Acta Mathematica Sinica, English Series April, 2001, Vol.17, No.2, pp. 305–312

Average Widths of Sobolev-Wiener Classes with Mixed Smoothness in Lq (Rd ) He Ping WANG Department of Mathematics, Capital Normal University, Beijing, 100037, P. R. China E-mail: [email protected]

r Abstract In this paper, the average σ-K width of Sobolev-Wiener classes Spq W with mixed smoothd ness in Lq (R ) is studied for 1 < q ≤ p < ∞, and the weak asymptotical behavior of these quantities is obtained.

Keywords Sobolev-Wiener classes with mixed smoothness, Average σ-K width, Step hyperbolic crosses 1991MR Subject Classification 41A46, 46E35

1 1.1

Introduction Sobolev-Wiener Classes with Mixed Smoothness

Let Rd be the d-dimensional real Euclidean space. For D ⊂ Rd , 1 ≤ p, q < ∞, we denote
Lp (D) :=

 f  f L

p (D)



 p1  |f (x)| dx <∞ , p

= D

 1q
   q d  f (· + v)Lp (I) <∞ . Lpq (R ) := f f pq = v∈Zd

where Zd is the set of all integer vectors in Rd , I = [0, 1]d . Lpq (Rd ) is the amalgams of Lp and

q , it is a Banach space with norm  · pq . When p = q, Lpq (Rd ) = Lq (Rd ) is the usual Lq (Rd ) space. When n = 1, these notions may be seen in [1]. For convenience, we write  · q instead of  · qq and  · Lq (Rd ) . When x = (x1 , . . . , xd ) ∈ Rd , we define that x ≥ 0 (or x > 0) ⇐⇒ xi ≥ 0 (or xi > 0), i = 1, . . . , d. Received July 21, 1998, Accepted May 31, 1999 Supported by Beijing Natural Science Foundation (Project No. 1982005)

H. P. Wang

306




xi , i ∈ e, Let e ⊂ ed := {1, 2, . . . , d}, we define xe := (xe1 , . . . , xed ), xei := 0, i ∈ e.  r Let Nd := {s ∈ Zd  s ≥ 0}. For r > 0, r ∈ Zd , then the Sobolev-Wiener space Spq L(Rd ) with a certain mixed smoothness is defined as    e r r L = L(Rd ) := f ∈ Lpq (Rd )  f Spq Dr f pq < ∞ , Spq e⊂ed

where Ds f (x) :=

∂ |s| f (x) s s ∂x11 ···∂xdd

(|s| = s1 + · · · + sd ), s ∈ Nd (in the following we always sup-

r L(Rd ) is the Sobolev space with mixed pose s ∈ Nd ). When p = q, the space Sqr L(Rd ) := Spq

smoothness, which was initiated and first studied by Lizorkin and Nikolskii. It is more elementary than the usual (isotropic or anisotropic) Sobolev space, since the usual Sobolev space may be looked on as the intersection of a sequence of Sobolev spaces with mixed derivatives r W are defined (see [2,3]). Correspondingly, Sobolev-Wiener classes with mixed smoothness Spq   r r L ≤ 1 . When 1 ≤ q ≤ p < ∞, it is easy to verify that as Spq W := f ∈ Lq (Rd )  f Spq r r W ⊂ Sqr W := Sqq W. Spq

1.2

Definitions of Average σ-K Width

Suppose C ⊂ Lq (D), L is a subspace of Lq (D), the quantity d(C, L, Lq (D)) := supf ∈C inf g∈L f − gLq (D) is the deviation of C from L; and the quantity dn (C, Lq (D)) :=

inf

dimL≤n

d(C, L, Lq (D)) =

inf

sup inf f − gLq (D)

dimL≤n f ∈C g∈L

is the Kolmogorov n-width of the set C in Lq (D). For α > 0, x(·) ∈ Lq (Rd ), we define Pα x(·) := χα (·)x(·), where χA (·) is the characteristic function of A(A ⊂ Rd ), χα (·) := χIαd (·), Iαd := [−α, α]d . Suppose L is a subspace of Lq (Rd ), for any ε > 0, let Kε (α, L, Lq (Rd )) := min{n ∈ Z+ | dn (Pα (L ∩ BLq (Rd )), Lq (Rd )) < ε}, where BLq (Rd ) is the unit ball of Lq (Rd ). The average dimension of L in Lq (Rd ) is defined to be dimL := lim lim inf ε→0 α→∞

