The Infinite-Dimensional Widths and Optimal Recovery of Generalized ...

JOURNAL OF COMPLEXITY 18, 815–832 (2002) doi:10.1006/jcom.2002.0639

The In¢nite-Dimensional Widths and Optimal Recovery of Generalized Besov Classes Liu Yongping1,2 and Xu Guiqiao1 Department of Mathematics, Beijing Normal University, Beijing 100875, People’s Republic of China E-mail: [email protected]

Received November 3, 2000; revised August 18, 2001; accepted August 24, 2001; published online July 2, 2002

The purpose of the present paper is to consider some weak asymptotic problems concerning the infinite-dimensional Kolmogorov widths, the infinite-dimensional linear widths, the infinite-dimensional Gel’fand widths and optimal recovery in Besov space. It is obvious to be found that the results obtained and methods used in Besov space are easily generalized, and hence in this paper these results in an extension of Besov spaces on Rd are stated and proved. Meantime, the representation theorem and approximation of these spaces by polynomial splines are discussed. # 2002 Elsevier Science (USA)

Key Words: generalized Besov classes; infinite-dimensional width; optimal recovery.

1. INTRODUCTION
 d Let Lp Rd denote the usual
d Lp -norm space on R : Suppose that k 2 N ; d and h 2 R : For each f 2 Lp R ; Dkh f ðxÞ

¼

k X

ð1Þ

lþk

l¼0

! k f ðx þ lhÞ l

ð1Þ

is the kth difference of the function f at the point x with step h: The k-order modulus of smoothness Ok ðf ; tÞp of f is defined to be Ok ðf ; tÞp ¼: sup jjDkh f jjp :

ð2Þ

jhj4t

1

Supported by the National Natural Science Foundation of China (10071007). Supported partly by the Foundation for University Key Teacher by the Ministry of Education of China and partly by the Scientific Research Foundation for Returned Overseas Chinese Scholars by the Ministry of Education of China. 2

815 0885-064X/02 $35.00 # 2002 Elsevier Science (USA) All rights reserved.

816

LIU AND XU

Similar to [8,9], we give the following definitions of some classes of smoothness functions defined on Rd : Definition 1.1. Let O denote the non-negative and univariate functions defined on Rþ ¼ ft : t50g: We say that OðtÞ 2 Fnk if it satisfies: (1) Oð0Þ ¼ 0; OðtÞ > 0 for any t > 0; (2) OðtÞ is continuous; (3) OðtÞ is almost increasing, i.e., for any two points t; t such that 04t4t; we have OðtÞ4COðtÞ; where C51 is a constant independent of t; (4) for any natural number n 2 Zþ ; OðntÞ4Cnk OðtÞ; where k51 is a fixed positive integer, C > 0 is a constant independent of n and t; (5) there exists a > 0 such that OðtÞ=ta is almost increasing; (6) there exists b; 05b5k such that OðtÞ=tb is almost decreasing, i.e., there exists C > 0 such that for any two points 05t4t it always holds OðtÞ=tb 5COðtÞ=tb : n Definition
d  1.2. Let k 2 N ; OðtÞ 2 Fk ; 14y41; and 14p51: We say O f 2 Bpy R if f satisfies the following conditions:
 (1) f 2 Lp Rd ; (2) 8

> R þ1 Ok ðf ;tÞp y dt 1=y > < 51; 14y51; 0 t OðtÞ jjf jjbO ðRd Þ ¼ py > > : sup Ok ðf ;tÞp 51; y ¼ 1: t>0 OðtÞ


d The space BO is a normed linear space with the norm py R jjf jjBO ðRd Þ :¼ jjf jjp þ jjf jjbO ðRd Þ : py

py

Set n o
d
 O Spy R :¼ f 2 Lp Rd : jjf jjBO ðRd Þ 41 : py


d
d
d r O When OðtÞ ¼ tr ; BO and Spy R the py
R  is the usual Besov space Bpy R r usual Besov class Spy Rd considered in [2]. Pustovoitov [8], Sun Yongsheng and Wang Heping [9] introduced the n n n periodic function classes HqO ðpd Þ and BO py ðpd Þ; respectively. When O ðxÞ ¼ oðx1    xd Þ for an univariate function o; the authors in [8, 9] discussed the n n Kolmogorov n-width [7] of HqO ðpd Þ and BO py ðpd Þ; respectively. Considering

