense of L (R d )-convergence. We let g := I >2 c n I ( f )ψ I which is a function in H n (L (R d )). We can estimate the remainder f g by f g = c I ( f )ψ I C c I ( f )H I I 2 n I 2 n ( ) 1/2 = C I r D r f,µ I H I C [ I r D r f,µ I χ I ] 2 I 2 n I 2 n ( ) 1/2 C2 nr [ D r f,µ I χ I ] 2 C2 nr I 2 n I D(R d ) D r f,µ I H I C2 nr D r f, where the last inequality uses the boundedness of the adjoint oerator T (which follows from (A2 )). We next consider the Bernstein inequality. In lace of (A2), we will assume that (A2 ) {(D r ψ) I } I D(R d ) {H I } I D(R d ). Theorem 3.5. Assume that (A1), (A2 ), (A3), and (A4) hold. Then, for each ositive integer r and each g H n (L (R d )),1< <,we have (3.9) D r g C2 nr g, n = 0, 1,..., with C indeendent of n and g.

14 14 R. A. DeVore, S. V. Konyagin, and V. N. Temlyakov Proof. Because of (A1), we can write (3.10) g = c I (g)ψ I. I >2 n In the roof of (3.9), it is enough to consider functions g for which the sum in (3.10) has only a finite number of terms; the general case follows by a limiting argument. In this case, we have D r g = c I (g)d r ψ I = I r c I (g)(d r ψ) I I >2 n I >2 n C I r c I (g)h I C I r c I (g)ψ I I >2 n I >2 n C2 nr c I (g)ψ I = C2 nr g. I >2 n 4. Sufficient Conditions for the Littlewood Paley Inequalities In the revious section, we have characterized the aroximation saces for hyerbolic wavelet aroximation under certain assumtions on the univariate functions ψ and ψ (r) relating to Littlewood Paley theory. For many functions ψ that occur in wavelet theory, it is ossible to utilize the existing Littlewood Paley theory to verify these assumtions. However, in some instances (e.g., for sline wavelets), the alication of the existing theory will not give the largest ossible value of r because this theory requires global smoothness of ψ (resectively ψ (r) ). The urose of the resent section is to rove a Littlewood Paley theorem which does not require global smoothness (rather it is enough to have certain iecewise continuity). We shall also address some related questions associated with Littlewood Paley theory. It is ossible to formulate our theorems without assuming that the family of functions under consideration are all shifted-dilates of a single function. We shall therefore revert back to our notation η(i, ), I D := D(R 1 ), to denote an arbitrary family of functions indexed on dyadic intervals. The strong Littlewood Paley inequalities (2.3) are the same as the equivalence {η(i, )} {H I }. We begin this section by discussing sufficient conditions in order that {η(i, )} {H I }. Let ξ I, I D, denote the center of the dyadic interval I. We will assume in this section that η(i, ), I D, is a family of univariate functions that satisfy the following assumtions: (A5) There is an ε>0, and a constant C 1 such that for all t R and all J D, we have η(j,ξ J +t J ) C 1 J 1/2 (1+ t ) 1 ε. (A6) There is an ε>0and a constant C 2 and a artition of [ 1 2, 1 ] into intervals 2 J 1,...,J m that are dyadic with resect to [ 1 2, 1 ], such that for any J D,any 2 j Z, and any t 1, t 2 in the interior of the same interval J k, k = 1,...,m,we

15 Hyerbolic Wavelet Aroximation 15 have η(j,ξ J +j J +t 1 J ) η(j,ξ J +j J +t 2 J ) C 2 J 1/2 (1+ j ) 1 ε t 2 t 1 ε, (A7) For any J D,wehave η(j, x) dx = 0. R In the case that η(j, ) = ψ J for a function ψ, it is enough to check these assumtions for J = [0, 1], i.e., for the function ψ alone. They follow for all other dyadic intervals J by dilation and translation. The condition (A5) is a standard decay assumtion and (A7) is the zero moment condition. The condition (A6) requires that the functions η(i, ) be iecewise in Li ε. Let T be the linear oerator which satisfies ( ) (4.1) T c I H I = c I η(i, ) for each finite linear combination c I H I of the H I. We wish to show that ( ) T c I H I C c I H I for each such sum. From this it would follow that T extends (by continuity) to a bounded oerator on all of L (R) and therefore {η(i, )} {H I }. We can exand η(j, ) into its Haar decomosition. Let (4.2) λ(i, J) := η(j, x)h I (x) dx, so that R η(j, ) = I D λ(i, J)H I. It follows that ( ) T c J H J = (4.3) λ(i, J)c J H I. J D J D We see that the maing T is tied to the bi-infinite matrix := (λ(i, J)) I,J D which mas the sequence c := (c J ) into the sequence (c I ) := c. One aroach to roving Littlewood Paley inequalities is to show that the matrix decays sufficiently fast away from the diagonal (see [FJ, 3]). Following [FJ], we say that a matrix A = (a(i, J)) I,J D is almost diagonal if for some ε>0, we have (4.4) with (4.5) ω(i, J) := a(i, J) Cω(I, J) ( 1 + ξ ) I ξ J 1 ε ( ( I min max( I, J ) J, J )) (1+ε)/2. I We will use the following secial case of a theorem of Frazier and Jawerth [FJ, Theorem 3.3] concerning almost diagonal oerators.

