Widths of weighted Sobolev classes on the ball

Journal of Approximation Theory 154 (2008) 126–139 www.elsevier.com/locate/jat

Widths of weighted Sobolev classes on the ballI Heping Wang ∗ , Hongwei Huang Department of Mathematics, Capital Normal University, Beijing 100037, China School of Mathematical Sciences, Beijing Normal University, Beijing 100087, China Received 3 December 2007; accepted 25 March 2008 Available online 1 November 2008 Communicated by Paul Nevai

Abstract r ,L r We study the Kolmogorov n-widths dn (BW p,µ q,µ ) and the linear n-widths δn (BW p,µ , L q,µ ) of r d weighted Sobolev classes BW p,µ on the unit ball B in L q,µ , where L q,µ , 1 ≤ q ≤ ∞, denotes the

weighted L q space of functions on B d with respect to weight (1 − |x|2 )µ− 2 , µ ≥ 0. Optimal asymptotic r ,L r orders of dn (BW p,µ q,µ ) and δn (BW p,µ , L q,µ ) as n → ∞ are obtained for all 1 ≤ p, q ≤ ∞ and µ ≥ 0. c 2008 Elsevier Inc. All rights reserved.

1

Keywords: n-widths; Weighted Sobolev classes; Unit ball

1. Introduction and main results Let B d = {x ∈ Rd : |x| ≤ 1} denote the unit ball in Rd , where x · y is the usual inner product and |x| = (x · x)1/2 is the usual Euclidean norm. Let Πnd be the space of all polynomials in d variable of degree at most n. For the weight Wµ (x) = (1 − |x|2 )µ−1/2 (µ ≥ 0), denote by L p,µ ≡ L p (Wµ ), 1 ≤ p < ∞, the space of measurable functions defined on B d with the finite norm Z 1/ p p k f k p,µ := | f (x)| Wµ (x)dx , 1 ≤ p < ∞, Bd

I Supported by Beijing Natural Science Foundation (Project no. 1062004) and by a grant from the Key Programs of Beijing Municipal Education Commission (KZ200810028013). ∗ Corresponding author at: Department of Mathematics, Capital Normal University, Beijing 100037, China. E-mail addresses: [email protected] (H. Wang), [email protected] (H. Huang).

c 2008 Elsevier Inc. All rights reserved. 0021-9045/$ - see front matter doi:10.1016/j.jat.2008.10.001

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and for p = ∞ we assume that L ∞,µ is replaced by the space C(B d ) of continuous functions on B d with the uniform norm. For f ∈ L p (Wµ ), we set E n ( f ) p,µ := inf{k f − Pk p,µ : P ∈ Πnd }. Let Vnd (Wµ ) denote the subspace of all polynomials of degree  n which  are orthogonal to lower n+d−1 d degree polynomials in L 2 (Wµ ). Note that dim Vn (Wµ ) =  n d−1 . It is well known n (see [1, p. 38 or p. 229]) that the spaces Vnd (Wµ ) are just the eigenspaces corresponding to the eigenvalues −n(n + 2µ + d − 1) of the second-order differential operator Dµd := 4 − (x · ∇)2 − (2µ + d − 1) x · ∇, where the 4 and ∇ are the Laplace operator and gradient operator respectively. Also, the spaces Vnd (Wµ ) are mutually orthogonal in L 2 (Wµ ) and L 2 (Wµ ) =

∞ M X

Vnd (Wµ ),

Πnd =

n M X

n=0

Vkd (Wµ ).

(1.1)

k=0

The orthogonal projector Projn : L 2 (Wµ ) 7→ Vnd (Wµ ) can be written as Z (Projn f )(x) = f (y)Pn (Wµ , x, y)Wµ (y)dy, Bd

where, for µ > 0, the kernel Pn (Wµ , x, y) has the representation   Z 1 q q 1 µ µ− n + λ Pn (x, y) = bd b1 2 Cnλ (x, y) + u 1 − |x|2 1 − |y|2 (1 − u 2 )µ−1 du. λ −1 γ