817

GENERALIZED BESOV CLASSES

that the radial function oðjxjÞ; x 2 Rd ; is completely different to
oðx1    xd Þ; d we will introduce another kind of generalized Besov spaces BO py R ; where OðtÞ is an univariate
d  function, and discuss some approximation problems on the space BO py R : Nikolskii (cf. [6, p. 217]) discussed the representation of Besov space

d Brpy Rd : In the following, we give the representation of the space BO py R :
d Theorem 1.1. If k 2 N ; OðtÞ 2 Fnk ; f 2 BO py R ; then f can be represented in the form of a series 1 X Qs ð3Þ f ¼ s¼0




 converging to it in the metric of Lp Rd ; whose terms Qs are entire functions of spherical exponential type 2s and whose equivalent norm is 8(   )1=y > > jjQs jp y P > > ; 14y51; < s2Zþ Oð2s Þ jjf jjBO ðRd Þ py > > jjQs jjp > > : sups2Zþ ; y ¼ 1: Oð2s Þ

  Let X Rd ; jj  jjX be a normed of the real functions on Rd : Let
dspace  a > 0 and Pa be the operator in X R defined by Pa f ðxÞ ¼ wa ðxÞf ðxÞ (where wa ðxÞ is the characteristic function of the cube Iad ¼ ½a; ad Þ: Let L be a
d subspace of X R : Set Pa L ¼ fPa f : f 2 Lg: Suppose that L is locally finitedimensional, i.e., dimðPa L; X Þ5 þ 1 for every a > 0: Then the following quantity is said to be the average dimension of L in X (in the sense of LFD).


d dim Pa L; X Rd g dimðL; X R Þ ¼ lim inf dim : a!1 ð2aÞd
 Let s > 0; and C be a centrally symmetric
subset of X Rd : The infinite dimensional Kolmogorov a-width of C in X Rd is defined to be

 d*s C; X Rd ¼ inf sup inf jjf  gjj ; L

f 2C

g2L

X



 g L; X Rd 4s: where the infimum is taken over all subspaces L with dim The above-mentioned concept of the infinite-dimensional width was introduced by Li Chun [3] and is different from that of the average width in [5]. In fact, if we denote the average dimension of a subspace
d

L dand  the average width of a set C in X R dim L; X R defined in [5] by



 and d% s C; X Rd ; respectively, then, we may see that d%s C; X Rd 4d* s




 g X Rd 4s implies dim L; X Rd 4s: dimðL; C; X Rd from
the
factthat dim

 g L; X Rd 4s is possibly untrue when dim L; X Rd 4s; Notice that dim

818

LIU AND XU

hence, the above-stated two kinds of widths are different. For example, when d ¼ 1; by letting Ba ðRÞ denote the class of the entire functions of the s g exponential type s > 0; we see that dimðBa ðRÞ; Lp ðRÞÞ42p from [1], and dim ðBs ðRÞ; Lp ðRÞÞ ¼ 1 by a simple computing.
 The infinite-dimensional linear s-width of C in X Rd is defined to be
 0 d* s ðC; X Rd :¼ inf sup jjf  Lf jjX ; L

f 2C

where the infimum is over all
taken 
L which
are  continuous linear operators g Im L; X Rd 4s; and Im L denotes the from spanðCÞ to X Rd and dim range of the operator
 L:
 Denote by X n Rd the dual space of X Rd : The average codimension can be different values for the same subspace according to the definition of Magaril-Il’yaev [5] and the
functionals that he used gðf Þ ¼ f ðxj Þ; 8f 2
  Lp Rd do not belong to Lnp Rd for 14p51: To improve them, we give the following definition
 of the infinite-dimensional Gelf’and width. Suppose that X n Rd is also a normed space of real functions on Rd : For s >
0; we  take the annulling class Rs which consists of all linear subspaces of X n Rd with the locally finite-dimensional property (LFD-property), and moreover, for each L 2 Rs it holds

lim inf

a!1


 dim Lj½a;ad ; X n Rd ð2aÞd

4s:


 The infinite-dimensional Gel’fand s-width of the set C in X Rd is given as
 s
d* C; X Rd :¼ inf

sup jjf jjX ;

L2Rs f 2C\L?


 where L? :¼ fx 2 X Rd : f ðxÞ ¼ 0; 8f 2 Lg: Jiang Yanjie and Liu Yongping [2] discussed the approximation problems of Besov classes by the entire functions of the exponential type s; and obtained the accurate order
of the average
 Kolmogorov s-widths and r average linear s-widths of Spy Rd in Lp Rd for 14p51: To consider the weak asymptotic results concerning the infinite-dimensional Kolmogorov s-widths linear s-widths (in the sense of LFD)
dand  the infinite-dimensional
 r of Spy R in Lp Rd for 14p51;
in this paper, we will construct a continuous linear operator from Lp Rd to the space of multivariate polynomial splines. Meantime, it is obvious to be seen that the results obtained and the proof’s methods used
in  Besov classes are easily O generalized to the more general class Spy Rd ; and hence
d in this present O paper we state and prove these results on the classes Spy R :