Kε (α, L, Lq (Rd )) . (2α)d

Let σ > 0 and C be a centrally symmetric subset of Lq (Rd ). Then the average Kolmogorov σ-width (average σ-K width, for short) of C in Lq (Rd ) is defined to be dσ (C, Lq (Rd )) := inf L supx(·)∈C inf y(·)∈L x(·) − y(·)q , where the infimum is taken over all linear subspace L ⊂ Lq (Rd ) which satisfies dim(L, Lq (Rd )) ≤ σ (see [4], [11]). 1.3

Entire Functions Whose Spectrals Lie in Step Hyperbolic Crosses

For s ∈ Nd , let 

 Q∗2s := λ ∈ Rd  η(sj )2sj −1 ≤ |λj | < 2sj , j = 1, · · · , d , ρ(s) := Q∗2s ∩ Zd .

Average Widths of Sobolev-Wiener Classes with Mixed Smoothness in Lq (Rd )

307

where η(t) = 1 or 0 according to whether t > 0 or not. Let γ = (γ1 , . . . , γd ), 1 = γ1 = · · · = γν < γν+1 ≤ · · · ≤ γd (1 ≤ ν ≤ d), without loss of generality for any r > 0, r ∈ Zd , we may assume r = r1 γ, r1 ∈ Z+ . We make use of the following Q∗2s , (s, γ) := s1 γ1 + · · · + sd γd , which is notations. For every n ∈ Z+ we define Qγn := (s,γ)≤n

called a step hyperbolic cross. Let f ∈ Lq (Rd ), denote by Ff the Fourier transformation of f (generally speaking, in the sense of the Schwarz distribution), by F −1 f the inverse Fourier  transformation of f . Supp Ff is called the spectral of f . Denote Eq (Qγn ) = {f ∈ Lq (Rd ) Supp Ff ⊂ Qγn }, which is a subspace of Lq (Rd ) consisting of exponential-type entire functions defined on Rd whose spectrals lie in step hyperbolic cross Qγn (see [5]).

For f ∈ Lq (Rd ), write δs∗ (f, x) := F −1 (Ff · χQ∗2s )(x), Snγ (f, x) := (s,γ)≤n δs∗ (f, x), where δs∗ (f, x) is a Littlewood-Paley block of f and Snγ (f, x) is a bounded linear operator from Lq (Rd ) into Eq (Qγn ). 1.4

Main Results

It is well known that the notion of average width was first proposed by Tikhomirov [6]. After that, many interesting results were obtained by Magaril-II’yaev, Yongping Liu et al. (see [4], [11]) However, only the problems of average width of the usual Sobolev classes are studied, for more elementary classes of functions—Sobolev classes with mixed smoothness, the problems of average width are not involved. The difficulty lies in that the norm with mixed smoothness is nonhomogeneous in essence for the dilation transformation, the usual methods do not work. In this paper, we use new methods and obtain the following results. Theorem

Suppose r = r1 γ, r ∈ Nd , γ = (γ1 , . . . , γd ), 1 = γ1 = · · · = γν < γν+1 ≤ · · · ≤

γd (1 ≤ ν ≤ d). When 1 < q ≤ p < ∞, r1 > 0, we have  r1 ν−1 r W, Lq (Rd ))  (log σ) , where A(σ)  B(σ) means that A(σ)  B(σ) and (I) dσ (Spq σ A(σ)  B(σ), and A(σ)  B(σ) means there exists a constant c > 0 independant of σ such that A(σ) ≤ cB(σ). (II) Eq (Qγn ) is a weakly asymptotically optimal subspace of average dimension ≤ σ for dσ (Spr W, Lq (Rd )), where n = n(σ) satisfies σ  2n nν−1 .