Here, Cnλ is the nth degree Gegenbauer polynomial as defined in [13], λ = µ + d−1 2 , bd := R 2 γ −1/2 −1 ( B d (1−|x| ) dx) . See [2] for the proof of the formula of Pn (x, y), including the limiting case of µ = 0. Denote by Sn the orthogonal projector of L 2 (Wµ ) onto Πnd in L 2 (Wµ ). Evidently, for any f ∈ L 2 (Wµ ), (1.1) can be rewritten in the form f =

∞ X

Projn f,

Sn f :=

n=0

n X

Projk f.

k=0

Given r > 0, we define the fractional power (−Dµd )r/2 of the operator −Dµd on f by (−Dµd )r/2 ( f ) =

∞ X (k(k + 2µ + d − 1))r/2 Projk ( f ), k=0

in the sense of distribution. Using this operator we define the weighted Sobolev space as follows: For r > 0 and 1 ≤ p ≤ ∞, n o r r r W p,µ := f ∈ L p (Wµ ) : k f kW p,µ := k f k p,µ + k(−Dµd ) 2 ( f )k p,µ < ∞ , r . The notation r while the weighted Sobolev class BW p,µ is defined to be the unit ball of W p,µ A  B means that A  B and A  B, and A  B means that there exists an inessential positive constant c such that A ≤ cB.

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Let X be a Banach space with the norm k · k X , and let A be a (convex, compact, centrally symmetric) subset of X . The Kolmogorov width of A in X is defined by dn (A, X ) := inf sup inf k f − gk X , X n f ∈A g∈X n

where X n runs over all subspaces of X of dimension n or less. The linear width of A in X is defined by δn (A, X ) := inf sup k f − Pn f k X , Pn f ∈A

where Pn varies over all linear operators of rank at most n that map X into itself. More information on Kolmogorov and linear widths can be found in [3,4]. In the one-dimensional case, the exact orders of the Kolmogorov and linear widths of unweighted Sobolev classes on [0, 1] and on the torus T = [0, 2π] in L q space can be found in [4] (or [3]) and [5] respectively. The Kolmogorov n-widths dn (Bsα (L p,µ ), L q,µ ) and the linear nwidths δn (Bsα (L p,µ ), L q,µ ) of the weighted Besov classes Bsα (L p,µ ) on [−1, 1] in L q,µ , where L q,µ , 1 ≤ q ≤ ∞, denotes the weighted L q space of functions on [−1, 1] with respect to weight (1 − t 2 )µ− 2 , µ > 0, are studied by Feng Dai in a recent paper [6], in which optimal asymptotic orders of dn (Bsα (L p,µ ), L q,µ ) and δn (Bsα (L p,µ ), L q,µ ) as n → ∞ are given for all 1 ≤ p, q ≤ ∞ and µ > 0. The main purpose of this paper is to establish multivariate analogues of these results on the unit ball and obtain optimal asymptotic estimates about the Kolmogorov r and linear widths of weighted Sobolev classes BW p,µ in L q,µ for all 1 ≤ p, q ≤ ∞ and µ ≥ 0. Our main result then can be formulated as follows: 1

Theorem 1. Let 1 ≤ p, q ≤ ∞, r > 0. Then r dn (BW p,µ , L q,µ )  n −r/d+λ , r δn (BW p,µ , L q,µ )  n −r/d+τ .

¯ if r > λ,

(1.2)

if r > τ¯ ,

(1.3)

where (1) λ = µ = λ¯ = τ¯ = 0 if p, q lie in region I: 1 ≤ q ≤ p ≤ ∞; (2) λ = τ = 1/ p − 1/q, λ¯ = τ¯ = (1 + 2µ)(d/ p − d/q) if p, q lie in region II: 1 ≤ p ≤ q ≤ 2; (3) λ = 0, τ = 1/ p −1/q, λ¯ = (1+2µ)d/2, τ¯ = (1+2µ)(d/ p −d/q) if p, q lie in region III: 2 ≤ p ≤ q ≤ ∞; (4) λ = 1/ p −1/2, τ = max(1/ p −1/2, 1/2−1/q), τ¯ = (1+4µ)d max(1/ p, 1−1/q)−2µd, λ¯ = (1 + 2µ)d/ p, if p, q lie in region IV: 1 ≤ p ≤ 2 ≤ q ≤ ∞. We organize this paper as follows. Section 2 contains some basic facts about positive cubature formulae and Marcinkiewicz–Zygmund inequalities on the unit ball and the discretization theorems of estimates of widths. Based on the results of Section 2, we prove Theorem 1 in Section 3. 2. Main lemmas Let η be a nonnegative C ∞ -function on [0, +∞) supported in [0, 2] and equal to 1 on [0, 1]. We also suppose that η(x) > 0 for x ∈ [0, 2). We define   ∞ X j L n (x, y) := η P j (x, y), x, y ∈ B d , n j=0

