Average onesided widths of Sobolev and Besov classes

Acta Mathematica Scientia 2010,30B(1):148–160 http://actams.wipm.ac.cn

AVERAGE ONESIDED WIDTHS OF SOBOLEV AND BESOV CLASSES∗

)

Yang Zhuyuan (

School of Mathematics and Computer Science, Yunnan Nationalities University, Kunming 650031, China E-mail: [email protected]

)

Yang Zongwen (

Department of Mathematics, Yunnan University, Kunming 650091, China E-mail: [email protected]

Liu Yongping (


)†

Department of Mathematics, Beijing Normal University, Beijing 100875, China E-mail: [email protected]

Abstract The article concerns the average onesided widths of the Sobolev and Besov classes and the classes of functions with bounded moduli of smoothness. The weak asymptotic results are obtained for the corresponding quantities. Key words average onesided widths; Sobolev classes; Besov classes 2000 MR Subject Classification

1

41A29; 41A25; 41A10

Introduction

In the 1990’s, G. Freud [1] first considered the onesided polynomial approximation. After that (1960–80’s), T. Ganelius, A. Meir, A. Sharma, A. Pinkus, V.A. Popov, and many other authors respectively studied the onesided approximation by the trigonometrical polynomials, splines and entire functions of exponential type, where the approximated function classes are defined on T = [0, 2π] or on the whole real axis R, and the approximative metric is L∞ or L1 . N.P. Korneichuk [2] systematically summarized the results of onesided approximation of univariate periodic Sobolev classes, including a few results of onesided widths. The onesided approximation of several variables was firstly studied by V.P. Popov. In [3], he discussed the periodic cases. V.H. Hristov and K.G. Ivanov [4] studied the onesided approximation by the trigonometrical and entire functions of exponential type and obtained the same approximative order as the classical cases for Sobolev classes. Y.P. Liu [5], Y.J. Jiang and Y.P. Liu [6], G.Q. Xu and Y.P. Liu [7], Y.P. Liu and G.Q. Xu [8, 9] studied the average ∗ Received June 15, 2007; revised April 15, 2008. Supported by the Foundation of Education Department of Yunnan Province (07Z10533), Supported partly by the National Natural Science Foundation of China (10471010) and partly by the project “Representation Theory and Related Topics” of the “985 program” of Beijing Normal University, Supported by the Science Foundation of Yunnan University (2008YB027). † Corresponding author: Liu Yongping.

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widths of Sobolev and Besov function classes. But little was achieved on the onesided widths of multivariate cases. In this article, we consider the average onesided widths of Sobolev and Besov functional classes and the classes of functions with bounded moduli of smoothness. Let (X(Rd ,  · X )) be a normed space of real functions on Rd . Denote by BX the unit ball in X. Let α > 0 and Pα be the continuous linear operator in X(Rd ) defined by Pα f (x) = χα (x)f (x), x ∈ Rd , where χα is the characteristic function of the cube Iαd = [−α, α]d . Let ε > 0 and L be a subspace of X(Rd ). Define K (α, L, X(Rd )) := min {n ∈ Z+ |dn (Pα (L ∩ BX(Rd )), X(Rd )) < ε}, where Z+ denotes the set of positive integers, and dn (A, X) the Kolmogorov n-width of A in X. The average dimension of the subspace L in X(Rd ) is defined by dim(L, X(Rd )) := lim lim inf →0 α→∞

k (α, L, X(Rd )) . (2α)d

Let σ > 0 and C be a centrally symmetric subset of X(Rd ). The average onesided Kolmogorov σ-width of C in X(Rd ) is defined by +

dσ (C, X(Rd )) := inf E + (C, L, X(Rd )), L

where the infimum is taken over all subspaces L ⊂ X(Rd ) with dim(L, X(Rd )) ≤ σ and E + (f, L, X(Rd)) := inf{f − gX : f ≤ g ∈ L}, E + (M, L, X(Rd)) := sup{E + (f, L, X(Rd )) : f ∈ M } for any subspace L of X(Rd ) and M a subset of X(Rd ). For 1 ≤ q < ∞, r ∈ N , the Sobolev classes are defined by       ∂αf  ˙ qr (Rd ) := f ∈ Lq : SW  α ≤ 1 , ∂x q |α|=r

      ∂αf  SWqr (Rd ) := f ∈ Lq :  α ≤ 1 , ∂x q |α|≤r

where

∂αf ∂xα

=

∂ |α| f α , α ∂x1 1 ···∂xd d

d α = (α1 , · · · , αd ) ∈ Z+ , |α| =

d  j=1

αj .

For 1 ≤ p, q ≤ ∞, denote by Lpq (Rd ) the linear normed space of locally Lp -integrable functions f with the finite norm ⎧  1/q q ⎪ ⎪ f (· + v) , 1 ≤ q < ∞, d ⎨ Lp [0,1] d v∈Z f pq := ⎪ ⎪ ⎩ sup f (· + v)Lp [0,1]d , q = ∞, v∈Z d

 where f Lp(E) = ( E |f (x)|p dx)1/p , 1 ≤ p < ∞, f L∞(E) = sup |f (x)| for the Lebesgue x∈E

measurable subset E of Rd . When p = q, Lqq (Rd ) = Lq (Rd ) is the usual Lq -space. When p > q, there hold the following relations [7]: f pq ≥ f p , f pq ≥ f q , Lpq (Rd ) ⊂ Lp (Rd ) ∩ Lq (Rd ),

(1)

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f qp ≤ f p , f qp ≤ f q , Lqp (Rd ) ⊃ Lp (Rd ) ∪ Lq (Rd ).