2

Some Lemmas

For simplicity, we may suppose ν = d. In the following, we always suppose that r = r1 γ, r1 ∈ Z+ , γ = (1, . . . , 1). Lemma 2.1 [7] Lemma 2.2

Let 1 ≤ q < ∞, r1 > 0. Then dim(Eq (Qγn ), Lq (Rd )) =

mes(Qγn ) (2π)d

Let 1 < q < ∞, r1 > 0. Then d(Sqr W, Eq (Qγn ), Lq (Rd )) ≤ sup Snγ (f, ·) − f (·)q  2−nr1 . f ∈Sqr W

 2n nd−1 .

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308

Proof

For any f ∈ Sqr W , from the Littlewood-Paley theorem we know that d(f, Eq (Qγn ), Lq (Rd )) ≤ Snγ (f, ·) − f (·)q      1/2      = δs∗ (f, ·)   |δs∗ (f, ·)|2  q

(s,γ)>n

q

(s,γ)>n

  1/2    ≤ sup 2−(r,s) ·  22(r,s) |δs∗ (f, ·)|2  (s,γ)>n

q

(s,γ)>n

 2−r1 n f Sqr L ≤ 2−r1 n , where we apply the representation theorem of the space Sqr (Rd ) (1 < q < ∞) [10], that is,   1/2    f Sqr L   22(r,s) |δs∗ (f, ·)|2  . q

(s≥0

Supposing s ∈ Nd , f ∈ Lq (πd ), we denote δs (f, x) =



ck ei(k,x)



ck = ck (f ) = (2π)−d



 f (x)e−i(k,x) dx .

πd

k∈ρ(s)

Lemma 2.3 [8] The isomorphism between the space of trigonometric polynomials of the

(s,1) form f (t) = k∈ρ(s) ck ei(k,t) and the space R2 (where 1= (1, . . . , 1) ∈ Rd ) is established by associating a function f (·) with the vector δs f j = {fn (τj )} ∈ R2 fn (t) =



(s,1)

, where

ck ei(k,t) , l = 1, . . . , d, n = (±1, . . . , ±1) ∈ Rd ,

sgn kl =sgn nl

τj = (π22−sj j1 , . . . , π22−sd jd ), ji = 1, 2, . . . , 2si −1 (i = 1, . . . , d). There also exists the following order equality between these spaces. 

−(s,1)

δs (f )Lp (πd )  2

(s,1) 2

|δs f j |p

 p1

, 1 < p < ∞.

j=1

:= Suppose 1 ≤ p < ∞, M ∈ Z+ . For x ∈ RM , we introduce the norm x&M p Then we obtain the normed linear space

M p

= {x ∈ R

M

 M i=1

|xi |p

| x&M < ∞} with the unit ball p

If S is a subset of integral lattice Zd , then we denote by |S| the number of elements in S. Lemma 2.4 [9] Lemma 2.5 [8] 1

1

|S|( 2 − p )

1

1

q−p . Let 1 ≤ q ≤ p < ∞, 1 ≤ k ≤ M . Then dk (BpM , M q ) = (M − k)

Let 1 < p < ∞, S ⊂ Nd , f (x) = s∈S δs (f, x). Then



δs (f )pLp (πd )

 p1

1

1

 f Lp (πd )  |S|( 2 − p )+

s∈S

where a := min{a, 0}, a+ := max{a, 0}.

 s∈S

δs (f )pLp (πd )

 p1

,

 p1

.

BpM .

Average Widths of Sobolev-Wiener Classes with Mixed Smoothness in Lq (Rd )

Let 1 < p < ∞, S ⊂ Nd , α ∈ Nd , f (x) =

Lemma 2.6 [8]

309

s∈S δs (f, x).

    1   2(α,s) δs (f )2 2  Dα f Lp (πd )   

Then

.

Lp (πd )

s∈S

Suppose a > 1, u > 0, n ∈ Z+ , 2n ≥ 2u · ad , let  n−u , i = 1, . . . , d , Qun = s ∈ Nd | |s| = n, si ≥ d

µ(n) = ∪s∈Qun ρ(s).



Then |Qun |  ud−1 , |µ(n)| = 2n |Qun |  2n ud−1 . Let T (Qun ) = f | f (x) = s∈Qu δs (f, x) = n

i(k,x) be the d-dimensional trigonometric polynomial space whose harmonics lie in k∈µ(n) ck e   e −|re | Dr gp  1 . We µ(n). Consider the class of functions Spr,a W := g ∈ T (Qun )  e⊂ed a want to find the lower estimate of the width of Spr,a W in Lq (πd ). We obtain: |µ(n)| 2 .