819

GENERALIZED BESOV CLASSES


 For s51; let r ¼ s1=d =k: The space Ss;k1 Rd of multivariate polynomial splines is defined to be (
d
 k2 i h i is a multivariate Ss;k1 R ¼ f : @ k2f 2 C Rd ; f jh j1 j1 þ1 jd jd þ1 r ; r  r ; r

@xj

)

polynomial of order not exceeding k  1 to xj ; 14j4d :


 It is easy to see that dim Ss;k1 Rd ; Lp Rd 4s for 14p51: Let Pn denote the collection of all univariate polynomials of degree not exceeding n: Denote by Ln ðxÞ the cardinal spline satisfying conditions: (1) Ln ðxÞ 2 Sn ¼ fs : s 2 C ðn1Þ ðRÞ; sjðv;vþ1Þ 2 Pn ; 8v 2 Zg; (2) Ln ðan þ vÞ ¼ d0v for v 2 Z; where an ¼ ð1 þ ð1Þn1 Þ=4; (3) jLn ðxÞj4AeBjxj ; x 2 R; for some positive constants A and B: Set j%i ¼ ji =r þ ak1 =r for j ¼ ðj1 ; . . . ; jd Þ 2 Z d and j% ¼ ðj%1 ; . . . ; j%d Þ:
 For f 2 Lp Rd ; let ðss;k1 f ÞðxÞ ¼

X

rd

Z

j2Z d

½0;1=rd

k % k f ðj%Þ dt D% t1    D td

d Y

Lk1 ðrxi  ji Þ;

ð4Þ

i¼1



 P r where D% h f ðxÞ ¼ ð1Þrþ1 Drh f ðxÞ þ f ðxÞ ¼ rl¼1 ð1Þlþ1 kl f ðx þ lhÞ:

 By [4], ss;k1 is a continuous linear operator from Lp Rd to Ss;k1 Rd ; and has the following approximation properties. Theorem 1.2. s51;

Let k 2 N ; OðtÞ 2 Fnk ; 14p51 and 14y41: Then for



 

 0 O O C1 Oðs1=d Þ 4d* s Spy Rd ; Lp Rd 4d* s Spy Rd ; Lp Rd 4 sup O ðRd Þ f 2Spy

jjf  ss;k1 f jjp 4C2 Oðs1=d Þ;



 s O Rd ; Lp Rd 4C4 Oðs1=d Þ; C3 Oðs1=d Þ 4d* Spy where and in the sequel Cj are absolute constants depending only on p; y; O and d: For s > 0; let Ys be the set of all sequence x ¼ fxv : v 2 Z d g of points in Rd satisfying the following conditions: (1) jxv j4jxv0 j; if and only if jvj4jv0 j for v; v0 2 Z d ; (2) xv =xv0 ; if and only if v=v0 for v; v0 2 Z d ;

820

LIU AND XU d

Þ (3) cardx :¼ lima!1 inf cardðx\½a;a 4s: Here j  j is the usual Euclidean ð2aÞd d norm and cardðx \ ½a; a Þ; for any a > 0; denotes the number of elements of the set x \ ½a; ad :


 Let K  X Rd : The quantity dðKÞ :¼

sup

jjxðÞ  yðÞjjX

xðÞ;yðÞ2K

is called the diameter of K: For any x 2 Ys ; the information of f 2 K is defined by Ix f ¼ ff ðxv Þgv2Z d : Ix is called a standard sampling operator of the average cardinality 4s: The quantity Ds ðK; X Þ :¼ inf sup dðIx1 Ix f \ KÞ x2Ys f 2K

is called the
net width or the minimum information diameter
of  the set K in the space X Rd : Here we have put the set Ix1
Ix f ¼ fg 2 X Rd : Ix f ¼ Ix gg: If K is a balanced and convex subset of X Rd ; then, from the book by Traub and Wozniakowski (cf. [10, p. 30]), we may see that Ds ðK; X Þ :¼ inf supfjjf jjX : Ix f ¼ 0; f 2 Kg: x2Ys

ð5Þ


 For any x 2 Ys ; a mapping j : Ix ðKÞ ! X Rd
is called an algorithm and j  Ix f is called a recovering function of f in X Rd : Denote by Fx the set of all algorithms on K: If j can be extended into a linear operator on the linearized set of K; we call the algorithm to be linear. Denote by FLx the set of all linear algorithms on the linearized set of K: The quantity Es ðK; X Þ :¼ inf inf sup jjf  jðIx f ÞjjX x2Ys j2Fx f 2K

ð6Þ

is called the minimum intrinsic error of the optimal recovery of the set K in the space X : Taking FLx in the place of Fx on the right-hand side of (6), we denote the obtained quantity by EsL ðK; X Þ and call it the minimum linear intrinsic error. If K is a convex and centrally symmetric subset of X ; then by [10, p. 11,12], there hold the following inequalities: L 1 2Ds ðK; X Þ4Es ðK; X Þ4Es ðK; X Þ:

ð7Þ


d For f 2 BO py R ; s > 1; let ðTs;k1 f ÞðxÞ ¼

X j2Z d

f ðj%Þ

d Y i¼1

Lk1 ðrxi  ji Þ:

ð8Þ

821

GENERALIZED BESOV CLASSES

Theorem 1.3. Then, for s51;

Let k 2 N ; OðtÞ 2 Fnk ; 14p51; 14y41 and a > d:

  

 

 O O C5 O s1=d 412Ds Spy Rd ; Lp Rd 4Es Spy Rd ; Lp Rd 

 O 4EsL Spy Rd ; Lp Rd 4 sup

O ðR d Þ f 2Spy

jjf  Ts;k1 f jjp

4C6 Oðs1=d Þ:

2. PROOF OF THEOREM 1.1
 Proof. Denote SEvp Rd the restriction to Rd of the
space  of all functions of spherical exponential type v which belong to Lp Rd : For s 2 Zþ and u 2 Rd ; set Ks ðuÞ ¼

l1 s



2s 2k sin 2k juj ; juj

where ls ¼

Z Rd




2s 2k sin 2k juj du: juj

 d

For f 2 Lp R ; define Ts f ðxÞ ¼ ð1Þ

kþ1

Z X k

ð1Þ

lþk

Rd l¼1

Q0 ¼ ðT0 f ÞðxÞ;

k l

! f ðx þ luÞKs ðuÞ du;

Qs ¼ ðTs f ÞðxÞ  ðTs1 f ÞðxÞ;

s ¼ 1; 2; . . . :
 Then, P from [6, p. 138], we know that Ts ðf ; xÞ belongs to SE2ps Rd
and the
 d O d series 1 s¼0 Qs converges to f in the metric of Lp R : For f 2 Spy R ; we will first establish the following estimate: (  )1=y X jjQs jjp y jjf jjBO ðRd Þ 5 ; 14y51; ð9Þ py Oð2s Þ s2Zþ jjQs jjp ; s Þ Oð2 s2Zþ

jjf jjBO ðRd Þ 5 sup py

y ¼ 1:

ð10Þ

822

LIU AND XU

When 15y5 þ 1; first, it is easy to verify that   Z þ1  Z 1 Ok ðf ; tÞp y dt Ok ðf ; 2s Þp y ¼ ln 2 ds t OðtÞ Oð2s Þ 0 0

¼ ln 2

þ1 Z X N

N¼0

Since f ¼

P1

s¼0

N þ1 O ðf ; 2s Þ y k p Oð2s Þ

ds5

y þ1  X Ok ðf ; 2N Þp Oð2N Þ

N¼0

:

ð11Þ

Qs ; it is obvious to see that Ok ðf ; 2N Þp 4

þ1 X

Ok ðQs ; 2N Þp :

ð12Þ

s¼0

Notice that Dkh Qs ðxÞ ¼

Z 0

1

!  Z 1 k X @ k  h Qs x þ uj h du1    duk @x 0 j¼1

ð13Þ

@ ¼ h1 @x@1 þ    þ hd @x@d : for h ¼ ðh1 ; . . . ; hd Þ 2 Rd ; where we have put h@x By the generalized Bernstein inequality [6, p. 116]  a  @ Qs  sk    @xa  42 jjQs jjp p

for jaj ¼ k; the definition of difference and Minkowskii integral inequality, we have   X @a Qs  N Nk ðsNÞk    Ok ðQs ; 2 Þp 52 jjQs jjp ; s4N : ð14Þ  @xa  52 p jaj¼k Ok ðQs ; 2N Þp 42k jjQs jjp ; s5N :

ð15Þ

By (12), (14) and (15), we have y þ1  X Ok ðf ; 2N Þp N ¼0

Oð2N Þ

5

8 y < X N

þ1  X N ¼0

þ

1 Oð2N Þ 1 X

s¼N þ1

:

jjQs jjp

!y 2

ðsN Þk

jjQs jjp

s¼0

!y 9 = ;

¼ J1 þ J2 :

ð16Þ

823

GENERALIZED BESOV CLASSES

Choosing d > 0 such that d þ b5k; then J1 ¼

þ1  X N ¼0

5

!y y X N 1 dðsN Þ ðsN ÞðkdÞ 2 2 jjQs jjp Oð2N Þ s¼0

þ1  X N ¼0

1 Oð2N Þ

y X N

2ðsNÞðkdÞy jjQs jjyp

s¼0

þ1 X 1  ðsN ÞðkdÞ y X 2 ¼ jjQs jjyp : Oð2N Þ s¼0 N ¼s

ð17Þ

Since OðtÞ=tb is almost decreasing, we know 1 2bðN sÞ 5 : Oð2N Þ Oð2s Þ Further, by (17), we have J1 5