16 16 R. A. DeVore, S. V. Konyagin, and V. N. Temlyakov Theorem 4.1. If (a(i, J)) I,J D is an almost diagonal matrix, then the oerator A defined by ( ) A c J H J := (4.6) a(i, J)c J H I J D J D is bounded on L (R) for each 1 < <. Proof. For the comleteness of the resent aer, we give the following roof of this theorem. Let W be the oerator defined as in (4.6) with (ω(i, J)) I,J D in lace of (a(i, J)) I,J D. From the Littlewood Paley inequalities for the Haar functions, we have ( ) A c J H J C a(i, J) c J H I C ω(i, J) c J H I J D J D J D C W c J H J C W c J H J J D J D with W the norm of W as an oerator from L (R) into itself and with the constants C deending only on. 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), J I, ω + (I, J) := 0, J < I, and we let W + and W be defined as in (4.6) for these two sequences. We shall show that W + and W are bounded on L (R) which will comlete the roof of the theorem. Since the roof of boundedness in these two cases is similar, we shall only consider W +. We will emloy the following inequality for nonnegative sequences (b l ): l=1 b l l τ C τ max m 1 1 m m b l, τ > 1, which is easily roved by summation by arts. To bound W +, it is enough to consider its action on J D c J H J where the c J are nonnegative and only a finite number of them are nonzero. We let l=1 c I := J D ω + (I, J)c J. From the above inequality, it follows that for each interval I with I =2 ν and µ ν, we have ( 1 + ξ ) ( ) I ξ J 1 ε c J C2 µ/2 M c J χ max( I, J ) J, x J =2 µ J =2 µ = C2 µ/2 M( f µ, x), x I,

17 Hyerbolic Wavelet Aroximation 17 with M the Hardy Littlewood maximal oerator, and with f µ := ( ) 1/2 c J χ J = c 2 J χ 2, J J =2 µ J =2 µ and with C here and later in this roof deending on ε. Hence, from (4.5) I 1/2 c I C µ ν2 µε/2 2 νε/2 M( f µ, x), x I. Using the Cauchy Schwartz inequality, we obtain c I 2 χ 2 (x) C 2 (ν µ)ε/2 (M( f µ, x)) 2. I µ ν Since the functions χ 2 have disjoint suorts, we have I c I 2 χ 2 C 2 (ν µ)ε/2 (M( f µ )) 2. I I =2 ν µ ν Summing over ν Z gives c I 2 χ 2 C 2 (ν µ)ε/2 (M( f µ )) 2. I ν Z µ ν We now aly the Littlewood Paley inequalities for the Haar system and the Fefferman Stein inequality (2.4) to find ( ) ( ) 1/2 ( W + 1/2 c I H I C c I 2 χ 2 C 2 (ν µ)ε/2 f I µ) 2 ν Z µ ν ( 1/2 C f 2 µ) C c I H I, µ Z where the last inequality follows from the Littlewood Paley inequalities (2.3) for the Haar functions and the definition of the functions f µ. Theorem 4.2. If η(i, ), I D, satisfy assumtions (A5) (A7), then the oerator T defined by (4.1) is bounded from L (R) into itself for each 1 < <. Proof. For an interval I D, let I ± denote the dyadic intervals with I ± = I, which are immediately to the right and left of I, resectively. We define I := I I I +.For the λ(i, J) of (4.2), we define { λ(i, J), I J, λ (I, J) := 0, else, { λ(i, J), I > J, J I λ, (I, J) := 0, else, { λ(i, J), I > J, J I λ, (I, J) := 0, else.