(2)

For 1 ≤ p, q ≤ ∞, r ∈ N , the Sobolev–Wiener classes are defined by       ∂αf  r ˙ pq SW (Rd ) := f ∈ Lpq : f ˙ rpq =  α ≤ 1 , ∂x pq |α|=r

      ∂αf  r (Rd ) := f ∈ Lpq : f rpq = SWpq  α ≤ 1 . ∂x pq |α|≤r

Theorem 1 Suppose 1 ≤ q ≤ p ≤ ∞, d < r ∈ N , then + ˙ r (Rd ), Lq (Rd )) σ −r/d (σ → +∞), dσ (S W q +

dσ (SWqr (Rd ), Lq (Rd )) σ −r/d (σ → +∞), +

˙ r (Rd ), Lq (Rd )) σ −r/d (σ → +∞), dσ (S W pq +

r dσ (SWpq (Rd ), Lq (Rd )) σ −r/d (σ → +∞). k  k d k l+k f (x + lt) is the k -th difference (−1) Suppose that k ∈ N and t ∈ R , Δt f (x) = l l=0 r of the function f at the point x with step t. Let k > r > 0, 1 ≤ θ ≤ ∞. We say f ∈ Bqθ (Rd ) if f satisfies the following conditions (i) f ∈ Lq (Rd ); (ii) f brqθ (Rd ) < ∞, where

f brqθ (Rd ) =

⎧ ⎪ ⎪ ⎪ ⎨

 Δk f  θ dt 1/θ q t , 1 ≤ θ < ∞, r |t| |t|d d R

⎪ Δkt f q ⎪ ⎪ , ⎩ sup |t|r t=0

θ = ∞.

Remark 1 There are some semi-norms equivalent to the above semi-norm. As an example we list one of them as follows:  ∞ 1/θ ωk (f, t)q (∗) (∗) f br = t−1−θr ωk (f, t)θq dt , 1 ≤ θ < ∞; f br = sup . q∞ qθ tr t>0 0 r (Rd ) if f satisfies the following conditions: We say f ∈ Bpqθ d (i) f ∈ Lpq (R ); (ii) f brpqθ (Rd ) < ∞, where

f brpqθ (Rd ) =

⎧ ⎪ ⎪ ⎪ ⎨

 Δk f  θ dt 1/θ pq t , 1 ≤ θ < ∞, r |t| |t|d d R

⎪ Δkt f pq ⎪ ⎪ , ⎩ sup |t|r t=0

θ = ∞.

r r Bqθ (Rd ) and Bpqθ (Rd ) are two Banach spaces with norms

r (Rd ) = f q + f Bqθ

d  j=1

r (Rd ) = f pq + f br f brqθ (Rd ) , f Bpqθ d , pqθ (R )

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respectively. r r When p = q, Bpqθ (Rd ) = Bqθ (Rd ). Set Sbrqθ (Rd ) = {f ∈ Lq (Rd ) : f brqθ (Rd ) ≤ 1}, r r (Rd ) ≤ 1}, SBqθ (Rd ) = {f ∈ Lq (Rd ) : f Bpθ

Sbrpqθ (Rd ) = {f ∈ Lpq (Rd ) : f brpqθ (Rd ) ≤ 1}, r r (Rd ) ≤ 1}. (Rd ) = {f ∈ Lpq (Rd ) : f Bpqθ SBpqθ

Theorem 2 Suppose 1 ≤ q ≤ p ≤ ∞, 1 ≤ θ ≤ ∞, d/q < r ∈ N , then +

dσ (Sbrqθ (Rd ), Lq (Rd )) σ −r/d (σ → +∞), +

r dσ (SBqθ (Rd ), Lq (Rd )) σ −r/d (σ → +∞), +

dσ (Sbrpqθ (Rd ), Lq (Rd )) σ −r/d (σ → +∞), +

r dσ (SBpqθ (Rd ), Lq (Rd )) σ −r/d (σ → +∞).