þ1 X 1 X s¼0

! 2

ðsN ÞðkdbÞy

jjQs jjyp

5 Oy ð2s Þ

N ¼s

þ1 jjQ jjy X s p

Oy ð2s Þ

s¼0

:

ð18Þ

Choosing 05g5a; then

J2 ¼

þ1  X

y 1 Oð2N Þ

N ¼0

5

þ1  X N ¼0

1 Oð2N Þ

!y

1 X

2

ðNsÞg ðN sÞg

2

jjQs jjp

s¼N þ1

y X þ1

2ðsN Þgy jjQs jjyp

s¼N þ1

! þ1 X s1 X 2ðsNÞgy ¼ jjQs jjyp : y N O ð2 Þ s¼1 N ¼0

ð19Þ

Since OðtÞ=ta is almost increasing, we know 1 2aðN sÞ 5 : Oð2N Þ Oð2s Þ Further, by (19), we have J2 5

þ1 X s1 X s¼1

N ¼0

! 2

ðsN ÞðgaÞy

jjQs jjyp

Oy ð2s Þ

5

þ1 jjQ jjy X s p s¼0

: Oy ð2s Þ

ð20Þ

824

LIU AND XU

Second, for f ¼

jjf jjp 4

Pþ1 s¼0

þ1 X

Qs ; it is easy to see that there hold þ1 X

jjQs jjp 5

s¼0

5

y0

!1=y0

O ð2 Þ

s¼0

!1=y0

þ1 X

2

say0

s¼0

s¼0

!1=y

þ1 jjQ jjy X s p s¼0

5

Oy ð2s Þ

1

1 O ðf ; tÞ y k p

OðtÞ

dt 5 t

Z

1 jjf jj

y

p

OðtÞ

1

dt 5 t

Oy ð2s Þ þ1 jjQ jjy X s p s¼0

Hence, by (1) and the almost monotonicity of Z

Z

!1=y

þ1 jjQ jjy X s p

s

1

OðtÞ ta ;

Oy ð2s Þ

:

we have

jjf jjyp tayþ1

1

!1=y

dt5

þ1 jjQ jjy X s p s¼0

: Oy ð2s Þ

ð21Þ

By (11), (16), (18), (20) and (21), we know that (9) holds for 15y5 þ 1: It is similarly proved that (9) holds for y ¼ 1: Suppose that K ¼ sup s2Zþ

jjQs jjp Oð2s Þ

P for f ¼ þ1 j¼0 Qj . Then, for any s > 0 and for y ¼ 1; (10) follows the following inequalities. P1 s s1 ðjsÞk þ1 X Ok ðf ; 2s Þp 2 jjQj jjp X jjQj jjp j¼0 Ok ðQj ; 2 Þp 4 4 þ s s s Oð2 Þ Oð2 Þ Oð2 Þ Oð2s Þ j¼s j¼0 ! s1 ðjsÞk þ1 X 2 Oð2j Þ X Oð2j Þ þ 5K; jjf jjp 5K Oð2s Þ Oð2s Þ j¼s j¼0 4

þ1 X

jjQs jjp 4K

þ1 X

s¼0

Oð2s Þ5K:

s¼0

Thus, we complete the proof of (9) and (10). Next, we will establish the following estimate: ( jjf jjBO ðRd Þ 4 py

X jjQs jjp y s2Zþ

Oð2s Þ

jjf jjBO ðRd Þ 4 sup py

)1=y

s2Zþ

jjQs jjp ; Oð2s Þ

;

14y51;

y ¼ 1:

ð22Þ

ð23Þ

825

GENERALIZED BESOV CLASSES

Notice that [3, p. 186] jjQs jjp 5Ok ðf ; 2sþ1 Þp 5Ok ðf ; tÞp ;

t 2 ð2s ; 2sþ1 Þ

and hence there hold jjQs jjp Ok ðf ; tÞp 5 ; Oð2s Þ OðtÞ

t 2 ð2s ; 2sþ1 Þ

for s ¼ 0; 1; 2; . . . : Therefore, it is easy to see that   1  1 Z 2sþ1  X Ok ðf ; tÞp y dt jjQs jp y X 5 t Oð2s Þ OðtÞ s s¼0 s¼0 2 y Z 1 Ok ðf ; tÞp dt ¼ jjf jjybO ðRd Þ 4jjf jjyBO ðRd Þ 4 py py t OðtÞ 0 for 14y51; and that sup s2Zþ

jjQs jjp 5jjf jjBO ðRd Þ py Oð2s Þ

for y ¼ 1: Thus, we obtain (22) and (23). Theorem 1.1 is complete.

