Representation and approximation of multivariate functions with mixed smoothness by hyperbolic wavelets

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J. Math. Anal. Appl. 291 (2004) 698–715 www.elsevier.com/locate/jmaa

Representation and approximation of multivariate functions with mixed smoothness by hyperbolic wavelets Wang Heping 1 Department of Mathematics, Capital Normal University, Beijing 100037, People’s Republic of China Received 29 May 2001 Submitted by R.H. Torres

Abstract In this paper, we study the representation theorems of multivariate functions with mixed smoothness by wavelet basis formed by tensor products of univariate wavelets, we also study the best approximation in the Lq (Rd ) metric for some function classes with mixed smoothness by hyperbolic wavelets and obtain some asymptotic estimates of approximating order.  2003 Elsevier Inc. All rights reserved. Keywords: Hyperbolic wavelets; Functions with mixed smoothness; Decomposition of functions; Best approximation

1. Introduction Let ϕ be a univariate scaling function that satisfies multiresolution analysis of L2 (R), i.e., a nested sequence of closed subspaces {Vm }m∈Zd of L2 (R) such that (i) (ii) (iii) (iv)

⊂ · · · ⊂ L2 (R), · · · ⊂ V−1 ⊂ V0 ⊂ V1
V = {0}, and j ∈Z j j ∈Z Vj is dense in L2 (R), For any j ∈ Z, f (·) ∈ Vj if and only if f (2 ·) ∈ Vj +1 , {ϕ(· − j )}j ∈Z is an orthonormal basis of V0 . E-mail address: [email protected].

1 Supported by National Natural Science Foundation of China (Project No. 10201021), Educational

Commissional Foundation of Beijing, and by Beijing Natural Science Foundation. 0022-247X/$ – see front matter  2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2003.11.023

W. Heping / J. Math. Anal. Appl. 291 (2004) 698–715

699

We define the subspace Wj as an orthogonal complement of Vj in Vj +1 , and the function ψ, the wavelet, such that the set of functions {ψ(· − j )}j ∈Z is an orthonormal basis of W0 . Then we can represent the space L2 (R) as a direct sum   Wj = V1 Wj . L2 (R) = j ∈Z

j
1

So we can construct efficient basis for L2 (R) and other function spaces by dilation and shifts. For example, the functions   (1.1) ψj,k (·) := 2k/2 ψ 2k · −j , j, k ∈ Z and

 ∗ ψj,s

:=

  ψj,s := 2s/2ψ 2s · −j ,   ϕj,1 := 21/2ϕ 21 · −j ,

if s > 0, if s = 0,

j, s ∈ Z, s > 0,

(1.2)

both form a orthonormal basis for L2 (R) (see [3,9]). We also suppose ϕ, ψ are l-regular, i.e., ϕ, ψ satisfy  (k)   (k)    ϕ (x), ψ (x)
Cp,k 1 + |x| −p , k = 0, 1, . . . , l, p ∈ Z+ , x ∈ R. (1.3) There exists a different indexing for the functions ψj,k . Let D(R) denote the set of dyadic intervals. Each such interval I is the form I = [j 2−k , (j + 1)2−k ]. We also denote by D+ (R) the subset of D(R) such that for each interval I ∈ D+ (R), the length |I | of I satisfies |I |
1. For each dyadic interval I ∈ D(R), we define  (1.4) ψI := ψj,k , I = j 2−k , (j + 1)2−k ∈ D(R) and ∗ ψI∗ := ψj,s ,

 I = j 2−s , (j + 1)2−s ∈ D+ (R),

(1.5)

∗ } thus the basis {ψj,k }j,k∈Z and the basis {ψj,s j,s∈Z,s
0 are the same as {ψI }I ∈D (R) and ∗ {ψI }I ∈D+(R) . For multivariate function space L2 (Rd ), we can construct multivariate wavelet basis by taking tensor products of the univariate basis functions. If ψ is a univariate wavelet and d  1, then for j, k, s ∈ Zd , j = (j1 , . . . , jd ), k = (k1 , . . . , kd ), s = (s1 , . . . , sd )  0 (i.e., si  0, i = 1, . . . , d), and x = (x1 , . . . , xd ) ∈ Rd , the functions

ψj,k (x) := ψj1 ,k1 (x1 ) · · · ψjd ,kd (xd )

(1.6)

∗ (x) := ψj∗1 ,s1 (x1 ) · · · ψj∗d ,sd (xd ) ψj,s

(1.7)

and

are both the orthonormal basis for L2 (Rd ). Again, we can use another indexing for the basis functions (1.6) and (1.7). Denote by D(Rd ) the set of all dyadic rectangles in Rd , any I ∈ D(Rd ) is the form I = I1 × · · · × Id with I1 , . . . , Id ∈ D(R). Also let   

  D+ Rd := I ∈ D Rd : I = I1 × · · · × Id , I1 , . . . , Id ∈ D+ (R) .

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W. Heping / J. Math. Anal. Appl. 291 (2004) 698–715

We define ψI (x) := ψI1 (x1 ) · · · ψId (xd ),

  I ∈ D Rd

(1.8)

ψI∗ (x) := ψI∗1 (x1 ) · · · ψI∗d (xd ),

  I ∈ D+ Rd .