Now, we introduce the definition of the class of functions with bounded moduli of smoothness, which is a generalization of the usual Besov class. Definition 1 [10] Let ω denote a non-negative function on R+ = {t : t ≥ 0}, we say that ω(t) ∈ Φ(k, α, β) if it satisfies: (i) ω(0) = 0, ω(t) > 0 for any t > 0; (ii) ω(t) is continuous; (iii) ω(t) is almost increasing, i.e., for any two points t, τ such that 0 ≤ t ≤ τ , we have ω(t) ≤ Cω(τ ), where C ≥ 1 is a constant independent of t; (iv) For any n ∈ Z+ , ω(nt) ≤ Cnk ω(t), where K ≥ 1 is a fixed positive integer, C > 0 is a constant independent of n and t; (v) There exists α > 0, such that ω(t)/tα is almost increasing; (vi) There exists β (0 < β < k) such that ω(t)/tβ is almost decreasing, i.e., there exists C > 0 such that, for any two points t, τ, 0 < t ≤ τ , there always holds ω(t)/tβ ≥ Cω(τ )/τ β . Definition 2 Let ω(t) ∈ Φ(k, α, β), k ∈ N, 0 < α, 0 < β < k, 1 ≤ θ ≤ ∞, 1 ≤ q < ∞. We ω say f ∈ Bqθ if f satisfies (i) f ∈ Lq (Rd ); (ii) ⎧   +∞  ⎪ ωk (f, t)q θ dt 1/θ ⎪ ⎪ < ∞, 1 ≤ θ < ∞, ⎨ ω(t) t 0 f bωqθ (Rd ) = ⎪ ωk (f, t)q ⎪ ⎪ < ∞, θ = ∞, ⎩ sup ω(t) t>0 where wk (f, δ)q(Rd ) = sup{Δkh f (·)q(Rd ) : |h| ≤ δ}. ω ω (Rd ) = f q + It is easy to see that Bqθ (Rd ) is a Banach space with the norm f Bqθ α ω d α d f bωqθ (Rd ) . When ω(t) = t , Bqθ (R ) is the usual Besov space Bqθ (R ). Definition 3 Let ω(t) ∈ Φ(k, α, β), k ∈ N, 0 < α, 0 < β < k, 1 ≤ θ ≤ ∞, 1 ≤ p, q < ∞. ω We say f ∈ Bpqθ if f satisfies (i) f ∈ Lpq (Rd );

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(ii)

⎧ ⎪ ⎪ ⎪ ⎨

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 ω (f, t) θ dt 1/θ k pq < ∞, 1 ≤ θ < ∞, ω(t) t 0 = ⎪ ωk (f, t)pq ⎪ ⎪ < ∞, θ = ∞. ⎩ sup ω(t) t>0

f bωpqθ (Rd )

+∞

ω ω (Rd ) = f pq + It is easy to see that Bpqθ (Rd ) is a Banach space with the norm f Bpqθ

f bωpqθ (Rd ) Set

d d Sbω d ≤ 1}, qθ (R ) = {f ∈ Lq (R ) : f bω qθ (R ) ω ω (Rd ) ≤ 1}, SBqθ (Rd ) = {f ∈ Lq (Rd ) : f Bqθ d d Sbω d ≤ 1}, pqθ (R ) = {f ∈ Lpq (R ) : f bω pqθ (R ) ω ω (Rd ) ≤ 1}. SBpqθ (Rd ) = {f ∈ Lpq (Rd ) : f Bpqθ

Theorem 3 β < k, then

Suppose 1 ≤ q ≤ p ≤ ∞, 1 ≤ θ ≤ ∞, ω(t) ∈ Φ(k, α, β), k ∈ N, d/q < α, 0 < +

d d −1/d dσ (Sbω ) (σ → +∞), qθ (R ), Lq (R )) ω(σ +

ω dσ (SBqθ (Rd ), Lq (Rd )) ω(σ −1/d ) (σ → +∞), +

d d −1/d dσ (Sbω ) (σ → +∞), pqθ (R ), Lq (R )) ω(σ +

ω dσ (SBpqθ (Rd ), Lq (Rd )) ω(σ −1/d ) (σ → +∞).

2

Preliminaries and Some Lemmas

To prove the theorems, we need to give the conception of the average modulus. We consider Rd as a normed vector space with elements x = (x1 , x2 , · · ·, xd ) and norm x = max{|xi | : i = 1, 2, · · ·, d}. U (x, δ) = {y ∈ Rd : y − x < δ} is the

δ neighbourhood of the point x. As k  k d k l+k f (x + lh) is the k -th difference of the above, let k ∈ N and h ∈ R , Δh f (x) = (−1) l l=0 function f at the point x with step h. We denote the local modulus of f by wk (f, x, δ) = sup{|Δkh f (y)| : y, y + kh ∈ U (x, kδ/2)}. Two global moduli of the function f are defined by wk (f, δ)q = sup{Δkh f (·)q : |h| ≤ δ}, the usual modulus of smoothness, and τk (f, δ)q = wk (f, ·, δ)q , the average modulus of smoothness. The properties of wk are assumed to be known. Some properties of τk were given in [11, 12]. The following properties will be employed later. 

δ |α| Dα f q ,

(3)

wk (f, u)q u−d/q−1 du (rq > d).