Let 1 < q ≤ p ≤ ∞, M ≤

Lemma 2.7

Then dM (Spr,a W, Lq (πd ))  2−nr1 ar1 d .

Proof Suppose µ = max{p, 2}, ν = min{q, 2}. Then from  · Lq (πd )   · Lν (πd ) and  · Lp (πd )   · Lµ (πd ) , we know that dM (Spr,a W, Lq (πd ))  dM (Sµr,a W, Lν (πd )) (1 < ν ≤ 2 ≤ µ < ∞). So we may assume 1 < q ≤ 2 ≤ p < ∞. Suppose Pnu is the projection operator

from Lq (πd ) to T (Qun ), that is, Pnu f (x) = s∈Qu δs (f, x). By the Littlewood-Paley theorem n

we know Pnu (f )Lq (πd )  f Lq (πd ) (1 < q < ∞). Hence, for any f ∈ Lq (πd ), t ∈ T (Qun ), we have t − f Lq (πd )  Pnu (t − f )Lq (πd ) = t − Pnu (f )Lq (πd ) . Then dM (Spr,a W, Lq (πd ))  dM (Spr,a W ∩ T (Qun ), Lq (πd ) ∩ T (Qun )). For g ∈ Spr,a W , by Lemma 2.5, Lemma 2.6 and Lemma 2.3 we know 

a−|r | Dr gLp (πd )  e

e

e⊂ed



1

1

a−|r | |Qun |( 2 − p ) e

e⊂ed 1



1

  

a−|r

e

|p

1

e

,s)p

δs gpLp (πd )

 p1



2(r

e

,s)p

δs gpLp (πd )

s∈Qu n

e⊂ed

= |Qun |( 2 − p )

2(r

s∈Qu n

1

 |Qun |( 2 − p )

 

a−|r

e

|p (re ,s)p

2

δs gpLp (πd )

 p1

 p1

.

e∈Qu n s∈ed

Since 2(r a−|r

e

e

,s)p

≥ 2r1 |e|p

|p (re ,s)p

2

n−u d

≥ ar1 |e|p = a|r

= a−r1 dp 2r1 np · a|r

ec

e

|p

, so

c

|p −(re ,s)p

2

≤ a−r1 dp 2r1 np · 2(r

ec

c

,s)p −(re ,s)p

2

= a−r1 dp 2r1 np ,

where ec = ed − e. Hence 

1

1

a−|r | Dr gLp (πd )  |Qun |( 2 − p ) e

e

 

a−r1 dp 2r1 np δs gpLp (πd )

s∈Qu n

e⊂ed

1

1

 a−r1 d 2r1 n 2− p |Qun |( 2 − p ) n

(s,1)   2

|δs g j |p

s∈Qu n j=1

−n p

= a−r1 d 2r1 n 2

1 ( 12 − p )

|Qun |

 j δs g 

2n |Qu n|

&p

 p1

.

 p1

H. P. Wang

310

By the preceding relation, we know each function from Spr,a W ∩ T (Qun ) is associated with an 1

2n |Qu n|

1

element of the ball of radius of c · ar1 d 2−r1 n 2 p |Qun |( p − 2 ) of the space p n

hand, if f ∈

T (Qun ),

. On the other

then from Lemma 2.5, Lemma 2.3 we get    1q 1 1 f Lq (πd )  |Qun |( 2 − q ) δs (f )qLq (πd ) s∈Qu n −n q

2

−n q

=2

1 1 |Qun |( 2 − q )

(s,1)   2



|δs f j |q

 1q

s∈Qu n j=1



1 1 |Qun |( 2 − q ) δs f j &2n |Qun | . q

After discretization, comparing and applying Lemma 2.4, we obtain 1

1

1

1

2n |Qu n|

dM (Spr,a W, Lq (πd ))  ar1 d 2−r1 n 2 p |Qun |( p − 2 ) 2− q |Qun |( 2 − q ) dM (Bp n

1 + q1 ) r1 d −n(r1 − p

a

2

n

2n |Qu n|

, q

)

1 1 1 1 |Qun |( p − q ) (2n |Qun |)( q − p )

= ar1 d 2−r1 n . Lemma 2.8 Proof

Let 1 < q ≤ p < ∞, g ∈ Spr,a W . Then gLq (πd )  ar1 d 2−r1 n .