(1.9)

and

Therefore, the wavelet basis (1.6) and (1.7) are the same as the set of functions {ψI }I∈D(Rd ) and {ψI∗ }I∈D+ (Rd ) . For k, s ∈ Zd , s  0, and 1
p
∞, let  

∗ Wk (p) := span ψj,k : j ∈ Z d , Ws∗ (p) := span ψj,s (1.10) : j ∈ Zd denote respectively the closed linear span of the finite linear combinations of the functions ∗ , j ∈ Zd , with the closure taken with respect to the L (Rd )-norm. We ψj,k , j ∈ Zd and ψj,s p also suppose that Dk and Ds∗ are the corresponding (orthogonal) projection operators, that is, for any f ∈ Lp (Rd ), we have

Dk f (x) := dj,k (f )ψj,k (x), dj,k (f ) := f (x)ψj,k (x) dx (1.11) j∈Zd

Rd

and Ds∗ f (x) :=



∗ ∗ dj,s (f )ψj,s (x),

∗ dj,s (f ) :=

j∈Zd

∗ f (x)ψj,s (x) dx.

Rd

For 1
p
∞, γ = (γ1 , . . . , γd ) ∈ Rd , γ > 0, n = 0, 1, . . . , let  

 γ γ Hn := Hn Lp Rd := span ψj,k : j, k ∈ Zd , (k, γ )
n and γ ,∗

γ ,∗ 

Hn := Hn

(1.12)

 

∗  Lp Rd := span ψj,s : j, s ∈ Zd , s  0, (s, γ )
n

(1.13)

(1.14)

denote respectively the Lp (Rd )-closure of linear span of the functions ψj,k , j, k ∈ Zd , ∗ , j, s ∈ Zd , s  0, (s, γ )
n, where (k, γ ) := k γ + · · · + k γ . (k, γ )
n and ψj,s 1 1 d d γ γ ,∗ We are interested in the approximation properties of the spaces Hn and Hn , we call γ γ ,∗ the approximation by Hn or Hn hyperbolic wavelet approximation in analogy with the approximation by trigonometric polynomial approximation (see [14]). For 1
p
∞, f ∈ Lp (Rd ), we define  γ γ En (f )p := E f, Hn p := inf γ f − gp , g∈Hn

γ ,∗ En (f )p

 γ ,∗  := E f, Hn p := infγ ,∗ f − gp g∈Hn

(1.15)

with  · p here and later the Lp (Rd ) norm. From the fact Hγ ,∗ ⊂ Hγ , we know that for any f ∈ Lp (Rd ), we have γ

γ ,∗

En (f )p
En (f )p .

(1.16)

W. Heping / J. Math. Anal. Appl. 291 (2004) 698–715

We also define the partial sum operators of hyperbolic wavelets as follows: γ γ ,∗ Dk f (x), Sn f (x) := Ds∗ f (x). Sn f (x) := (k,γ )n

701

(1.17)

s
0, (s,γ )n

For 1 < p < ∞, by Littlewood–Paley theory about hyperbolic wavelets (see [4]), we can γ γ ,∗ γ γ ,∗ show that the operators Sn , Sn are bounded from Lp (Rd ) to Hn , Hn , and also   γ γ  γ ,∗ γ ,∗  En (f )p  f − Sn f p , (1.18) En (f )p  f − Sn f p , where A  B means A  B and B  A, and A  B means there exists a positive constant c such that A
cB. We are interested in the approximating properties of the function classes with mixed smoothness by hyperbolic wavelets. The investigation of function classes with mixed smoothness defined on Rd as well as on T d was initiated by S.M. Nikolskii [12, p. 390]. In 1963–1965, P.I. Lizorkin and S.M. Nikolskii [7], S.M. Nikolskii [11], T.I. Amanov [1] defined three types of function spaces with mixed smoothness, i.e., the Sobolev-type space r B. They obtained a series Spr L, the Hölder-type space Spr H , and the Besov-type space Sp,θ of fundamental results about these spaces, such as the representation theorem, imbedding theorem, trace theorem, etc. In [8] P.I. Lizorkin and S.M. Nikolskii studied these spaces again from the view point of function decomposition. They obtained the representation theorem by Littlewood–Paley blocks, introduced the class of entire functions whose spectrals lie in a step hyperbolic cross, and proposed to study the approximation by hyperbolic cross. It is well known that in the periodic case approximation by trigonometric polynomials whose spectrals lie in some hyperbolic crosses was first considered by K.I. Babenko [2]. In the decades of the 1970s and 1980s, N.S. Nikolskaja [10], E.M. Galeev [6], D. Zung [5], V.N. Temlyakov [14], etc., systematically investigated the multivariate approximation by trigonometric polynomials of hyperbolic crosses in the periodic case. As to the Rd (non-periodic) case, Wang Heping and Sun Yongsheng [15] studied the approximation of multivariate functions with mixed smoothness by entire functions whose spectrals lie in a step hyperbolic cross; and there are very few works about the d  2 case as far as we know. In this paper, we will study the best approximation in Lq (Rd ) metric for the function r B(Rd ), 1 < p
q < ∞ by hyperbolic wavelets. Let us classes Spr L(Rd ), Spr H (Rd ), Sp,θ recall the definitions of the spaces with mixed smoothness (see [8] or [15]). For ∀e ⊂ ed := {1, 2, . . . , d}, x = (x1 , . . . , xd ) ∈ Rd , we define    e xi , i ∈ e, e e e , x x := x1 , . . . , xd i := 0, i ∈ e. For r = (r1 , . . . , rd )  0, let D r f (x) :=

∂ |r| f (x) ∂x1r1 · · · ∂xdrd

  |r| := r1 + · · · + rd

be the generalized derivative of f in the sense of Liouville (see [7]). Then for r = (r1 , . . . , rd ) > 0, the Sobolev space Spr L(Rd ) with mixed smoothness is defined as follows:    e      D r f  < ∞ . (1.19) Spr L Rd := f ∈ Lp Rd : f Spr L := p e⊂ed

702

W. Heping / J. Math. Anal. Appl. 291 (2004) 698–715

For f ∈ Lq (Rd ), t = (t1 , . . . , td ) > 0, e ⊂ ed , we set  e   e Ω l f, t e q := sup ∆lhe f (·)q ,

(1.20)

|hj |tj

where l ∈ Z+ is a fixed positive integer, h = (h1 , . . . , hd ) ∈ Rd , and ∆0hj ,j f (x) := f (x), ∆lhj ,j f (x) :=

l k=0

e

∆lhe f (x) :=

 j ∈e

(−1)l−k

  l f (x1 , . . . , xj + khj , . . . , xd ), k

 ∆lhj ,j f (x),

∆lh f (x) := ∆lh1 ,1 · · · ∆lhd ,d f (x).