(4)

τ1 (f, δ)q ≤ c(d)

0≤αi ≤1 |α|>0

 τk (f, t)q ≤ c · td/q

0

t

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In what follows we denote by SBσ,q (Rd ) the restriction to Rd of the space of all functions d   sin(σxi /2) 2  π 2 of spherical exponential type σ, which belongs to Lq (Rd ). Set Φσ (x) = . σxi /2 2 i=1

It is easy to verify that Lemma 1 (i) Φσ (x) ≥ 0, x ∈ Rd ; (ii) Φσ (x) ≥ 1, x ≤ πσ ;  (iii) Rd Φσ (x)dx ≤ σc1d ;  (iv) Φσ (x − 2πj σ ) ≤ c2 , where c1 , c2 are two positive constants independent of σ; j∈Z d

(v) Φσ (x) ∈ SBσ,q (Rd ). Using the Jensen inequality, we have     1/q   2π d  Lemma 2  aj Φσ (x − 2πj |aj |q , c3 > 0. σ ) ≤ c3 ( σ ) q

j∈Z d

j∈Z d

For the proofs of theorems, following the thoughts of [4], we construct the next two onesided operators. ˙ r (Rd ). Denoted by gal a function of entire spherical degree al , such Suppose that f ∈ S W q that     ∂αf  f − gal q ≤ 2a−lr  α , ∂x q |α|=r

and put Qa0 = ga0 , Qal = gal − gal−1 (l = 1, 2, · · ·). Then, in the sense of Lq , f (x) =

∞  l=0

Qal (x).

For given σ > 1 , set σ < aN < 2σ. The onesided operators will be employed in the proof of Theorem 1 as follows: Sσ+ (f, x) =

N 

Qal (x) +

l=0

∞     2πj    sup Φσ x − Qal (y).  σ y∈U ( 2πj , π ) d

 j∈Z

σ

σ

l=N +1

Next, we construct the onesided operators which will be employed in the proofs of Theorem 2 and Theorem 3 . Let  sin σt 2s Jσ,s (t) = (t ∈ R), Kσ,s (x) = λ−1 σ,s Jσ,s (|x|), t where  d  λσ,s = Jσ,s (|x|)dx σ 2s−d (2s > d), |x|2 = x2i . Rd

It is well known that  Rd

Set

Kσ,s (x)|x|l dx σ −l (l > 0, 2s > d + l).

 Tσ (f, x) = −

i=1

k 

(−1)(j+k)

Rd j=1

(5)

k f (x + jt)Kσ,s (t)dt, ∀f ∈ Lp , j

then Tσ (f, ·) ∈ SB2sσ,q (Rd ), 1 ≤ q ≤ ∞. The onesided operators will be employed in the proofs of Theorem 2 and Theorem 3 as follows:   2πj  Tσ+ (f, x) = Tσ (f, x) + sup Φσ x − |f (y) − Tσ (f, y)|. (6) σ y∈U ( 2πj , π ) d j∈Z

σ

σ

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Lemma 3 (a) Sσ+ (f, ·) ∈ SB2σ,q (Rd ), Tσ+ (f, x) ∈ SB2sσ,q (Rd ), (b) f (x) ≤ Sσ+ (f, x), f (x) ≤ Tσ+ (f, x). N  Qal (x) ∈ SB2σ,q (Rd ), Tσ (f, ·) ∈ SB2sσ,q (Rd ) Proof (a) follows from Lemma 1(v) and l=0

and (b) follows from Lemma 1 (i), (ii). Lemma 4 [13] dim(SBσ,q (Rd ), Lq (Rd )) = σ d /π d ≤ σ d .

3

Proofs of Theorems

Now, we begin the estimate of the approximative order. Noticing the definitions of the average width and average onesided width, we easily know that +

dσ (C, X(Rd )) ≤ dσ (C, X(Rd )).

(7)

From Y.P. Liu [5], Y.J. Jiang, and Y.P. Liu [6], G.Q. Xu and Y.P. Liu [7], Y.P. Liu and G.Q. Xu [8], the lower estimates of these functions classes of Theorems 1, 2, 3 are given. So, we only give the upper estimates of Theorems 1, 2, 3. In what follows by ci or Ci we denote positive constants which might only depend on k, r, d, q, p, θ. + Proof of Theorem 1 The upper estimate of dσ of Sobolev classes       ∂αf  ˙ qr (Rd ) := f ∈ Lq : SW  α ≤ 1 . ∂x q |α|=r

˙ qr (Rd ), from Lemma 3, Sσ+ (f ) is a onesided operator, next, we begin the Suppose f ∈ S W estimate of the approximative order. ∞ ∞          2πj       f − Sσ+ (f )q ≤  sup Qal  +  Φσ x − Qal (y) := I1 + I2 ,  σ q 2πj π q y∈U ( , ) d l=N +1

j∈Z

σ

l=N +1

σ

∞ ∞ ∞            I1 =  Qal  =  (gal − gal−1 ) ≤ gal − gal−1 q q

l=N +1 ∞ 

≤

q

l=N +1

(f − gal q + f − gal−1 q ) ≤ 2

l=N +1 ∞  

≤2

l=N +1

∞ 

f − gal q

l=N

2a−lr

     a−N r    ∂αf   ∂αf   α = 4  α  ≤ C1 σ −r . −r ∂x q 1−a ∂x q

|α|=r

l=N

Let aj =

sup

|α|=r

∞      Qal (y) := 

π y∈U ( 2πj σ , σ ) l=N +1

sup π y∈U ( 2πj σ ,σ )

|h(y)|,

π 2π d and noticing that mes{U ( 2πj σ , σ )} = ( σ ) . From Lemma 2, we get

     1/q ∞      2π d 2πj      sup I2 =  Φσ x − Qal (y) ≤ c3 aqj  σ σ 2πj π q y∈U ( , ) d d j∈Z