18 18 R. A. DeVore, S. V. Konyagin, and V. N. Temlyakov Then, λ(i, J) = λ (I, J) + λ (I, J) + λ (I, J). We let T 1, T 2, T 3 be the oerators defined as in (4.3) with λ relaced by λ, λ, λ, resectively. We will show that each of these oerators is bounded from L (R) into itself. We first show that T 1 satisfies the conditions of Theorem 4.1. Let 2 α be the length of the smallest dyadic interval aearing in the statement of condition (A6). We first consider intervals I such that I 2 α J. Then from roerty (A6), there is a constant a such that ( η(j, x) a C 2 J 1/2 1+ ξ ) I ξ J 1 ε ( ) I ε, x int (I ). J J Indeed, for one of the intervals J k of assumtion (A6) and an aroriately chosen j, any x I can be written as x = ξ J + j J +t J with t J k. Using the last inequality, we obtain λ (I, J) H I (x)(η(j, x) a) dx I ( C 2 I 1/2 J 1/2 1+ ξ I ξ J J ) 1 ε ( ) I ε. J 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 relaced by 1. This shows that (λ (I, J)) I,J D is almost diagonal and hence the boundedness of T 1 on L (R) follows from Theorem 4.1. A similar calculation shows that (λ (I, J)) I,J D is almost diagonal and hence T 2 is also bounded on L (R) because of Theorem 4.1. The roof of the boundedness of T 3 will require more care since this oerator is not necessarily almost diagonal. We can decomose this oerator into a sum of four oerators corresonding to whether J is contained in the left- or right-half of I,orJis contained in I + or I. We can show that each of these oerators is bounded on L (R) in a similar way. We consider in detail the oerator A 0 corresonding to the left-half of I and show that it is bounded on L,1< <. Later we shall note the modifications that need to be made to handle the remaining three cases. We denote the left-half of intervals I D by I. Then A 0 has the associated matrix { λ(i, J), J I, λ 0 (I, J) : 0, else. We can write A 0 = A m, m=1 where A m has the associated matrix { λ(i, J), J =2 m I,J I, λ m (I, J) := 0, else.

19 Hyerbolic Wavelet Aroximation 19 We will show that for a certain δ>0 (deending on ), (4.7) A m C2 mδ, with denoting the norm of the oerator A m from L (R) into itself, and with C indeendent of m. This will then comlete the roof of the theorem. We recall that an oerator on L (R) has the same norm as its adjoint on L,1/+ 1/ =1. It is therefore enough to show that the adjoint oerators A m satisfy (4.7) for each 1 < <. The oerator A m is defined by A m ( ) c J H J = J D J D I =2 m J I J λ m (I, J)c I H J. We first estimate λ m (I, J), I =2 m J,J I. Let C(I ) denote the comlement of I. Using assumtions (A7) and (A5), we have (4.8) λ m (I, J) = η(j, x)h I (x) dx R ( ) I I 1/2 η(j, x) dx I\I + η(j, x) dx 2 I 1/2 η(j, x) dx C(I ) 2C 1 I 1/2 J 1/2 C J 1/2 I 1/2 k(i, J) ε, C(I ) where k(i, J) 1 is the largest integer k such that dist(ξ J, C(I )) k J /2. ( 1+ x ξ ) 1 ε J dx J Let c J be the H J -coefficient of A m ( c I H I ). Then, there is at most one interval I with J I and I =2 m J, and for that interval I, we have from (4.8), (4.9) c J λ m(i,j) c I Ck(I, J) ε 2 m/2 c I. This give 2 m c J 2 C2 m k 2ε c I 2 C2 2εm c I 2. J I k=1 We now sum over all I D to find c J 2 C2 2εm c I 2. J D