]

3. PROOF OF THEOREM 1.2 Lemma 3.1(cf. Liu Yongping and Xu Guiqiao [4]). holds

For 14p41; there

jjss;k1 f jjp 5jjf jjp : Let 8 9     @k f   < =

   Wpk Rd ¼ f : f 2 Lp Rd ;  k  41; j ¼ 1; . . . ; d :  @xj  : ; p

Lemma
3.2  (cf. Liu Yongping and Xu Guiqiao [4]). f 2 Wpk Rd ; there holds   d  k  X @ f  k jjss;k1 f  f jjp 5r  k  :  @xj  j¼1 p

Next, we prove the Theorem 1.2.

For 14p41 and

826

LIU AND XU


d O Proof (Upper Estimate). For f 2 Spy R ; let Qs be defined in Theorem 1.1 and the nature number N satisfy r52N 52r: Here r ¼ 1k s1=d defined in the Introduction section. For 04s4N  1; by Lemma 3.2 and the generalized Bernstein inequality [6, p. 116], we have    d  k X @ Q s  k jjQs  ss;k1 Qs jjp 5r ð24Þ  k  5rk 2sk jjQs jjp :  @xj  j¼1 p

For s5N ; Lemma 3.1 implies the following inequalities: jjQs  ss;k1 Qs jjp 5jjQs jjp :

ð25Þ

For 15y51; by the equivalent norm of jjf jjBO in Theorem 1.1 and (24), we py have N 1 X s¼0

( sk

2 jjQs jjp 5

N 1 jjQ jjy X s p

)1=y (

N 1 X

sy0 k

y0

)1=y0 s

2 O ð2 Þ Oy ð2s Þ s¼0 ( )1=y0 N 1 Oð2N Þ X sy0 ðkbÞ 5jjf jjBO bN 2 52Nk Oð2N Þ: py 2 s¼0 s¼0

Hence, by (24) and (26), we obtain that   X  N 1 N 1 X   Qs  ss;k1 Qs  5Oðs1=d Þ:   s¼0  s¼0

ð26Þ

ð27Þ

p

By (25) and the equivalent norm of jjf jjBO in Theorem 1.1, we have py    X  1 1 1 X  X  Qs  ss;k1 Q s  5 jjQs jjp   s¼N  s¼N s¼N p

( 4

1 X jjQs jjyp s¼N

5jjf jjBO

py

)1=y (

Oy ð2s Þ Oð2N Þ 2aN

(

þ1 X

2

)1=y0

Oy ð2s Þ

s¼N 1 X

0

)1=y0

say0

s¼N

5Oð2N Þ5Oðs1=d Þ:

ð28Þ

Similarly, we may verify that (27) and (28) hold for y ¼ 1; 1: *0  Thus,
d  by
(27)  and (28), we obtain the upper estimate of d s O d Spy R ; Lp R for 14p51:

827

GENERALIZED BESOV CLASSES

Define

( A ¼ span


 gj 2 Lnp Rd : gj ðf Þ ¼




 d

8f 2 Lp R ; j 2 Z

 d

Z ½0;1=rd


 % k f j% dt; %k D D t1 td

:

It is easy to verify that A
is reasonable and A 2 Rs : From (3), (27) and (28), O we know that if f 2 Spy Rd \ A? ; then jjf jjp 5Oðs1=d Þ:

ð29Þ  
d
d s O From (29), we obtain the upper bound of d* Spy R ; Lp R for 14p 51: Lower estimate: From the method of [9, p. 234] and the similar
d
d O computation, we may obtain the lower estimate of d* s ðSpy R ; Lp R : 
d 
d  s O * Here, we will establish only the lower bound of d S R ; Lp R for py

14p51: Mn :¼ dim  If L 2 Rs
; then  there exists a natural number
n with  Lj½n;nd ; Lnp Rd 4ð2nÞd 2s: Hence, there exist gi 2 Lnp Rd ; i ¼ 1; . . . ; Mn ; such that f 2 L? $ hgi ; f i ¼ 0;

i ¼ 1; . . . ; Mn ;

ð30Þ


 d for f 2 Lp R
d and ! supp  f  ½n; n : 1=d Let l ¼ 4 s þ 1 (where ½t denotes the integer part of t 2 R) and for ln þ 14l4ln; jl ðtÞ ¼ sink ð2pltÞw½ðl1Þ=l;l=l ðtÞ;

t 2 R;