From Lemma 2.6 we know   12    (r,s)  re 2   D gLp (πd )   |2 δs (g, x)|  s∈Qu n

   12    r1 n  2  =2  |δs (g, x)| 

Lp (πd )

 2r1 n gLp (πd ) .

Lp (πd )

s∈Qu n

Since g ∈ T (Qun ), then gLq (πd )  gLp (πd )  2−r1 n Dr gLp (πd )  e e  ar1 d 2−r1 n a−|r | Dr gLp (πd )  ar1 d 2−r1 n . e⊂ed d , 0 ≤ φ ≤ 1, and Let φ be the function in C ∞ (Rd ) satisfying supp φ ⊂ [−2π, 2π]d = I2π  r,a when x ∈ πd , we have φ(x) ≡ 1. Set Sp,∗ W := {f  f (x) = φ(x)g(x), g ∈ Spr,a }. For any

g ∈ Lp (Rd ), let g −→ δa−1 g, δa−1 g(x) := g(a−1 x1 , . . . , a−1 xd ) be the dilation transformation. Then we have: Lemma 2.9 Proof

d

r,a r L  aq . Let f ∈ Sp,∗ . Then δa−1 f Spq

d It is obvious that Supp δa−1 f ⊂ I2πa . Then for any α ∈ Nd , we have    1q Dα δa−1 f pq = Dα δa−1 f (· + v)qLp (I) d v∈I4πa ∩Zd 1

1

≤ (8πa + 2)d( q − p )



Dα δa−1 f (· + v)pLp (I)

v∈Zd

a

d d q−p

d

D δa−1 f p = a q Dα f p , α

 p1

Average Widths of Sobolev-Wiener Classes with Mixed Smoothness in Lq (Rd )

so r L = δa−1 f Spq



e

d

Dr δa−1 f pq  a q

e⊂ed



311

a−|r | Dr f p . e

e

e⊂ed e

From the Libnitz formula we know Dr f p  d

r L  aq δa1 f Spq



a−|r

e

a a

d q



|

 

e ⊂ed



Dr gLp (πd ) . So e

Dr gLp (πd )

e ⊂e

e⊂ed d q

e

e ⊂e

a−|r

e

|

 e Dr gLp (πd )

e⊃e e

e

a−|r | Dr gLp (πd )  a q . d

e ⊂ed

3

Proof of Theorem

(I) Upper Estimate First, from Lemma 2.1 and Lemma 2.2 we see that r r dσ (Spq W, Lq (Rd )) ≤ d(Spq W, Eq (Qγn ), Lq (Rd )) ≤ d(Sqr W, Eq (Qγn ), Lq (Rd ))

≤ sup f (·) − Snγ (f, ·)q  2−r1 n , f ∈Sqr W

  mes(Qγ ) where n = n(σ) satisfies n = sup k ∈ Z+  (2π)dk ≤ σ , so we can infer that 2−n   d−1 r1 r hence dσ (Spq W, Lq (Rd ))  log σ σ .

logd−1 σ , σ

(II) Lower Estimate Let L be a linear subspace of average dimension σ(σ = 2u ud−1 ) of Lq (Rd ). By definition, for any ε > 0, there exist {am }∞ m=1 such that am > 0, am  ∞ and lim inf a→∞

kε (a, L, Lq (Rd )) kε (am , L, Lq (Rd )) = lim . m→∞ (2a)d (2am )d

For every m > 0 there exists a linear subspace M = M (ε, am , L) of finite dimension of Lq (Rd ) such that dim(M ) ≤ kε (am , L, Lq (Rd )) and d(Pam (L ∩ BLq (Rd )), M, Lq (Rd )) < ε. r W , then for any g ∈ L, we have If f ∈ Spq