20 20 R. A. DeVore, S. V. Konyagin, and V. N. Temlyakov Since (H I ) is an orthonormal system, it follows that ( ) (4.10) c J H J C2 2 εm c J H J. 2 A m J D In other words, the oerator A m has norm at most C2 εm when acting on L 2 (R). We now consider 1 < q < and bound A m on the sace L q(r). We use (4.9) and the fact that k(i, J) 1 to find that c J 2 χ 2 J C2 m c I 2 χ 2 = C c I 2 χ 2. J I J I J I Therefore, summing over I D, wefind (I) c J 2 χ 2 C c I 2 χ 2. J I J D I D From this, and the strong Littlewood Paley inequalities for the Haar functions, we obtain ( ) (4.11) A m c J H J C c J H J. J D q J D q In other words, the oerator A m has norm at most C when acting on L q(r). Suose that 1 < 2. We choose a value of q with 1 < q <. If0<θ<1 satisfies 1/ = (1 θ)/q + θ/2, then (4.10), (4.11), and the Riesz Thorin interolation theorem for linear oerators gives A m L L A m 1 θ L q L q A m θ L 2 L 2 C2 δm, δ := θε. A similar argument alies when 2 < <. Thus, we have shown that the oerator A 0 is bounded from L to L for all 1 < <. In the same way, we can show that the oerator A 0 corresonding to the case J is contained in the right-half of I is bounded on L,1< <. To show the boundedness of the oerator A 0 corresonding to the case J I ± we will need a slight modification of the above roof. The same argument as above gives the inequality λ m (I, J) I 1/2 with k(i, J) the largest integer such that I J D η(j, x) dx C J 1/2 k(i, J) 1/2 I 1/2 dist(ξ J, I ) k J /2. As in the revious case, we obtain (4.10) and in lace of (I) we obtain c J 2 χ 2 C c I 2 χ 2. J I ± J D Since χ I± CM(χ I ) with M the Hardy Littlewood maximal function, we can use the Fefferman Stein inequality together with the strong Littlewood Paley inequalities for Haar functions to arrive at (4.11). The remainder of the roof of the boundedness of A 0 in this case is the same as above. This comletes the roof of the theorem.

21 Hyerbolic Wavelet Aroximation 21 Corollary 4.3. {H I }. If η(i, ), I D, satsify the assumtions (A5) (A7), then {η(i, } We can use a duality argument to give sufficient conditions that the oerator T of (4.1) is boundedly invertible. For this, we assume that η(i, ), I D, is a family of functions for which there is a dual family λ(i, ), I D, that satisfies η(i, ), λ(j, ) =δ(i, J), I, J D. Theorem 4.4. If the functions λ(i, ), I D, satisfy the assumtions (A5) (A7), then {H I } {η(i, }. Proof. have (4.12) We need to show that for each 1 < < and each sequence (c I ),we c I H I C c I η(i, ) with a constant C indeendent of the sequence (c I ). We can assume that the sequence (c I ) has at most a finite number of nonzero entries. We have c I H I = su c I H I, (d d I H I = su c I η(i, ), d I λ(i, ) I ) (d I ) with the su taken over all sequences (d I ) with at most a finite number of nonzero entries which satisfy d I H I = 1. From Hölder s inequality, we have c I H I c I η(i, ) d I λ(i, ) I D C c I η(i, ) d I H I = C c I η(i, ), I D I D where in the last inequality we used Theorem 4.2 for the sequence (λ(i, )). We now aly Theorems 4.2 and 4.4 to the setting of Section 3. Let ψ be a univariate function and let (ψ I ) be its univariate shifted-dilates. We also suose that (ψ I ) has a dual basis (λ I ) satisfying ψ I (x)λ I (x) dx = δ(i, J). R Corollary 4.5. If the functions ψ and λ satisfy conditions (A5) (A7) in the case J = [0, 1], then (ψ I ) satisfies the strong Littlewood Paley roerty (2.3). Proof. If follows from Theorem 4.2 that (ψ I ) (H I ) and from Theorem 4.4 that (H I ) (ψ I ). Thus, (ψ I ) (H I ) and the theorem follows from Lemma 2.2.

