THE GEL'FAND n-WIDTHS OF MULTIVARIATE PERIODIC BESOV CLASSES

2003,23B(3) :399-404

.Atat~cta.9'cientia

If.~JJ!fm THE GEL'FAND n-WIDTHS OF MULTIVARIATE PERIODIC BESOV CLASSES 1 )

Xu Guiqiao ( *-t~ Department of mathematics, Tianjin Normal University, Tianjin 300074, China Yu Chunwu ( ~#.A ) Department of Compter, Wuhan University, Wuhan 430072, China Abstract The Gel'fand n-widths of Besov classes are considered. The weak asymptotic behavior is established for the corresponding quantity. Key words

Besov classes; Gel'fand n-widths; weak asymptotic behavior

2000 MR Subject Classification

1

41A30; 41A63

Introduction

Let Lp(T d ) , 1 ::; p ::; 00 denote the usual periodic Lp-norm space on t» . Suppose that kEN, and hERd. For each f E Lp(T d) ,

fj.~f(x) = i: _1)1+k ( 1=0

k ) f(x l

+ lh)

(1.1)

is the k t h difference of the function f at the point x with step h. The k-order modulus of smoothness nk(f, t)p of f is defined to be

(1.2)

e::;

d If k > 0: > 0 and 1 ::; p, 00, then the periodic Besov spaces B;() = B;() (T ) consists of d all functions f defined on T such that the norm

is finite, where

IflBa

p6

:= {

1 ::;

e < 00,

e=

00.

Let

1 Received October 8, 2001. The project was supported by the Development Foundation of Science and Technology of Tianjin Universities (52LD47)

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ACTA MATHEMATICA SCIENTIA

Vol.23 Ser.B

For C C Lp(Td), denote by a;«: Lp(Td)), dn(C, Lp(Td)), and d~(C, Lp(T d)) the Kolmogorov n-width, the Gel'fand n-width and the linear n-width of C in Lp(Td), respectively. (see [1] for above definitions). . Many authors such as Romaniuk[2], [3],[4], Pustovoitov, N.N[5J and Sun Yongsheng, Wang Heping[6J discussed the Kolmogorov n-widths of periodic Besov type function classes in the metric of Lp(T d). We will consider the Gel'fand n-widths of periodic Besov classes in the metric of Lp(T d). Our results are Theorem If 1 ~ q, p ~ 00, and 0: > 2d, then for n large enough

e,

where l/p + l/p' = l/q that A/C ~ B ~ CA.

2

+ l/q'

n-;t ,

1

~

~

q

~

n-;r+~-~ ,

1

~ p ~

2

~ q ~ 00,

p

2,

= 1, A ::::: B means there exists C independent of

f and n such

Some Lemmas Let

= 1 + 2 L cos kt + 2

L

m

Vm(t)

2m

k=l

((2m - k)/m) cos kt

k=m+1

be the de la Vallee Poussin kernel. Then the multidimensional de la Vallee Poussin kernel are defined by d

Vm(x) :=

II Vm(Xj) j=l

for mEN. For functions f on Td consider the convolution operator Vmf := f * Vm defining the de la Vallee Poussin sums of f. The differences of successive de la Vallee Poussin sums are defined by if?of:= Vi], if?sf:= V2 . f - V2 . - 1f , S = 1,2,···. Lemma 1[7J

If 0:

> 0, 1 ~ p, e ~

00,

then 1~

e < 00,

e = 00. Let Tm denote the set of trigonometric polynomials of order

1,'" ,d. Lemma 2[8J

If 1 ~ p ~

00,

then for every

Ilfllp

:::::

m- d / p {

f

L

< 2m,

m in each variable x i» j =

E r;

If(hkW }

l/P

p~

where Pm := {k E Zd : -2m ~ kj

~

(2.1)

j = 1, ... ,d}, h = 2:'

(2.2)

Xu & Yu: GEL'FAND n-WIDTHS OF MULTIVARIATE PERIODIC BESOV CLASSES

No.3

For 1

~

p ~

00,

401

Rm can be equipped by the norm

with the usual change to the max norm when p = 00. Let B(l;;') denote the unit ball in l;;'. Lemma 3 Let p < q, a > d(i - ~). Then

(2.3) where nk are non-negative integers for which E~o nk ~ n. Proof First, we list two fact about n-widths (see [2]). Let L(X, Y) denote the set of continuous maps from X to Y. A) Denote dn(Tj Xj Y) the Gel'fand n-width of the set {Tx : IIxll ~ I} as a subset of Y. Let T; E L(X, Y), i = 1,2. Then ~+m(Tl +T2;X;Y)

s ~(TljXjY) +r(T2;X;Y).