f − gq ≥ Pam (f − g)q ≥ d(Pam f, M, Lq (Rd )) − d(Pam g, M, Lq (Rd )) ≥ d(Pam f, M, Lq (Rd )) − εPam gq ≥ d(Pam f, M, Lq (Rd )) − εPam f q − ε(f − g)q . Hence (1 + ε)f − gq ≥ d(Pam f, M, Lq (Rd )) − εPam f q . It follows that (1 + ε)d(Spr W, L, Lq (Rd )) ≥ d(Pam f, M, Lq (Rd )) − εPam f q r,a r holds for any f ∈ Spq W . For any g ∈ Sp,∗ W , then by Lemma 2.9 we know there exists a r W (Rd ). When a = constant A1 > 0 such that A1 · a− q · δa−1 g(x) ∈ Spq d

am 2π ,

r,a g ∈ Sp,∗ , then

H. P. Wang

312

Pam δa−1 g(x) = δa−1 g(x). So  −d  r (1 + ε)d(Spq W, L, Lq (Rd ))  am q d(δa−1 g, M, Lq (Rd )) − εδa−1 gLq (Rd )  d(g, da (M ), Lq (Rd )) − εgLq (Rd )  d(g, da (M ), Lq (πd )) − εgLq (πd ) . n d−1 ≤ |µ(n)|, n ∈ Z+ . Choose A2 > 1 large enough such that 2n = A2 2u adm , 3.2d A−1 2 2 u

Then for a sufficiently large m and a sufficiently small ε > 0, we have Kε := Kε (am , L, Lq (Rd )) ≤

3 3 |µ(n)| n d−1 σ(2am )d = 2d A−1 = 2n−1 |Qun |. ≤ 2 2 u 2 2 2

r,a Since the restriction of Sp,∗ W in πd is the same as Spr,a W and dim(da (M )) = dim(M ) ≤

Kε ≤

|µ(n)| 2 ,

then sup r,a g∈Sp,∗ W



d(g, δa (M ), Lq (πd )) − εgLq (πd )

 d(Spr,a W, δa (M ), Lq (πd )) − ε



sup g∈Spr,a W

gLq (πd )

 dKε (Spr,a W, Lq (πd )) − εA3 2−r1 n arm1 d r1 d  2−r1 n am (A4 − εA3 ),

hence r (1 + ε)d(Spq W, L, Lq (Rd ))  2−r1 n arm1 d (A4 − εA3 )  2−ur1 (A4 − εA3 ).

where A1 , A2 , A3 , A4 are positive numbers independent of ε, n, u, am . Letting ε → 0, we obtain the lower estimate. The theorem is proved. References [1] J. J. F. Fournier, J. Stewart, Amalgams of Lp and q , Bulletin(New Series) of the Amer. Math. Soc., 1985, 13(1): 1–22. [2] P. I. Lizorkin, S. M. Nikolskii, A classification of differentiable functions on the basis of spaces with dominant mixed derivative, Trudy Mat. Inst. Steklov, 1965, 77: 143–167, English transl in Proc. Steklov Inst. Math., 1965, 77. [3] S. M. Nikolskii, Approximation of Functions of Several Variables and Imbedding Theorems, Berlin, Heidelberg, New York: Springer-Verlag, 1975. [4] G. G. Magaril-II’yaev, Average widths of Sobolev classes on Rn , Journal of Approximation Theory, 1994, 76: 65–76. [5] H. P. Wang, Y. S. Sun, Approximation of multivariate functions with a certain mixed smoothness by entire functions, North Math. J., 1995, 11(4): 454–466. [6] V. M. Tikhomirov, Theory of extremal problems and approximation theory(in Chinese), Advances in Math., 1990, 19: 449–451. [7] D. Zung, Average ε-dimension of functional class BG,p , Mat. Zametki, 1980, 28(5): 727–736. ˜ pα in the ˜ pα and H [8] E. M. Galeev, Kolmogorov diameters of classes of periodic functions of many variables W space Lq , Izv. Akad. Nauk. SSSR Ser. Mat., 1985, 49(5): 916–934. [9] A. Pinkus A. N.-Widths, in Approximation Theory, Berlin: Springer-Verlag, 1985. [10] P. I. Lizorkin, S. M. Nikolskii, Spaces of functions with a mixed smoothness from decomposition point of view, Trudy Mat. Inst. Steklov, 1989, 187: 143–161. [11] Y. P. Liu, Average σ-K width of class of Lp (Rd ) in Lq (Rd ), Chin Ann. of Math., 1995, 16B(3): 351–360.