22 22 R. A. DeVore, S. V. Konyagin, and V. N. Temlyakov We shall use the remainder of this section to give an examle which shows that, in a certain sense, the assumtion of iecewise Lischitz ε continuity of the η(i, ) in (A6) cannot be removed. Namely, we will show that there is a continuous function ψ suorted on [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 in C[0, 1] which vanish at the endoints ( f (0) = f (1) = 0) and have mean value zero ( 1 0 f (x) dx = 0). This is a closed subsace of C[0, 1]. For the formulas that follow, we consider f (x) := 0 for x R\[0, 1]. Let us consider the oerator R defined for any function suorted on [0, 1] by (4.13) Rf(x):= f(2x)+ f(2x 1), x [0, 1]. Then, R has norm one on L 2 [0, 1]. We will use the following lemma: Lemma ε/4 and (4.14) For each ε>0, there is a function f = f ε in X such that f L2 [0,1] f Rf L2 [0,1] <ε. Proof. We choose an integer N > 1 such that ( ) 2N + 1 1/2 (4.15) 2 N <ε/4. N Each oint in x (0, 1) has a dyadic exansion with digits x 1, x 2,... We require that infinitely many of the x i are zero; then these digits are unique. We define { f 1, if 2N+1 i=1 x i N, (x) := 1, else. Then f is iecewise constant taking the values ±1 on dyadic intervals of length 2 2N 1. We have { R f 1, 2N +2 i =2 x i N, (x ) = 1, else. It follows that R f (x) = f (x) excet for the set E of oints x for which 2N+1 i=1 x i = N, N + 1. Since E has measure (( ) ( )) ( ) 2N + 1 2N + 1 2N N 1 + = 2 2N. N N + 1 N We have (using (4.15)) ( ) f R f 2N + 1 1/2 L2 [0,1] 2 N+1 <ε/2. N The function f has a finite number of discontinuities which occur at oints j2 2N 1, j = 1,...,2 2N+1. We can adjust f near its oints of discontinuity to obtain a function f X with f C[0,1] = 1 and f f L2 [0,1] <ε/4.

23 Hyerbolic Wavelet Aroximation 23 Then, f L2 [0,1] f L2 [0,1] ε/4 = 1 ε/4. Since the oerator R has norm one on L 2 [0, 1], it follows that f Rf 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 function ψ suorted on [0, 1] with mean value zero such that the Littlewood Paley inequalities (2.2) and (2.3) do not hold for ψ I, I D. Proof. Consider again the oerator R defined by (4.13) and define for each n = 1, 2,...,the oerator S n f := 1 n 1 R k f, f X. n k=0 Let D n be the set of dyadic intervals in [0, 1] of length 2 n 1, it follows that S n f = 1 I 1/2 f I. n n For the Haar function H, we have by orthogonality that S n H L2 [0,1] = 1. It is therefore enough to show that there is a function ψ X such that su S n ψ L2 [0,1] =. n 1 By the Banach Steinhaus theorem, we need only show that the oerators S n are unbounded as maings from X into L 2 [0, 1]. To this end, let ε>0and let f = f ε be the function in X of Lemma 4.6. Since R has norm one on L 2 [0, 1], we have R k f f L2 [0,1] k R l f R l 1 f L2 [0,1] l=1 Therefore, S n f nf L2 [0,1] = 1 n 1 (R k f f ) n It follows that k=0 k f Rf L2 [0,1] kε. l=1 L2 [0,1] 1 n 1 n(n 1)ε R k f f L2 [0,1]. n 2 k=0 S n f L2 [0,1] n f L2 [0,1] S n f nf L2 [0,1] n(n 1)ε n f L2 [0,1]. 2 Since f L2 [0,1] 1 ε/4, by letting ε 0, we see that the norm of S n is n.

24 24 R. A. DeVore, S. V. Konyagin, and V. N. Temlyakov 5. Further Remarks The results we have develoed in the revious sections can be alied to any of the known wavelets. We shall consider in detail only the examle of the Daubechies wavelets. This will indicate how the results of the revious sections are alied. Other examles such as the biorthogonal wavelets of Cohen, Daubechies, and Feauveau [CDF] can be treated in a similar way. Let ψ := D k, k > 1, be the kth univariate, orthogonal wavelet of Daubechies (see 6.4 of [Da]). We define, as usual, the multivariate family {ψ I } (R d ) by (1.4). We first consider how one verifies the Jackson inequality (J) of Section 1 for the sace H n.wedo not want to use Theorem 3.2 to establish the Jackson inequality because it requires too much smoothness for ψ. Instead, we will use Theorem 3.4. This requires us to establish conditions (A1), (A2 ), (A3), (A4) of Section 3 for the functions η(i, ) = ψ I. The usual theory of multiresolution tells us that the ψ I, I D(R d ), san L (R d ), 1 < <. It is also well known (see, e.g., [M]) that ψ I, I D(R), is an unconditional basis for L (R), 1< < (this also follows from our Section 4). The results in Section 2 then tell us that ψ I, I D(R d ), is an unconditional basis for L (R d ) and the Littlewood Paley relations (2.3) hold for this basis. Hence, condition (A1) is satisfied. Since λ = ψ, the moment condition (A3) holds for any r k. The decay condition (A4) obviously holds for any r because ψ has comact suort. We are left with showing (A2 ). We can show that this condition holds for r = k 1. We need to show that {µ I } {H I } I D where µ is the rth integral of ψ. For this, we can use Corollary 4.3. Since D k has k vanishing moments, the functions µ I, I D(R), have comact suort and therefore satisfy (A5). The function µ is at least Lischitz continuous since it is the integral of a bounded function. Hence, condition (A6) is valid. Since D k has k vanishing moments, µ has at least one vanishing moment and so (A7) is satisfied. Theorem 4.3 now imlies that (A2 ) is satisfied. We therefore have the Jackson inequality for hyerbolic aroximation with the Daubechies wavelets (5.1) E n ( f ) C2 nr f W r (L (R d )), n = 0, 1,..., 1< <, with r = k 1 and with C indeendent of n and f. It is imortant 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 for r = k. The reason for this is that in the univariate case one does not need to assume r moments are zero as in (A3) but only that r 1 moments are zero. However, in the multivariate case, we cannot make this less restrictive assumtion as can be seen already for the Haar function D 1. For examle, if we define F(x) := d j=1 f (x j), with f a univariate function from W 1 (L 2 (R)) which has comact suort and for which f (t) = t, t ( 1 4, 3 ). Then 4 for any dyadic rectangle I ( 1 4, 3 4 )d,wehave and therefore E n ( f ) 2 2 C f, H I C I 3/2 f, H I 2 C I 3 C ( n (d 1)/2 2 n ) 2. I <2 n I <2 n,i (1/4,3/4) d This examle shows, in articular, that the relation (5.1) is not correct for the case