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129

and Vn ( f )(x) :=

Z Bd

L n (x, y) f (y)Wµ (y)dy,

x ∈ Bd .

(2.1)

It is well known that Vn (g) = g for g ∈ Πnd , and for any f ∈ L p,µ , 1 ≤ p ≤ ∞, we have d , and Vn ( f ) ∈ Π2n kVn ( f )k p,µ  k f k p,µ ,

and

k f − Vn ( f )k p,µ  E n ( f ) p,µ .

(2.2)

We will keep the notations η, L n and Vn for the rest of this paper. r , 1 ≤ p ≤ ∞, Lemma 1 (See [7, Corollary 4.4]). For f ∈ W p,µ

E n ( f ) p,µ  n −r k(−Dµd )r/2 f k p .

(2.3)

We introduce the metric ρ on B d : r q q ρ(x, y) := |x − y|2 + ( 1 − |x|2 − 1 − |y|2 )2

on B d .

Note that in [8], the following metric d on B d was introduced:   q q 2 2 d(x, y) := arccos (x, y) + 1 − |x| 1 − |y| on B d . It turns out that the metric d is equivalent to the metric ρ. Indeed, we have ρ 2 (x, y) = 2(1 − cos d(x, y)) = 4 sin2

d(x, y) 2

and

2 d(x, y) ≤ ρ(x, y) ≤ d(x, y). π

For r > 0, x ∈ B d and a positive integer n, we set Bρ (x, r ) := {y ∈ B d : ρ(x, y) ≤ r }

q and Wµ (n; x) := ( 1 − |x|2 + n −1 )2µ .

It is straightforward to show that (see also [8, Lemma 5.3]) q |Bρ (ξ, r )|  r d 1 − |ξ |2 and m(Bρ (ξ, r )) :=

Z Bρ (ξ,r )

Wµ (x)dx  r d (r +

q

1 − |ξ |2 )2µ .

For δ > 0, n ∈ N, we say that a finite subset Λ ⊂ B d is maximal ( nδ , ρ)-separable if   [ δ δ Bd ⊂ Bρ y, and min ρ(y, y 0 ) ≥ . n n y6= y 0 ∈Λ y∈Λ The following cubature formulae and Marcinkiewicz–Zygmund (MZ) inequalities on the ball were given in [9]. Lemma 2. There exists a constant γ > 0 depending only on d and µ such that for any δ ∈ (0, γ ), any positive integer n, and any maximal ( nδ , ρ)-separable subset Λ ⊂ B d , there

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exists a sequence of positive numbers λξ 

R

Bρ (ξ, nδ )

Wµ (x) dx  n −d Wµ (n; ξ ), ξ ∈ Λ, for

d, which the following quadrature formula holds for all f ∈ Π6n Z X λξ f (ξ ). f (x)Wµ (x) dx = Bd

(2.4)

ξ ∈Λ

d , we have Moreover, for any 1 ≤ p < ∞, f ∈ Π2n Z Z X X | f (x)| p Wµ (x) dx  λξ | f (ξ )| p  | f (ξ )| p Bd

ξ ∈Λ

ξ ∈Λ

Bρ (ξ, nδ )