(1.21)

e

As we know, ∆lhe f (x) is the l-order mixed difference with vector-valued step h of e f (x), Ω l (f, t e )q is the l-order mixed moduli of smoothness in Lq norm. For r = (r1 , . . . , rd ) > 0, we choose l ∈ Z+ such that l > max{r1 , . . . , rd }, then the Hölder space r B(Rd ) with mixed smoothness are defined in the folSpr H (Rd ) and the Besov space Sp,θ lowing way respectively [8,15]:     r Spr H Rd := Sp,∞ B Rd    r := f ∈ Lp Rd : f Spr H := f Sp,∞ B   −rj e   l e f, t := sup tj Ω <∞ , p e⊂ed t >0 j ∈e



    r r B := Sp,θ B Rd := f ∈ Lp Rd : f Sp,θ e⊂ed

 θ   e · Ω l f, t e p dtj

1/θ

(1.22)

 2 ··· 0

2 

−θrj −1

tj

0 j ∈e



<∞ ,

(1
θ < ∞).

(1.23)

j ∈e

It is to be noticed that when e is empty set, the corresponding terms in (1.20), (1.22), (1.23) r B are norms. From the definitions are f p by definition, and f Spr ,L , f Spr H , f sp,θ we know that the Hölder space is the special case of the Besov space when θ = ∞, so it is r B the unit enough to consider Sobolev spaces and Besov spaces only. Denote by Spr L, Sp,θ r B(Rd ). For 1
p, q
∞, 1
θ
∞, r = r · γ > 0, balls of the spaces Spr L(Rd ) and Sp,θ 1 r1 > 0, γ ∈ Rd , n  1, we introduce the quantities   γ γ γ ,∗  γ ,∗ En Spr L q = sup En (f )q , En Spr L q = sup En (f )q , (1.24)  γ r B q En Sp,θ

f S r L 1

f S r L 1

p

=

sup

f S r B 1 p,θ

γ En (f )q ,

 γ ,∗  r En Sp,θ B q

p

=

sup

f S r B 1 p,θ

γ ,∗

En (f )q . (1.25)

W. Heping / J. Math. Anal. Appl. 291 (2004) 698–715

703 γ

γ ,∗

In Section 4, we will find out the asymptotically exact order of En (Spr L)q , En (Spr L)q , γ r γ ,∗ r and En (Sp,θ B)q , En (Sp,θ B)q when 1 < p
q < ∞, 1
θ
∞, n → ∞. These results may be compared with their periodic analogy obtained by V.N. Temlyakov [14] and Romaniuk [13]. The main tool we use is the representation theory of functions with mixed smoothness by hyperbolic wavelets, which we will show in Section 3. In Section 2, we shall give some properties of operators Ds∗ and Dk which will be used in Section 3. In this paper, we restrict our development to approximation on Rd , we could also give similar results for the case of approximation on a compact subset of Rd or on the torus T d . 2. Properties of operators Dk and Ds∗ (s  0) In the following, we always suppose that ϕ, ψ are l-regular, k, s ∈ Zd , s  0, r = (r1 , . . . , rd ) ∈ Rd , 0
r1 , . . . , rd
l. Lemma 2.1. Let k, r ∈ Z d fixed, 1
p
∞. Then for any finite sum f (x) = ψj,k (x), we have  1/p  r  p D f   2(1/2−1/p)|k|2(r,k) |d | . j,k p



j dj,k ×

(2.1)

j

Proof. From d    i) ψj(ri ,k (xi ), D r (ψj,k )(x) = 2(r,k) D r ψ j,k (x) := 2(r,k) i j =i

we know that D r f (x) =



dj,k D r (ψj,k )(x) =



j

  dj,k 2(r,k) D r ψ j,k (x).

j

By a simple variable transformation, we know that it suffices to prove the lemma in the case k = 0 = (0, . . . , 0). Upper estimates of D r f p . Cr (ψ) := sup

As ψ is l-regular, we have

d      −2  D r ψ (x)
C sup 1 + |xi − ji | j,0

x∈Rd j∈Zd


2C sup

d  

x∈Rd j∈Zd i=1

1 + |ji |

−2

< ∞.

x∈Rd i=1 ji ∈Z

For 1
p
∞, let p satisfy 1/p + 1/p = 1. Then  r   1/p  1/p

  D f (x)
|dj,0 | ·  D r ψ j,0 (x) ·  D r ψ j,0 (x) j

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W. Heping / J. Math. Anal. Appl. 291 (2004) 698–715






   |dj,0 | ·  D r ψ j,0 (x) p

j


Cr (ψ)

1/p

·



1/p  

 1/p  r    D ψ j,0 (x) · j

   |dj,0 | ·  D r ψ j,0 (x)

1/p

p

,

j

since

Cr (ψ) < ∞,

d     r   (r )     D ψ (x) dx =  D r ψ  = ψ j  j,0 0,0 1 L j =1

Rd

we get   r   1/p  · D f 
Cr (ψ)1/p ·  D r ψ p 0,0 1



1 (R)

< ∞,

1/p |dj,k |

p

.