B) Let T 1 E L(X, Z) and T 2 E L(Z, Y). Then T

(2.4)

= T2T1 E L(X, Y), and (2.5)

where

Since f

IIT1 x llz II T11I (x ,z ) = sup II x II X . 0:#0

= Et:'o ~kf for

f E B~IJ it follows that 00

dn(S~IJ,Lq(Td))

s L~k(~k;B~IJ;Lq(Td)). k=O

We wish to prove that

for k E Z+. We factor the operator ~k : B~IJ --t T2k+1 n Lq(T d) as follows: ~k is ~k! from B~IJ to Lp(Td), then by Lemma 1

lI~kll(B;9,Lp(Td»

where Pk:= {m E

z«, _2k+2 ~

mj

< 2k+2,

j

«2-

ak.

= 1,···,d}.

(2.6)

By Lemma 2

(2.7)

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ACTA MATHEMATICA SCIENTIA

.. (k+3)d . "2(k+3)d 1 I denotes the identity map of l; mto lq ,LJ; T2k+1 n Lq(T d) by

.

IS

Vol.23 Ser.B . 2(k+3)d . the operator takmg from lq mto

By computation we know

It is well known that for

f E T2k+1, LJ;IL k (f)

=

f. By Lemma 2 and (2.8) we know (2.9)

By (2.4)-(2.9) we obtain Lemma 3.

3

Proof of Theorem Upper estimate

(1) We will prove that for 1 ::; p, q ::; d d'n (S'" pO' L q (T ) ) «

+00,

n -~+(i-~)+ .

(3.1)

From Ilfllp « Ilfllq for p ::; q and the imbedding theorem of Besov classes we know that we only need to consider the case p = q. Let 2m ~ n I / d and Tm(f, x) = f * V2~ (x). Then T m is a continuous linear operator from S;o to T2~+1, dim(T2~+1, t; (Td)) << nand +00

f(x) - Tm(f, x) =

L


(3.2)

j=m+I

For 1

< () < +00, f

E

S;O, by Lemma 1 we know

Similarly, we can verify that (3.3) also hold for () (2) We will prove that

= 1,

Q

d" (S;" L. (T'))

By

Ilfllt ::; Ilflls for

for 2 ::; q ::;

00.

«{

00.

(3.1) is proved.

n-d:,

1 ::; p ::; q ::; 2,

n-~+i-!,

1 ::; p ::; 2 ::; q ::;

(3.4) 00.

1 ::; t ::; s ::; 00 we know that we only need to prove

No.3

Xu & Yu: GEL'FAND n-WIDTHS OF MULTIVARIATE PERIODIC BESOV CLASSES

nk = {

2(k+3)d

o ~ k ~ m,

2(2m-k;d,

m

0,

< k < 2m, 2m < k.

403

(3.5)

<< n. For 0 ~ k ~ m, d2(k+3 jd(B(Zf+ 3)d), Z~(k+3jd) = 0 2(k+3jd 2(k+3)d and for 2m ~ k, u: (B(ll ), Zq ) = 1. For m < k < 2m, from [1] we know Thus 2::=0 2(k+ 3)d + 2:~:~~1 2(2m-k)d JO

( 3jd) d2(2m- kjd(B(lr k+3)d), Zt+

«

2(k+3)d

T(m-k/2)d(1+ln 2(2m-k)d )3/2

«

2-(m-k/2)d(k_m)3/2. (3.6)

By Lemma 3 and (3.6) we obtain 2m-l

L:

dn(Sfe, Lq(T d)) «

«

2(-"'+d+~)kT(m-~)d(k - m)3/2

+

k=m+l

= L:

2(-"'+d-~)k

k=2m

n-~+!-t.