≤ c3

   j∈Z d

π U ( 2πj σ ,σ)

σ

σ

l=N +1

  1/q q aj dx = c3 j∈Z d

π U( 2πj σ ,σ)

j∈Z

 sup π y∈U ( 2πj σ ,σ )

q 1/q |h(y)| dx

(8)

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≤ c3

 

≤ c4 ≤ c4

sup π y∈U ( 2πj σ ,σ)

  j∈Z d



 = c4

sup

π U ( 2πj σ ,σ)

  j∈Z d

π y∈U ( 2πj σ ,σ)



π U ( 2πj σ ,σ)

 sup

Rd

q 1/q |h(y) − h(x)| + |h(x)| dx



π U ( 2πj σ ,σ)

j∈Z d

y∈U (x, 3π σ )

155

sup y∈U (x, 3π σ )

 1/q q q |h(y) − h(x)| + |h(x)| dx

 1/q q q |h(y) − h(x)| + |h(x)| dx

 1/q q q |h(y) − h(x)| + |h(x)| dx

  1 ≤ c5 τ1 (h, )q + hq . σ Suppose that r > d, from (3) we have τ1 (h,

∞ ∞    1 )q ≤ τ1 (Qal , 1/σ)q ≤ c(d) σ −|α| Dα Qal q σ 0≤α ≤1 l=N +1

≤ c(d)

≤ c(d)

≤ c(d)

≤ c(d)

l=N +1

∞ 



l=N +1

0≤αi ≤1 |α|>0

∞ 



l=N +1

0≤αi ≤1 |α|>0

∞ 



l=N +1

0≤αi ≤1 |α|>0



σ



σ −|α|

0≤αi ≤1 |α|>0



≤ c6

σ −|α|

∞ 

al|α| (f − gal q + f − gal−1 q )

l=N +1 ∞ 

∞ 

σ −|α|

0≤αi ≤1 |α|>0

≤ c8

σ −|α| al|α| (f − gal q + f − gal−1 q )

l=N

 

σ −|α| al|α| gal − gal−1 q

al|α| f − gal q )

l=N

0≤αi ≤1 |α|>0

≤ 2c6

σ −|α| al|α| Qal q

−|α|

0≤αi ≤1 |α|>0

≤ c6

i |α|>0

      ∂αf  al|α| 2a−lr  α ∂x q

∞ 

|α|=r



al|α| a−lr = c7

l=N



σ −|α| aN (|α|−r) ≤ c9

0≤αi ≤1 |α|>0

0≤αi ≤1 |α|>0

σ −|α|

∞ 

al(|α|−r)

l=N

σ −|α| σ (|α|−r)

0≤αi ≤1 |α|>0

≤ C2 σ −r (r > d).

(9)

Identify with the estimate of I1 , we know hq ≤ C1 σ −r . ˙ qr (Rd ). From (8)–(10), we obtain f − Sσ+ (f )q ≤ Cσ −r for any f ∈ S W

(10)

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+ ˙ qr (Rd ), Lq (Rd )) ≤ Cσ −r/d . From Lemma 4, we know that dσ (S W ˙ r (Rd ) and from (1) we know, when 1 ≤ q ≤ p ≤ ∞, SW r (Rd ) ⊂ Since SWqr (Rd ) ⊂ S W q pq r d r d ˙ ˙ S Wpq (R ) ⊂ S Wq (R ), and the upper estimates of these functional classes are given. From [5] and noticing (7), the lower estimates are given. The proof of Theorem 1 is completed. Using the method of next Theorems 2 and 3, we can get the same results of theorem under condition r > d/q, but not r > d. Here, we just for purpose given a new method. + Proof of Theorem 2 The upper estimate of dσ of Besov classes

Sbrqθ (Rd ) = {f ∈ Lq (Rd ) : f brqθ (Rd ) ≤ 1}. We use the following operator as the approximative operator as listed before: Tσ+ (f, x) = Tσ (f, x) +

j∈Z

Then,

 2πj  sup Φσ x − |f (y) − Tσ (f, y)|. σ y∈U ( 2πj , π ) d



σ

σ

q 1/q     I1 = f − Tσ (f )q = kt f (x)Kσ,s (t)dt dx  Rd Rd    1/q k q ≤ |t f (x)| dx Kσ,s (t)dt = kt f (·)q Kσ,s (t)dt Rd Rd Rd 1/θ    k f (·) θ 1/θ    q t (r+ d θ θ )θ |K dt |t| (t)| dt ≤ σ,s d |t|r+ θ Rd Rd   1   1/θ  d θ = |t|(r+ θ )θ Kσ,sθ (t)dt λ−1 σ,s λσ,sθ  f brqθ (Rd ) Rd  1 −(r+ d θ )θ · θ

≤ c10 σ



1

σ d−2s σ −(d−2sθ ) θ f brqθ (Rd )