j

Lower estimate of D r f p . Since the wavelet function ψ is l-regular, then ψ has up to l order of vanishing moments (see [9]), so we can integrate the univariate function ψ, (r ) rj times to find a function µj ∈ Lp (R) which satisfies (−1)rj µj j = ψ. It follows that µj (x)(j = 1, . . . , d) are rapidly decreasing and   d  r r D (µj,k )(x) := D (µi )ji ,ki (xi ) = (−1)|r| 2(r,k)ψj,k (x), j, k ∈ Zd . i=1

Integration by parts then shows that

  D r ψj,k (x) · µj ,k (x) dx = 2(r,k)δ( j, j )δ(k, k ), Rd

where δ is the Kronecker delta. Hence,

dj,k = f (x)ψj,k (x) dx = 2−(r,k) D r f (x)µj,k (x) dx. Rd

Rd

Since µj (x)(j = 1, . . . , d) are rapidly decreasing, then Cr (µ) := sup

d     −2 µj,0 (x)
C sup 1 + |xi − ji | < ∞.

x∈Rd j∈Zd

So we have

|dj,0 |


x∈Rd j∈Zd i=1

  r D f (x)µj,0 (x) dx =

Rd

   

 r D f (x)µj,0 (x)1/p · µj,0 (x)1/p dx,

Rd

then by Hölder inequality, we get  1/p  r    1/p

D f (x)p µj,0 (x) dx |dj,0 |
µ0,0 1 . Rd

W. Heping / J. Math. Anal. Appl. 291 (2004) 698–715

Hence

| dj,0 |

p

p/p


µ0,0 1

j

    D r f (x)p µj,0 (x) dx · j

p/p


µ0,0 1

705

Rd

 p · Cr (µ) · D r f p .

Lemma 2.1 is proved. 2 Remark 2.1. Lemma 2.1 holds for any f ∈ Wk (p). Remark 2.2. For 1
p
∞, f (x) =

 j∈Zd

∗ ψ ∗ (x) ∈ W ∗ (p), s  0, we have dj,s s j,s

   r   ∗ p 1/p D f   2(1/2−1/p)|s|2(r,s) d  . j,s p

(2.2)

j

Furthermore, if when i ∈ ce(s) := {j ∈ ed : sj = 0}, ri = 0, then we also have    r   ∗ p 1/p D f   2(1/2−1/p)|s|2(r,s) d  . j,s p

(2.3)

j

The proof is similar, we omit it. From Lemma 2.1 and the following remarks, we get the corresponding Bernstein inequality and Nikolskii inequality in Wk (p) or Ws∗ (p). Lemma 2.2. Let 1
p
q
∞, f (x) ∈ Wk (p) or f (x) ∈ Wk∗ (p), k  0. Then  r  D f   2(r,k)f p , f q  2(1/p−1/q)|k|f p . p

(2.4)

Lemma 2.3. Let 1
p
∞, and when i ∈ ce(s), ri = 0. Then for any f, D r f ∈ Lp (Rd ), we have  ∗    D f   2−(r,s)D r f  . (2.5) s p p Proof. By Remark 2.2, we get    ∗ p d ∗ (f )p , D f   2p(1/2−1/p)|s| s j,s p j

where ∗ dj,s (f ) :=

∗ f (x)ψj,s (x) dx = 2−(r,s)

Rd

µ∗j,s

D r f (x) · µ∗j,s dx,

Rd

is defined as µj,s in a same way. Using the same methods as Lemma 2.1, we can get   ∗ p    D f   2p(1/2−1/p)|s| d ∗ (f )p  2−(r,s)p D r f p . 2 s j,s p p j

706

W. Heping / J. Math. Anal. Appl. 291 (2004) 698–715

Lemma 2.4. For any f ∈ Lp (Rd ), 1
p
∞, s  0, e := e(s) := {j ∈ ed : sj > 0}, then  ∗    D f   Ω l e f, 2−se , (2.6) s p p where 2−s = (2−s1 , . . . , 2−sd ). Proof. Without loss of generality, we suppose that s > 0. For any G(x) ∈ Lp (Rd ), we define l Gm l,j (x) := l



1/ l 1/ l l l ··· (−1)i+l i 0 0 i=1   × G x1 , . . . , xj + i2−m (u1 + · · · + ul ), . . . , xd du1 · · · dul .

Then we have l Gm l,j (x) − G(x) = l

1/ l 1/ l · · · ∆l2−m (u1 +···+ul ),j G(x) du1 · · · dul 0

and

Since

0

 l ∂l m l ml i+l l −l l i ∆(i/ l)2−m,j G(x). G (x) = l 2 (−1) l,j i ∂xjl i=1   s1 s1 (x) + fl,1 (x) := f11 (x) + f12 (x), f (x) = f (x) − fl,1

s1 s2 where f11 (x) := f (x) − f1,1 (x), f12 (x) := f1,1 (x). We can also decompose f11 (x), f12 (x) as follows:  s2  s2 (x), f22 (x) = f11 l,2 (x), f21 (x) = f11 (x) − f11 l,2     s2 s2 f23 (x) = f12 (x) − f12 l,2 (x), f24 (x) = f12 l,2 (x). d

Continuing this process, we get sequences of functions fd1 (x), fd2 (x), . . . , fd2 (x), such that d

f (x) = fd1 (x) + fd2 (x) + · · · + fd2 (x) j

j

j

and for every j = 1, 2, . . . , 2d , there exits a β j = (β1 , . . . , βd ), βi = 0 or 1, such that  lβ j j    D f (·)  2l(s,β j ) Ω l f, 2−s . d p p By Lemma 2.3, we get  ∗ j      D f (·)  2−l(s,β j ) D lβ j f j (·)  Ω l f, 2−s . s d d p p p Hence d