By (3.1) and (3.4) we obtain the upper estimate. Lower estimate Let A = 4([n 1 / d ] + 1) and for 1 ~ Z~ A, denote

'Pl(t) = sink (At)x[27T(l-1)/A,27Tl/Aj (t), t E R, where

X[a,b]

(3.7)

denote the characteristic function of the interval [a, b]. For s = Ad, let the set

{h,'" ,fs} denote the all functions of the set {n:=l 'Pl;(Xj) : 1 ~ Zj ~ A,1 ~ j ~ d}. Define 1> the periodic function set defined by

(3.8) For any 9 E (Lq(T d))*, define

9 E (zg)* by s

g(al,'" ,as) = g(L:ajfj). j=l For any L

(3.9)

= {gl' ... , gn} C (L q (T d ))* , there exists a E Z~ such that = 0, i = 1,"',nj

(3.10)

2:: dn(B(Z;), zg)lI a llp,

(3.11)

gi(a) Ilall q

where Iiali p = Ilalll~ for 1 ~ p ~ 00. Let f = 2:;=1 ajIi- Then by (3.9), (3.10) and (3.11), we know f E L1- (where L1- = {x E Lq(T d) : f(x) = 0, 'Vf E L}). It is easy to verify (3.12)

and for r E

zt, Irl = k, II ~:~

L= r: g(1

1 I(sink)(r i ) (t)IPdt) l/Pllallp'

(3.13)

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Vo1.23 Ser.B

ACTA MATHEMATICA SCIENTIA

(3.14)

By Minkowski inequality, (3.12), (3.13), (3.14) and (1.1) we can obtain

For 1

~

() <

00,

f2k(f, t)p « t k Ak-d/Pllall p •

(3.15)

f2 k(f,t)p« A-d/Pllallp'

(3.16)

from (3.15) and (3.16), we know

(3.17)

By (3.12), and (3.17)

11111B;9

«Aa-d/Pllall p •

Similarly, we can verify that (3.18) hold for () = 00. Let 117118'p9> = 1, (3.11), (3.12), (3.13) and (3.18) we know

11111 q » By [1] we know that dn(B(l~),l~)

cf'(B(l;), l~)A

(3.18)

7 = lI111I1B;9' then 7 E L1. n B;o'

-a+:-:.

By

(3.19)

= dn(B(l:t), l{f) and 1

nq-

1

p,

1

p,

1

1

n q-

1

1

~

q

~

p

~ 00,

2

~

p

~

q

~ 00,

n q - 2 , 1 ~ p ~ 2 ~ q ~ 00, 1,

1

~

p

~

q

~

(3.20)

2.

By (3.19), (3.20) and A ~ n 1 / d we can obtain the lower estimate of dn(S;o, Lq(T d ) . References 1 Pinkus A. N-widths in approximation theory. New York, 1985 2 Romaniuk A S. Approximation for Besov-type classes of periodic functions in L q space. Ukrain Mat Zh, 1991, 43(10):1398-1408 3 Romaniuk A S. Optimal trigonometric approximation and Kolmogorov widths for Besov classes of multivariate functions. Ukr Mat Zh, 1993, 45(5): 663-675 4 Romaniuk A S. Kolmogorov widths for classes B;,o of multivariate periodic functions with small smoothness in the space L q . Ukr Mat Zh, 1994, 46(7): 915-926 5 Pustovoitov N N. Representation and approximation of multivariate periodic functions with a given modulus of smoothness. Analysis Math, 1994, 20: 35-48 6 Sun Yongsheng, Wang Heping. Representation and approximation of multivariate functions with bounded mixed moduli of smoothness. Proceedings of the Steklov Institute of Mathematics, 1997, 219: 350-371 7 Nikol'skii S M. Approximation of Functions of Several Variables and Imbedding Theorems. New York: Spring-Verlag, 1975 8 Zygmund A. Trigonometric Series II. Cambridge: Cambridge Univ Press, 1959