= c10 σ −r f brqθ (Rd ) ≤ C3 σ −r , where

1 θ

+

1 θ

= 1(1 < θ < ∞). When θ = ∞, q 1/q     kt f (x)Kσ,s (t)dt dx I1 = f − Tσ (f )q =  Rd Rd   1/q ≤ |kt f (x)|q dx Kσ,s (t)dt Rd Rd    kt f (·)q  r |t| |Kσ,s (t)|dt kt f (·)q Kσ,s (t)dt ≤ = |t|r Rd Rd  kt f (·)q |t|r |Kσ,s (t)|dt ≤ c11 σ −r f brq∞ (Rd ) . ≤ sup |t|r t=0 Rd

When θ = 1,



  

q 1/q  kt f (x)Kσ,s (t)dt dx I1 = f − Tσ (f )q = d d R R    1/q k q ≤ |t f (x)| dx Kσ,s (t)dt = kt f (·)q Kσ,s (t)dt d d d R R R    kt f (·)q kt f (·)q  r+d |t| |K (t)|dt ≤ dt sup |t|r+d Kσ,s (t) ≤ σ,s |t|r+d |t|r+d t=0 Rd Rd   sin σ|t| 2s k  f (·) q t = sup |t|r+d Kσ,s (t) dt = sup |t|r+d λ−1 f brq1 (Rd ) . σ,s |t|r+d |t| t=0 t=0 Rd

(11)

(12)

No.1

Z.Y. Yang et al: AVERAGE ONESIDED WIDTHS OF SOBOLEV AND BESOV CLASSES

When |t| ≤

1 σ,

157

it is easily to know

the last quality ≤ c12 σ −r−d σ −2s+d σ 2s f brq1 (Rd ) ≤ c13 σ −r . When |t| >

1 σ , 2s

> r + d, −2s

the last quality ≤ sup |t|r+d λ−1 σ,s |t| t=0

f brp1 (Rd ) ≤ c14 σ −r−d σ −2s+d σ 2s f brq1 (Rd ) ≤ c15 σ −r .

Thus, when θ = 1, we get I1 ≤ cσ −r .

(13)

Noticing that aj = ≤

sup π y∈U ( 2πj σ ,σ)



y∈U ( 2πj σ

,π σ)



Rd

ωk f,

sup π y∈U ( 2πj σ ,σ)



sup 

≤

|f (y) − Tσ (f, y)| =

  

Rd

|kt f (y)|Kσ,s (t)dt ≤

 2πj 2π , + 2|t| Kσ,s (t)dt σ σ

Rd

  kt f (y)Kσ,s (t)dt

sup Rd

y∈U ( 2πj σ

,π σ)

|kt f (y)|Kσ,s (t)dt

π 2π 2d and mes{U ( 2πj σ , σ )} = ( σ ) . From Lemma 2, we get     2πj    sup I2 =  Φσ x − |f (y) − Tσ (f, y)| σ 2πj π q y∈U ( σ , σ ) j∈Z d  2π d  1/q   1/q ≤ c16 aqj ≤ c16 aqj dx 2πj π σ j∈Z d j∈Z d U( σ , σ )    q 1/q     2πj 2π   , + 2|t| Kσ,s (t)dt dx ≤ c16 ωk f,  2πj π σ σ d R j∈Z d U ( σ , σ )  q 1/q      4π   ≤ c16 ωk f, x, + 2|t| Kσ,s (t)dt dx  2πj π σ d R j∈Z d U ( σ , σ ) q 1/q     4π   + 2|t| Kσ,s (t)dt dx ≤ c16 ωk f, x,  σ Rd Rd q 1/q    4π + 2|t| ≤ c16 ωk f, x, dx Kσ,s (t)dt σ Rd Rd     4π + 2|t| ≤ c16 τk f, Kσ,s (t)dt σ q Rd   1 k ≤ c17 τk f, (4π + 2σ|t|) Kσ,s (t)dt. σ q Rd

From (5) and (4),  1/σ 1 1 1 −d/q the last inequality ≤ c18 τk f, ≤ c19 σ wk (f, u)q u− θ −r u θ +r u−d/q−1 du σ q 0   1/σ 1/θ  1/σ 1   ≤ c19 σ −d/q wkθ (f, u)q u−1−rθ du { u( θ +r−d/q−1)θ du}1/θ 

0

≤ c20 σ −d/q ≤ c21 σ

−r



1/σ



1

u( θ +r−d/q−1)θ du

0

f brqθ ≤ C4 σ

−r

1

1/θ  

0 ∞

0

wkθ (f, u)q u−1−rθ du

 1 +  = 1, 1 < θ < ∞, r > d/q . θ θ

1/θ

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158

ACTA MATHEMATICA SCIENTIA

Vol.30 Ser.B

When θ = ∞, r > d/q, 1 τk (f, )q ≤ c22 σ −d/q σ



1/σ

0

wk (f, u)q u−r u+r u−d/q−1 du

wk (f, u)q ur u>0

≤ c22 σ −d/q sup



1/σ

0

ur u−d/q−1 du ≤ c22 σ −r f brqθ ≤ c23 σ −r .