2  ∗ j   ∗    D f (·)  Ω l f, 2−s . D f (·)
s s d p p p j =1

2

W. Heping / J. Math. Anal. Appl. 291 (2004) 698–715

707

r B(R Rd ) and Spr L(R Rd ) by hyperbolic 3. Representation theorems of the spaces Sp,θ wavelets r B(Rd ). For every f ∈ L (Rd ), 1
p
∞, we have First we consider the space Sp,θ p the decomposition ∗ ∗ Ds∗ f (x), Ds∗ f (x) = dj,s (f )ψj,s (x). (3.1) f (x) = j∈Zd

s
0

r B(Rd ). The next theorem (called the We use the formulas (3.1) to renorm the space Sp,θ representation theorem by hyperbolic wavelets) is fundamental. r B(Rd ), Theorem 3.1. Let 1
p
∞, r = (r1 , . . . , rd ), 0 < r1 , . . . , rd < l. Then f ∈ Sp,θ if and only if     1/θ (r,s)θ  ∗ θ Ds f p 2
∞, 1
θ
∞. s
0

In this case, we have   θ 1/θ  r B  f Sp,θ 2(r,s)θ Ds∗ f p ,

1
θ < ∞.

(3.2)

s
0

When θ = ∞, we have

  (r,s)  ∗  r Ds f p . f Spr H = f Sp,∞ B  sup 2

(3.3)

s
0

Proof. We only give the proof when 1
θ < ∞. The proof of the case θ = ∞ is similar. Necessity. For any s > 0, t = (t1 , . . . , td ) > 0, we define  0, if 2si ti  1, β := (β1 , . . . , βd ), βi := i = 1, . . . , d. 1, otherwise, Let e1 = {j ∈ ed : βj = 0}, e2 = {j ∈ ed : βj = 1}. Then from (2.4), we get           sup ∆l D ∗ f (·)
sup 2l|e1 | |hj |l D lβ D ∗ f (·) |hj |tj

h

s

p

|hj |tj





j ∈e2

=

d  j =1

Since r B = f Sp,θ

 2 e⊂ed

0

s

j ∈e2

   |tj | 2(lβ,s)Ds∗ f (·)p l

 

min 1, |tj |l 2sj l Ds∗ f (·)p .

1/θ

2   θ  −θrj −1  l e  e Ω f, t p ··· tj dtj , 0 j ∈e

j ∈e

p

708

W. Heping / J. Math. Anal. Appl. 291 (2004) 698–715

without loss of generality, we may consider the term e = ed only. Then we have  2 ···

2  0 j ∈e

0

 =



1/θ dtj

j ∈e

1/θ 2−k1 +1 2−k

d +1 d d θ  −θrj −1  l Ω (f, t)p ··· tj dtj k
0



 θ −θr −1  l e  Ω f, t e p tj j

2−k1



j =1

2−kd

j =1

 l  θ Ω f, 2−k p 2θ(r,k)

1/θ .

k
0

Then 

  θ Ω f, 2−k p




l



sup |hj |2

 

=

−kj

  l ∆ f (·) h p

θ




sup

s
0 |hj |2

−kj

 l ∗   ∆ D f (·) s h p

θ

 d  ∗  

(s −k )l  θ j j D f (·) min 1, 2 s p j =1

s
0



d    

D ∗ f (·) min 1, 2(sj −kj )l s p

θ ,

j =1

e⊂ed s∈G(e,k)

where G(e, k) := {s  0: sj
kj , if j ∈ e; and sj  kj , if j ∈ e}. Choose αj , βj such that 0 < αj < rj < βj < l, j = 1, . . . , d. Below we suppose e0 = {1, . . . , m}, 1
m
d. It suffices to consider the term e0 only. In this case, we have 

 d   

(s −k )l  θ j j D ∗ f  min 1, 2 s p j =1

s∈G(e0 ,k)





= 

2



m 



(sj −kj )l

sj kj j =1 j =1,...,m

= 

m 




m 

2

βj sj −kj l

·2

·





(l−βj )sj

d  ∗   D f  2αj sj · 2−αj sj s p j =m+1

sj >kj j =m+1,...,d

2(βj sj −kj l)θ

sj kj j =1 j =1,...,m



θ

sj >kj j =m+1,...,d

sj kj j =1 j =1,...,m



 ∗  D f  s p

m 

sj kj j =1 j =1,...,m



d  ∗ θ  D f  2αj sj θ s p j =m+1

sj >kj j =m+1,...,d

2(l−βj )sj

θ



d 

sj >kj j =m+1 j =m+1,...,d

θ/θ

2−αj sj

θ



θ

W. Heping / J. Math. Anal. Appl. 291 (2004) 698–715





m 



2(sj −kj )βj θ

sj kj j =1 j =1,...,m

 ∗ θ D f  s

d 

p

sj >kj j =m+1,...,d

709

2(sj −kj )αj θ .

j =m+1

Hence

2(k,r)θ

  D ∗ f θ s p s
0



m 





2(sj −kj )βj θ

sj kj j =1 j =1,...,m

k
0

=





d  ∗ θ  D f  2(sj −kj )αj θ s p j =m+1

sj >kj j =m+1,...,d

2(k,r)θ

m 

2(sj −kj )βj θ

j =1

kj
sj j =1,...,m



d 

2(sj −kj )αj θ

kj
θ  2(r,s)θ Ds∗ f p .

s
0

Sufficiency. From Lemma 2.4, we know for any s  0, we have  ∗      D f   Ω l e f, 2−s e , s p p where e := e(s) := {j ∈ ed : sj > 0}. Hence     θ 1/θ θ 1/θ   e 2(s,r)θ Ds∗ f p = 2(s ,r)θ Ds∗e f p e⊂ed se >0

s
0



 



2(s

e ,r)θ

 e θ e Ω l f, 2−s p

1/θ

e⊂ed se >0

e⊂ed

2

···

2  0 j ∈e

0

r B.  f Sp,θ

 θ −θr −1  l e  Ω f, t e p tj j



1/θ dtj

j ∈e

2

Remark 3.1. We can use the decomposition

∗ ∗ ∗ ∗ f (x) = dj,s (f )ψj,s (x), dj,s (f ) = f (x)ψj,s (x) dx s
0 j∈Zd

(3.4)

Rd

r B(Rd ). From the above theorem and Lemma 2.1, to give a characterization of the space Sp,θ we obtain the following theorem.