ð31Þ

where w½a;b denotes the characteristic function of ½a; b: For s ¼ ð2nlÞd ; let the n Qd set ff1 ; . . . ; fs g denote the oset of all the functions j¼1 jlj ðxj Þ :  nl þ 14lj 4nl; 14j4d : From s > Mn ; we know that there exists at least a non-zero function 1=p P Ps p f ¼ sj¼1 aj fj 2 L? : Denote jjajjp ¼ : By computation, we j¼1 jaj j obtain jjf jjp ¼ l

d=p

Z

1 k

p

jsin ð2ptÞj dt

d=p jjajjp

ð32Þ

0

P and for any a 2 Zþd ; jaj ¼ di¼1 ai ¼ k;  a  1=p d Z 1 Y @ f  k ðai Þ p   ¼ ð2pÞk lkd=p jðsin Þ ð2ptÞj dt jjajjp 5lk jjf jjp :  @xa  0 p i¼1

ð33Þ

828

LIU AND XU

For h ¼ ðh1 ; . . . ; hd Þ 2 Rd ; by Minkowskii inequality, (33) and !  Z 1 Z 1 k X @ k k Dh f ðxÞ ¼  h f xþ uj h du1    duk ; @x 0 0 j¼1 @ where the notation h@x is defined in (13), we obtain ! k Z 1 Z 1   k X @    jjDkh f jjp 4  f þ uj h  du1    duk  h    @x 0 0 j¼1 p  a  X     @ f 4 Ca jhjk  a  5jhjk lk jjf jjp ; @x p jaj¼k

which implies Ok ðf ; tÞp 5tk lk jjf jjp :

ð34Þ

Ok ðf ; tÞp 42k jjf jjp :

ð35Þ

By (1), we know

For 14051; from (34) and (35), we know þ1 O

Z

k ðf ; tÞp

OðtÞ

0

y

dt 5 lky t

Z

1=l ky1

t dt þ Oy ðtÞ

0

Z

þ1 1=l

! dt jjf jjp : Oy ðtÞt

ð36Þ

Since OðtÞ=tb is almost decreasing and OðtÞ=ta almost increasing, we obtain Z 0

t ð1=lÞby dt5 y y O ðtÞ O ð1=lÞ

1=l ky1

Z

þ1

1=l

Z

1=l

tðkbÞy1 dt5

0

dt ð1=lÞay 5 y y O ðtÞt O ð1=lÞ

Z

þ1

1=l

1 Oy ð1=lÞlky

dt 1 : 5 tayþ1 Oy ð1=lÞ

;

ð37Þ

ð38Þ

Hence, by (36)–(38), we have jjf jjbO ðRd Þ 5jjf jjp =Oð1=lÞ; py

which implies jjf jjBO ðRd Þ 5jjf jjp =Oð1=lÞ: py

Similarly, we may prove that (39) also holds for y ¼ 1:

ð39Þ

829

GENERALIZED BESOV CLASSES


d O Let f% ¼ f =jjf jjBO ðRd Þ : Then, f% 2 Spy R \ L? : From jjf%jjBO ðRd Þ ¼ 1 and py py (39), we obtain jjf%jjp 4Oð1=lÞ4Oðs1=d Þ: Thus, we obtain the lower bound of the quantities in Theorem 2.

]

4. PROOF OF THEOREM 1.3 Lemma 4.1 (cf. Nikol’skii [6, p. 124]). Suppose that 14p51; h > 0; xðlÞ k ¼ khðl ¼ 1; . . . ; d; k 2 ZÞ; g is entire functions of spherical exponential type v > 0; ððgÞÞp ¼ sup hd ul

X



l1 2Z

X

!1=p ðdÞ p jgðxð1Þ l1  u1 ; . . . ; xld  ud Þj

51

ld 2Z

or jjgjjp 51: Then, jjgjjp 4ððgÞÞp 4ð1 þ hvÞd jjgjjp :

ð40Þ

Next, we prove Theorem 1.3.
d O Proof (Upper Estimate). For f 2 Spy R ; let Qs be defined in Theorem 1.1. It is easy to verify that Qs ðxÞ  ðTs;k1 Qs ÞðxÞ ¼

d X

Gm;s ðxÞ;

ð41Þ

m¼1

where for 14m4d; Gm;s ðxÞ ¼

X



j1 2Z



X

X

Qs ðj%1 ; . . . ; j%m1 ; xm ; . . . ; xd Þ

jm1 2Z

Qs ðj%1 ; . . . ; j%m ; xmþ1 ; . . . ; xd Þ

jm 2Z

 Lk1 ðrxm  jm Þ

m 1 Y i¼1

Lk1 ðrxi  ji Þ:

ð42Þ

830

LIU AND XU

By Ho! lder inequality and jLn ðtÞj4AeBt ; we obtain X

jGm;s ðxÞj 5

j1 2Z



X Qs ðj% ; . . . ; j% ; xm ; . . . ; xd Þ 1 m1



jm1 2Z

X

Qs ðj%1 ; . . . ; j%m ; xmþ1 ; . . . ; xd Þ 

jm 2Z

 Lk1 ðrxm  jm Þj

p

m 1 Y

!1=p jLk1 ðrxi  ji Þj

:

ð43Þ

i¼1

Let the nature number N satisfy r52N 42r: For 04s4N  1; notice that @k Qs ðxÞ @xkm

is entire functions of spherical exponential type 2s : From (40), (43)

and Lemma 3.2, we obtain Z



R

Z

jGm;s ðxÞjp dxm    dx1 R

5rkpmþ1

Z X



R j1 2Z

5rkp

Z R



 p X  @k Q s   %1 ; . . . ; j%m1 ; xm ; . . . ; xd Þ dxm ð j  @xk  m jm1 2Z

 Z  k @ Q s p   dxm    dx1 : ðxÞ  k  R @xm

Further, by the generalized Bernstein inequality [6, p. 116], we know jjGm;s jjp 5rk 2sk jjQs jjp :

ð44Þ

By (44) and the proof of (27), we obtain   X  N 1 N 1 X   Qs  Ts;k1 Qs  5Oðs1=d Þ:   s¼0  s¼0

ð45Þ

p

For s5N ; by Ho! lder inequality and jLn ðtÞj4AeBt ; we obtain 0 jðTs;k1 Qs ÞðxÞj5@

X j2Z d

jQs ðj%Þj

p

d Y i¼1

11=p jLk1 ðrxi  ji ÞjA

:

831

GENERALIZED BESOV CLASSES

Therefore, by Lemma 4.1, we obtain 0 jjðTs;k1 Qs Þjjp 5@

X

11=p jQs ðj%Þjp =rd A

5

j2Z d

2sd jjQs jjp : s

ð46Þ

Hence, by (46) and the equivalent norm of jjf jjBO in Theorem 1.1, we have py

1 X

1 X 2sd jjQs jjp s s¼N s¼N )1=y0 , (   )1=y (X 1 1 X jjQs jjp y sdy0 y0 s 5 2 O ð2 Þ s Oð2s Þ s¼N s¼N ( )1=y0 1   X Oð2N Þ sðdaÞy0 1=d 5O s 2 ; 5jjf jjBO ðRd Þ py 2aN s s¼N

jjQs  Ts;k1 Qs jjp 5

ð47Þ

for a > d: By (3), (45) and (47), we obtain the upper bound of the quantities in Theorem 1.3. Lower estimate: For any x 2 Ys ; there exists n 2 N with Mn ¼ cardðx \ ½n; nd Þ4ð2nÞd 2s: For s > Mn ; there exist 14j4s such that x \ Intðsupp ðfj ÞÞ ¼ F (here the set IntðQÞ and F denote the interior of the set Q  Rd and
 the empty set, respectively). Let f% ¼ fj =jjfj jj O d : Then, f% 2 S O Rd and Bpy ðR Þ

Ix f% ¼ 0: From the proof of Theorem 1.2, we obtain jjf%jjp 4Oðs1=d Þ:

py

ð48Þ

Hence, the lower bound of these quantities in Theorem 3 follows (5) and (48). ]

ACKNOWLEDGMENTS We are grateful for the valuable comments from the referee.

REFERENCES 1. Din Zung, Average e-dimension of functional class BG;p ; Mat. Zametki 28 (1980), 727–736. 2. Jiang Yanjie, Liu Yongping, Average widths and optimal recovery of multivariate Besov classes in Lp ðRd Þ; J. Approx. Theory 102 (2000), 155–170. 3. Li Chun, Infinite dimensional widths of function classes, J. Approx. Theory, 69 (1992), 14–34.

832

LIU AND XU

4. Liu Yongping, Xu Guiqiao, Some extremal properties of multivariate polynomial splines in the metric Lp ðRd Þ; Sci. China 31 (2001), 307–313. 5. G. G. Magaril-Il’yaev, Average dimension, widths and optimal recovery of Sobolev classes on the real axis, Math Sbornik 182 (1991), 1635–1656. [In Russian] 6. S. M. Nikol’skii, ‘‘Approximation of Functions of Several Variables and Imbedding Theorems,’’ Springer-Verlag, New York, 1975. 7. A. Pinkus, ‘‘N-widths in Approximation Theory,’’ Springer-Verlag, New York, 1985. 8. N. N. Pustovoitov, Representation and approximation of multivariate periodic functions with a given modulus of smoothness, Anal. Math. 20 (1994), 35–48. 9. Sun Yongsheng, Wang Heping, Representation and approximation of multivariate functions with bounded mixed modules of smoothness, Proc. Steklov Inst. Math. 219 (1997), 850–371. 10. J. F. Traub and H. A. Wo!zniakowski, ‘‘A General Theory of Optimal Algorithms,’’ Academic Press, New York, 1980.