(15)

When θ = 1, r > d/q,  1/σ  1/σ wk (f, u)q r−d/q wk (f, u)q −(r−d/q) 1 −d/q τk (f, )q ≤ c24 σ −d/q u du ≤ c σ σ du 24 r+1 σ u ur+1 0 0  ∞ wk (f, u)q ≤ c24 σ −r du = c24 σ −r f brqθ ≤ c25 σ −r . (16) ur+1 0 From (11)–(13) and (14)–(16), we obtain f − Tσ+ (f )q ≤ I1 + I2 ≤ Cσ −r for any f ∈ Sbrqθ (Rd ). +

From Lemma 4, we know that dσ (Sbrqθ (Rd ), Lq (Rd )) ≤ Cσ −r/d . r r Since SBqθ (Rd ) ⊂ Sbrqθ (Rd ) and from (1) we know when, 1 ≤ q ≤ p ≤ ∞, SBpqθ (Rd ) ⊂ r d r d Sbpqθ (R ) ⊂ Sbqθ (R ), and the upper estimate of these functional classes are given. From [6, 7] and noticing (7), the lower estimates are given. The proof of Theorem 2 is completed. + Proof of Theorem 3 The upper estimate of dσ of the classes of functions with bounded d d moduli of smoothness classes Sbω d ≤ 1}. qθ (R ) = {f ∈ Lq (R ) : f bω qθ (R ) We still use the following operator as the approximative operator:   2πj  sup Tσ+ (f, x) = Tσ (f, x) + Φσ x − |f (y) − Tσ (f, y)|, σ y∈U ( 2πj , π ) d j∈Z

σ

σ

q 1/q     I1 = f − Tσ (f )q = kt f (x)Kσ,s (t)dt dx  Rd Rd    1/q k q ≤ |t f (x)| dx Kσ,s (t)dt = kt f (·)q Kσ,s (t)dt Rd Rd Rd   ∞  sin σt 2s ≤ ωk (f, |t|)q Kσ,s (t)dt = |Ωd−1 | ωk (f, t)q td−1 λ−1 dt σ,s t Rd 0    sin σt 2sθ 1/θ  ∞  ω (f, t) θ dt 1/θ  ∞   k q ≤ |Ωd−1 | ω θ (t)tdθ −1 dt · λ−1 σ,s . ω(t) t t 0 0 Because ω(t)/tα is almost increasing, we know  1/σ  1/σ 2sθ   sin σt 2sθ   θ dθ  −1 sin σt θ αθ  ω (t)t dt ≤ c26 ω (1/σ)σ tαθ tdθ −1 dt t t 0 0  1/σ     sin σt 2sθ ≤ c27 σ −dθ +1 ω θ (1/σ) dt t 0 



≤ c28 σ (2s−d)θ ω θ (1/σ). Because ω(t)/tβ is almost decreasing, we know  ∞  ∞  sin σt 2sθ  sin σt 2sθ       ω θ (t)tdθ −1 dt ≤ c29 σ βθ ω θ (1/σ) tβθ tdθ −1 dt t t 1/σ 1/σ

No.1

Z.Y. Yang et al: AVERAGE ONESIDED WIDTHS OF SOBOLEV AND BESOV CLASSES





= c29 σ (2s−d)θ ω θ (1/σ) 





∞





uβθ udθ −1

1

 sin u 2sθ u

159

du

≤ c30 σ (2s−d)θ ω θ (1/σ) (2s > β + d). From the above discussion, we obtain I1 ≤ C5 ω(1/σ).

(17)

It is trivial when θ = ∞, 1. Similar to the above discussion or the proof of Theorem 2, we can get the results. Noticing that     aj = sup |f (y) − Tσ (f, y)| = sup kt f (y)Kσ,s (t)dt  π y∈U ( 2πj σ ,σ)

≤

sup y∈U ( 2πj σ



≤



,π σ)

Rd

π y∈U ( 2πj σ ,σ)



|kt f (y)|Kσ,s (t)dt ≤

  2πj 2π , + 2|t| Kσ,s (t)dt, ωk f, σ σ Rd

Rd

sup π Rd y∈U ( 2πj σ ,σ )