Theorem 3.1 . Let 1
p
∞, r = (r1 , . . . , rd ), 0 < r1 , . . . , rd < l. Then for any f ∈ r B(Rd ), 1
θ < ∞, we have Sp,θ r B  f Sp,θ

 s
0

2((r,s)+(1/2−1/p)|s|)θ

 j∈Zd

  ∗ d (f )p j,s

θ/p 1/θ .

(3.5)

710

W. Heping / J. Math. Anal. Appl. 291 (2004) 698–715

When θ = ∞, we have f Spr H  sup 2



(r,s)+(1/2−1/p)|s|)

s
0

  ∗ d (f )p j,s

1/p .

Remark 3.2. We cannot use the decomposition

f (x) = dI (f )ψI (x), dI (f ) = f (x)ψI (x) dx, I∈D (Rd )

or f (x) =



(3.6)

j∈Zd

(3.7)

Rd

Dk f (x),

Dk f (x) =

k∈Zd



dj,k (f )ψj,k (x)

(3.8)

j∈Zd

r B , since the space S r B(Rd ) is not a to give an equivalent description of the norm f Sp,θ p,θ homogeneous space.

We now turn to the decompositional description of spaces Spr L(Rd ), 1 < p < ∞. First, we introduce some notations. Let D ⊂ D(R) be an index set. Then for univariate functions  f 1 , . . . , f d , and I = I1 × · · · × Id ∈ Dd , we use the notation fI (x) := di=1 fIii (xi ) the L2 (Rd ) normalized shifted dilate of functions f 1 , . . . , f d (or f if f 1 = · · · = f d = f ). For 1 < p < ∞, we say that a family of real valued functions fI (x), I ∈ Dd , satisfies the Littlewood–Paley property (we briefly write LPP) for p, if for any finite sequence (cI ) of real number, we have          2 1/2    .     c f (·)  f (·) (3.9) c I I  I I    I∈D d

p

p

I∈D d

We also say fI (x), I ∈ Dd satisfies the strong Littlewood–Paley property (we briefly write SLPP) for p, if for any finite sequence (cI ) of real number, we have    1/2      2  ,    cI fI (·)   |cI χI | (3.10)   I∈D d

p

p

I ∈D d

where χ is the characteristic function of [0, 1], and χI is the L2 (Rd ) normalized shifted dilates of function χ . From [4], we know that if  −1 f i (x)
c 1 + |x| , a.e. x ∈ R and f i is a nonzero function, i = 1, . . . , d, (3.11) then fI , I ∈ D satisfies LPP if and only if fI , I ∈ D satisfies SLPP.  If for any f ∈ Lp (Rd ), we have a unique representation f (x) = I∈Dd cI fI (x), and for any assignment εI = ±1, I ∈ Dd , we have            cI fI (·)   εI cI fI (·) (3.12)   , I∈D d

p

I ∈D d

p

W. Heping / J. Math. Anal. Appl. 291 (2004) 698–715

711

then we say fI , I ∈ Dd is a unconditional basis of Lp (Rd ). Applying Khinchine’s inequality, we can get that (3.9) and (3.12) are equivalent (see [4]). Using the same methods in the proof of [4, Lemma 2.1, p. 7–8], we obtain the following lemma. Lemma 3.1. If {fIi }I ∈D (i = 1, . . . , d) satisfy LPP in Lp (R) for some p, 1 < p < ∞, then the multivariate family fI , I ∈ Dd also satisfies LPP for this p. Since ψ is an l-regular wavelet, we take for granted the known facts that for 1 < p < ∞ and any r = (r1 , . . . , rd ), 0
ri < l, the families of functions (ψ (ri ) )I , I ∈ D(R) and ((ψ ∗ )(ri ) )I , I ∈ D+ (R) (i = 1, . . . , d) satisfy LPP in Lp (R) (see [9]). Hence by Lemma 3.1, we know that (D r ψ)I , I ∈ D(R d ) and(D r ψ ∗ )I , I ∈ D+ (R d ) satisfy LPP. It is obvious that ψ (ri ) and (ψ ∗ )(ri ) satisfy (3.11), so (D r ψ)I , I ∈ D(R d ) and (D r ψ ∗ )I , I ∈ D+ (R d ) satisfy SLPP. From this and the definition of the norm of f Spr L , using the decomposition (3.7), we get the following results. Theorem 3.2. Let ψ be an l-regular univariate wavelet, r = (r1 , . . . , rd ), 0 < ri < l. Then for any f ∈ Spr L(Rd ), we have  1/2  d           2 1 + |Ij |−2rj · dI (f )χI (·) f Spr L    .   d I∈D (R ) j =1

(3.13)

p

Remark 3.3. Similarly, we can use (3.4) to express the equivalent norms of f Spr L . And we get the following theorem. Theorem 3.2 . Let ψ be an l-regular univariate wavelet, r = (r1 , . . . , rd ), 0 < ri < l. Then for any f ∈ Spr L, we have      (r,s) ∗  2 1/2   .    dj.s (f ) χj,s (·) f Spr L   2 

(3.14)

p

s
0 j∈Zd

 In the following, we use the decomposition f (x) = s
0 Ds∗ f (x). Since the family of ∗ , j, s ∈ Zd , s  0 satisfies LPP for p, 1 < p < ∞, then for any assignment functions ψj,s εs = ±1, we have          ∗    ∗ 2 1/2  ∗ ∗  . (3.15)        εs Ds f (·)   Ds f (·)   dj,s (f )ψj,s (·)   s
0

p

p

s
0

p

s
0 j∈Zd

From (3.15) and Khinchine’s inequality, we can get the following result:         2 1/2   ∗   ∗  .     f (·) =  Ds f (·)   Ds f (·)   p s
0

p

s
0

p

Hence, by (3.14) and (3.16), we obtain the following theorem.