25 Hyerbolic Wavelet Aroximation 25 r = 1 and aroximation using the Haar system. However, we do not know if (5.1) is correct for the Haar system in the case 0 < r < 1. Note that the class W r (L (R d )) can be defined for 0 < r < 1 using fractional derivatives. We next discuss the Bernstein inequalities for Daubhechies wavelets. Let again ψ = D k, k > 1. We can use Theorem 3.5 to establish a Bernstein inequality. To aly this theorem, we need to verify assumtions (A1), (A2 ), (A3), (A4). We have already noted that (A1), (A3), (A4) hold for any r k. Let ρ = ρ(k)be the maximum of all α such that D k has Hölder smoothness of order α. There are several aers dealing with the values of ρ. A discussion of this toic can be found, for examle, in Chater 7 of Daubechies book [Da]. It is known that (5.2) c 0 k ρ(k, ) c 1 k for constants 0 < c 0, c 1 < 1. For examle, it is known that for sufficiently large k, the constant c 0 = suffices. We can again use Corollary 4.3 to show that( A2 ) holds for allr <ρ, i.e., {D r ψ I } (R d ) {H I } I D(R d ). The decay ssumtion (A5) and the moment condition (A7) of that corollary is satisfied because D r ψ has comact suort. The definition of ρ and r <ρgives that D r ψ has Lischitz smoothness and therefore (A6) follows. So all the conditions of Corollary 4.3 are satisfied and roerty (A2 ) follows. In summary, we have shown Theorem 5.1. Let ψ = D k, k = 2, 3,...,be one of the Daubechies wavelets. Then: (i) The Jackson inequality (3.8) holds for r = k 1. (ii) If r c 0 k with c 0 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 any 0 <α<r,the aroximation class Aq α(l (R d )), 1 < <, of (1.5) is identical with the class of functions satisfying (1.2) with K ( f, t) the K-functional of (1.6). Acknowledgments. The authors wish to thank Professor Pencho Petrushev for several valuable discussions concerning the material in this aer. The authors would also like to thank the referee for many constructive comments about this aer. This research was suorted by the Office of Naval Research Contract N J1343 and the National Science Foundation Grant EHR and was comleted while S. V. Konyagin was a visiting scholar at the University of South Carolina. S. V. Konyagin was suorted in art by the Russian Fund of Fundamental Investigations Contract N References [BDR] [CW] [CDF] [Da] C. DE BOOR, R. A. DEVORE, A. RON (1993): On the construction of multivariate (re) wavelets. Const. Arox., 9: C. K. CHUI, J. Z. WANG (1992): On comactly suorted sline wavelets and a duality rincile. Trans. Amer. Math. Soc., 330: A. COHEN, I. DAUBECHIES, J.-C. FEAUVEAU (1992): Biorthogonal bases of comactly suorted wavelets. Comm. Pure Al. Math., 45: I. DAUBECHIES (1992): Ten Lectures on Wavelets. CBMS NSF Regional Conference Series in Alied Mathematics, vol. 61. Philadelhia, PA: SIAM.