|kt f (y)|Kσ,s (t)dt

π 2π 2d and mes{U ( 2πj σ , σ )} = ( σ ) . From Lemma 2, we get

    2πj    sup I2 =  Φσ x − |f (y) − Tσ (f, y)| σ y∈U ( 2πj , π ) q j∈Z d σ σ  2π d  1/q   1/q ≤ c31 aqj ≤ c31 aqj dx π σ U( 2πj σ ,σ ) j∈Z d j∈Z d    q 1/q     2πj 2π   , + 2|t| Kσ,s (t)dt dx ≤ c31 ωk f,  2πj π σ σ d R j∈Z d U ( σ , σ )  q 1/q      4π   + 2|t| Kσ,s (t)dt dx ≤ c31 ωk f, x,  2πj π σ d R j∈Z d U ( σ , σ ) q 1/q     4π   + 2|t| Kσ,s (t)dt dx ≤ c31 ωk f, x,  σ Rd Rd q 1/q    4π + 2|t| ≤ c31 ωk f, x, dx Kσ,s (t)dt σ Rd Rd     4π + 2|t| ≤ c31 τk f, Kσ,s (t)dt σ q Rd   1 r ≤ c32 τk f, (4π + 2σ|t| Kσ,s (t)dt. σ q Rd From (5) and (4),  1/σ  1 wk (f, u)q 1 −d/q−1 θ the last inequality ≤ c32 τk f, ≤ c33 σ −d/q du 1 ω(u)u u σ q ω(u)u θ 0   1/σ  w (f, u) θ du 1/θ   1/σ  1/θ  1 k q ω θ (u)u( θ −d/q−1)θ du ≤ c34 σ −d/q ω(u) u 0 0   ∞    1/σ   1/θ  1 wk (f, u)q θ du 1/θ ≤ c34 σ −d/q ω θ (u)u( θ −d/q−1)θ du ω(u) u 0 0

160

ACTA MATHEMATICA SCIENTIA

≤ c35 ω(1/σ)σ α−q/d



1/σ

1



Vol.30 Ser.B

u(α+ θ −d/q−1)θ du

0

≤ c35 ω(1/σ)f brqθ ≤ C6 ω(1/σ) (α > d/q,

1/θ

f brqθ

1 1 +  = 1, 1 < θ < ∞). θ θ

(18)

When θ = ∞, α > d/q,  1/σ  1 wk (f, u)q ω(u)u−d/q−1 du τk f, ≤ c37 σ −d/q σ q ω(u) 0  wk (f, u)q 1/σ ≤ c37 σ −d/q sup ω(u)u−d/q−1 du ω(u) u>0 0  1/σ ≤ c38 σ α−d/q ω(1/σ) uα−d/q−1 duf brqθ ≤ c39 σ −r .

(19)

0

When θ = 1, α > d/q,  τk

1 f, ≤ c40 σ −d/q σ q



1/σ

wk (f, u)q ω(u)u−d/q du ω(u)u 0  1/σ wk (f, u)q (α−d/q) u ≤ c40 σ α−d/q ω(1/σ) du ω(u)u 0  ∞ wk (f, u)q ≤ c41 ω(1/σ) du = c42 ω(1/σ)f brqθ ≤ c42 ω(1/σ). ur+1 0

(20)

From (17)–(20), we obtain f − Tσ+ (f )q ≤ I1 + I2 ≤ Cω(1/σ) for any f ∈ Sbrqθ (Rd ). +

From Lemma 4, we know that dσ (Sbrqθ (Rd ), Lq (Rd )) ≤ Cω(σ −1/d ). ω d ω d Becuase SBqθ (Rd ) ⊂ Sbω qθ (R ) and from (1) we know when 1 ≤ q ≤ p ≤ ∞, SBpqθ (R ) ⊂ d ω d Sbω pqθ (R ) ⊂ Sbqθ (R ), and the upper estimates of these functional classes are given. From [8] and noticing (8), the lower estimates are given. The proof of Theorem 3 is completed. References [1] Freud G. Uber einseitige approximation durch polynome I. Acta Sci Math Szeged, 1955, 16: 12–18 [2] Korneichuk Π H, et al. Approximation with restrictions (in Russian). ΠYMKA, KucB, 1982 [3] Popov V A. Onesided approximation of periodic functions of serveral variables. Comp Rend Acad Bulg Sci, 1982, 35(12): 1639–1642 [4] Hristov V H, Ivanov K G. Operators for Onesided Approximation of Functions. Sofia: Constructive Theory of Functions’87, 1988 [5] Liu Y P. Average σ − K width of class of Lp (Rd ) in Lq (Rd ). Chin Ann Math, 1995, 16B(3): 351–360 [6] Jiang Y J, Liu Y P. Average width and optimal recovery of multivarivariate Besov-Wiener classes. J Math Study, 1988, 331(4): 353–361 [7] Xu G Q, Liu Y P. The average widths of Sobolev-Wiener classes and Besov-Wiener classes. Acta Mathematica Sinica, English Series, 2004, 20(1): 81–92 [8] Liu Y P, Xu G Q. The average widths and non-linear widths of the classes of multivariate functions with bounded moduli of smoothness. Acta Mathematica Sinica, English Series, 2002, 18(4): 663–674 [9] Liu Y P, Xu G Q. Widths and average widths of sobolev classes. Acta Mathematica Scientia, 2003, 23B(2): 178–184 [10] Pustovoitov N N. Representation and approximation of multivariate periodic functions with agiven modulus of smoothness. Analysis Math, 1994, 20: 35–48 [11] Ivanov K G. On the behaviour of two moduli of functions II. Serdica Bulg Math Publ, 1986, 12: 196–203 [12] Popov V A. On the one-sided approximation of multivariate functions//Chui C K, Schumaker L L, Ward J D, eds. Approximation Theory IV. New York: London, 1983: 657–661 [13] Magaril-Π yaev G G. Average widths of Sobolev classes on Rn . J Approx Theory, 1994, 76(1): 65–76