(3.16)

712

W. Heping / J. Math. Anal. Appl. 291 (2004) 698–715

Theorem 3.3. Let ψ be an l-regular univariate wavelet, r = (r1 , . . . , rd ), 0 < ri < l. Then for any f ∈ Spr L(Rd ), 1 < p < ∞, we have      (r,s) ∗ 2 1/2   .    f Spr L   (3.17) 2 Ds f (·)  p

s
0

Remark 3.4. We may use the decomposition (3.8) to give an equivalent norm of f Spr L . Using (3.13) and the above induction, we get the following theorem. Theorem 3.3 . Let ψ be l-regular univariate wavelet, r = (r1 , . . . , rd ), 0 < ri < l. Then for any f ∈ Spr L(Rd ), 1 < p < ∞, we have  1/2  d        2   f Spr L   1 + 4ri ki · Dk f (·)  .   d k∈R i=1

(3.18)

p

4. Approximation by hyperbolic wavelets In this section, we always suppose that ϕ, ψ are l-regular, r = (r1 , . . . , rd ), r = r1 γ , 0 < r1 , . . . , rd < l, γ and γ satisfy 1 = γ1 = γ1 = · · · = γv = γv and 1 < γj < γj , j = v + 1, . . . , d (1
v
d), for the purpose of simplification. Theorem 4.1. Let 1 < p
q < ∞, 1
θ
∞. Then    γ r γ ,∗  r 2−nr1 n(d−1)(1/q−1/θ)+ , 1 < q
2, En Sq,θ B q  En Sq,θ B q  −nr1 (d−1)(1/2−1/θ)+ 2 n , 2 < q
∞,   γ r γ ,∗  r −n(r1 −1/p+1/q) (v−1)(1/q−1/θ)+ En Sp,θ B q  En Sp,θ B q  2 n , p < q, γ r  γ ,∗  r  −n(r1 −1/p+1/q) En Sp L q  En Sp L q  2 ,

(4.1) (4.2) (4.3)

where a+ = a, a > 0, a+ = 0, a
0. Proof. We can proceed in the same line as in [15] or in [13,14] to obtain the upper esγ ,∗ r γ ,∗ r γ ,∗ timates of En (Sq,θ B)q , En (Sp,θ B)q , En (Spr W )q , we omit its details. By (1.16), we γ r γ r γ B)q , En (Sp,θ B)q , En (Spr W )q . know it is enough to get the lower estimates of En (Sq,θ First we construct two functions gs (x) and hs (x). For any s  0, ρ(s) := {k ∈ Zd : 0
kj
2sj − 1, j = 1, . . . , d}, ω = (1, . . . , 1) ∈ Rd , we define gs (x) := 2|s|/2 ψω,s (x), hs (x) := 2−|s|/2 ψj,s (x). j∈ρ(s)

It is easy to know that we have

gs p  2|s|/p ,

gs , hs ∈ Ws (p) ⊂ Ws∗ (p) hs p  1.

2

and that for 1
p
∞, 1/p + 1/p = 1, (4.4)

W. Heping / J. Math. Anal. Appl. 291 (2004) 698–715

Lemma 4.1. Let 1 < p < ∞ and f (x) = f p 





s
0 cs hs (x).

713

Then

1/2 |cs |2

.

(4.5)

s
0

Proof. As ψI , I ∈ D(Rd ) satisfies SLPP, we have      2 1/2    2 −|s|   .  f (·)   χ |c | 2 (·) s k,s   p p

s
0 k∈ρ(s)

Since 2−|s|

   d χk,s (x)2 = 1, x ∈ [0, 1] , 0, otherwise.

k∈ρ(s)

Hence, we obtain 1/2  |cs |2 . f p 

2

s
0

Lemma 4.2. Suppose that 1 < p < ∞, and that f (x) = f p 







s
0 ds gs (x).

Then

1/p

|ds |p 2(p/p )|s|

.

(4.6)

s
0

Proof. As ψI , I ∈ D(Rd ) satisfies SLPP, we get      2 1/2    2 |s|  f (·)     . | ds | 2 χω,s (·)   p p

s
0

Since the support set of the functions χω,s is disjoint, so we obtain   f (·)p  p



Rd

=

|ds | 2

2 χω,s (x)

s
0



Rd



2 |s| 



p p|s|/2 

|ds | 2

p/2 dx

 p

 |ds |p 2p/p |s| . χω,s (x) dx 

s
0

s
0

We begin our discussion above the lower estimates. First we give the proof of (4.1) for 1 < q
2, q
θ
∞. Consider the function

frqθ (x) = n−(d−1)/θ · 2r1 n 2−2(r,s)−|s|/q gs (x). (4.7) (γ ,s)>n

By Theorem 3.1 and (4.4), we know

714

W. Heping / J. Math. Anal. Appl. 291 (2004) 698–715

frqθ 

r B Sq,θ

−(d−1)/θ

n

·2

r1 n



2

(r,s)θ

·2

(−2(r,s)−|s|/q )θ

(γ ,s)>n

2

nr1

−(d−1)/θ

·n

·



2

−(r,s)θ

1/θ · gs θq

1/θ  1.

(γ ,s)>n

Consequently, by (4.6), we get    γ r γ γ En Sq,θ B q  En (frqθ )q  frqθ − Sn frqθ q = frqθ q  1/q −(d−1)/θ r1 n (−2(r,s)−|s|/q )q (|s|/q )q n ·2 · 2 ·2 (γ ,s)>n

−(d−1)/θ

=n

·2

r1 n

·



2

−2q(r,s)

1/q

 2−r1 n · n(d−1)(1/q−1/θ).

(γ ,s)>n

For the other case, the proof is similar. We only give the extreme functions which attain the corresponding approximating order. γ r Case 2. Let 2 < q < ∞, 2
θ
∞. The extreme function of En (Sq,θ B)q is

ψr,θ (x) = 2r1 n · n−(d−1)/θ

2−2(r,s)hs (x).

(4.8)

(s,γ )>n

Case 3. Let 1 < q
2, 1
θ < q, or 2 < q < ∞, 1
θ
2. The extreme function of γ r En (Sq,θ B)q is r (x) = 2−r1 n hs˜ (x), ψ

(4.9)

where s˜ = (n + 1, 0, . . . , 0). γ r B)q is Case 4. Let 1 < p < q < ∞, 1
θ
q. The extreme function of En (Sp,θ

g˜pr (x) = 2−r1 n−|˜s|/p gs˜ (x).

(4.10)

Case 5. Let 1 < p < q < ∞, q
θ
∞. The extreme function of

grpθ (x) = 2r1 n · n−(v−1)/θ

γ r B)q En (Sp,θ



2−2(r,s)−|s|/p gs (x).

is (4.11)

(γ ,s)>n

Case 6.

γ

Let 1 < p
q < ∞. The extreme function of En (Spr L)q is

g˜pr (x) = 2−r1 n−|˜s|/p gs˜ (x). The theorem is proved. 2 Using the same methods, we can get the following theorem. Theorem 4.1 . Let 1 < p
q < ∞, 1
θ
∞. Then

(4.12)

W. Heping / J. Math. Anal. Appl. 291 (2004) 698–715

715

   γ  r γ ,∗  r 2−nr1 n(v−1)(1/q−1/θ)+ , 1 < q
2, En Sq,θ B q  En Sq,θ B q  −nr1 (v−1)(1/2−1/θ)+ 2 n , 2 < q
∞,   γ  r γ ,∗  r −n(r1 −1/p+1/q) (v−1)(1/q−1/θ)+ n , p < q, En Sp,θ B q  En Sp,θ B q  2    



γ γ ,∗ En Spr L q  En Spr L q  2−n(r1 −1/p+1/q).

References (r)

∗(r)

[1] T.I. Amanov, Representation and imbedding theorems for the function spaces Sp,θ B(Rn ) and Sp,θ B (0
xj
2π , j = 1, . . . , n), Trudy Mat. Inst. Steklov 77 (1965) 5–34, English transl. in Proc. Steklov Inst. Math. 77 (1965). [2] K.I. Babenko, Approximation for classes of multivariate functions by trigonometric polynomials, Dokl. Akad. Nauk USSR 132 (5) (1960) 982–985 (in Russian). [3] I. Daubechiles, Ten Lectures on Wavelets, in: CBMS–NSF Regional Conf. Ser. in Appl. Math., vol. 61, SIAM, Philadelphia, PA, 1992. [4] R.A. Devore, S.V. Konyagin, V.N. Temlyakov, Hyperbolic wavelet approximation, Constr. Approx. 14 (1998) 1–26. [5] D. Zung, Approximation of multivariate functions on torus by trigonometric polynomials, Mat. Sb. 131 (2) (1986) 251–271. [6] E.M. Galeev, Approximation for classes of functions with several dominant mixed derivatives by Fourier sums, Mat. Zametki 22 (2) (1978) 197–211. [7] P.I. Lizorkin, S.M. Nikolskii, A classification of differentiable functions on the basis of spaces with dominant mixed derivative, Trudy Mat. Inst. Steklov 77 (1965) 143–167, English transl. in Proc. Steklov Inst. Math. 77 (1965). [8] P.I. Lizorkin, S.M. Nikolskii, Spaces of functions with a mixed smoothness from decomposition point of view, Trudy Mat. Inst. Steklov 187 (1989) 143–161. [9] Y. Meyer, Ondelettes et Opérateurs, vols. 1 and 2, Hermann, Paris, 1990. [10] N.S. Nikolskaja, Approximation for class SH rρ ∗ of periodic functions by Fourier sums, Sibirsk. Mat. Zh. 4 (1975) 761–780. [11] S.M. Nikolskii, Functions with dominant mixed derivative satisfying a multiple Hölder condition, Sibirsk. Mat. Zh. 4 (1963) 1342–1364, English transl. in Amer. Math. Soc. Transl. 102 (2) (1973). [12] S.M. Nikolskii, Approximation of Functions of Several Variables and Imbedding Theorems, SpringerVerlag, Berlin, 1975. [13] A.S. Romaniuk, Approximation for Besov type classes of periodic multivariate functions in Lq space, Ukraïn. Mat. Zh. 43 (10) (1991) 1398–1408. [14] V.N. Temlyakov, Approximation of functions with bounded mixed derivative, Proc. Steklov Inst. Math. 178 (1989) 1–122. [15] W. Heping, S. Yongsheng, Approximation of multivariate functions with a certain mixed smoothness by entire functions, Northeast. Math. J. 11 (4) (1995) 454–466.