Wµ (x) dx,

(2.5)

where the constants of equivalence depend only on d, µ and p. Remark 1. An equality like (2.4) with positive weights is generally called a positive cubature formula of degree 6n, while an equivalence like (2.5) is called a Marcinkiewicz–Zygmund (MZ) inequality. Note that a similar result about positive cubature formulae on the ball was also given in [8, Theorem 5.4]. d Remark 2. In the case p = ∞, the MZ inequalities also hold R for f ∈ Πn . This can be obtained d by duality. In fact, if f ∈ Πn , then for any g, kgk1,1/2 = B d |g(x)|dx ≤ 1, we have X Z Z = = f (x)V (g)(x)dx λ f (ω)V (g)(ω) f (x)g(x)dx (by (2.4)) n ω n d d B B ω∈Λ Z X ≤ max | f (ω)| λω |Vn (g)(ω)|  max | f (ω)| |Vn (g)(x)|dx (by (2.5)) ω∈Λ

ω∈Λ

ω∈Λ

 max | f (ω)|

Z

ω∈Λ

Bd

|g(x)|dx ≤ max | f (ω)|, ω∈Λ

Bd

(by (2.2))

which gives max | f (ω)| ≤ k f k∞ = ω∈Λ

sup kgk1,1/2 ≤1

Z

Bd

f (x)g(x)dx  max | f (ω)|. ω∈Λ

Remark 3. Note that if µ = 0, then all weights λξ , ξ ∈ Λ in (2.5) are equivalent, which means d and all 1 ≤ p < ∞, for all f ∈ Π2n  1/ p X 1 | f (ξ )| p   k f k p,0 . n d ξ ∈Λ Note also that if µ > 0, then there exist two weights λξ and λω in (2.5) such that λξ and λω are not equivalent. For µ > 0, it should be pointed out that for any p ∈ [1, ∞) and the space d TL Π2n p,µ , the weights in MZ type inequalities cannot be taken to be equal for any choice of d  n d , i.e., the following equivalency cannot distinct points x1 , . . . , x N in B d with N  dim Π2n d be true for all f ∈ Π2n , N 1 X | f (xk )| p N k=1

!1/ p  k f k p,µ

(µ > 0).

(2.6)

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131

To prove this assertion, we suppose that the contrary were true. That is, we assume that d d/(µ+d) and (2.6) is true for some points x1 , . . . , x N in B d with q N  n . Set m = n p θ = max1≤i≤N 1 − |xi |2 . We assume that θ = 1 − |xi0 |2 for some i 0 ∈ [1, N ] and set ξ = xi0 /|xi0 |. We will arrive at the contradiction by considering the following two cases: Case I. θ ∈ [0, m1 ]. In this case we may take f (x) = L m (x, xi0 ) and apply the inequality (see [8, Proposition 5.9]) L n (x, x) ≥

n X

Pk (x, x)  n d (Wµ (n; x))−1

k=0

to get k f k p,µ 

N 1 X | f (xi )| p N i=1

 n −d/ p

!1/ p  n −d/ p L m (xi0 , xi0 )

md  m −(µ+d)/ p+d+2µ . Wµ (m; xi0 )

(2.7)

On the other hand, by (4.18) in [8], we get k f k p,µ = kL m (·, xi0 )k p,µ  m d(1−1/ p) (Wµ (m; xi0 ))−(1−1/ p)  m (d+2µ)(1−1/ p) . This combined with (2.7) gives m (d+2µ)(1−1/ p)  m −(µ+d)/ p+d+2µ , namely m (d+µ)/ p  m (d+2µ)/ p , which is impossible when n is sufficiently large. Case II. θ 6∈ [0, m1 ]. In this case, we take f (x) = L n (x, ξ ). Note that for i = 1, . . . , N , ρ(xi , ξ ) ≥ 1/m and Wµ (n; xi ) ≥ m −2µ . Then on the one hand, by (1.6) in [8], we have for any positive integer k, |L n (xi , ξ )|  p

nd p  n d+µ m µ (m/n)k , Wµ (n; xi ) Wµ (n; ξ )(1 + nρ(xi , ξ ))k

and k f k p,µ 

N 1 X |L n (xi , ξ )| p N i=1

!1/ p  n d+µ−k m k+µ .

(2.8)

On the other hand, by (2.3) in [10], we have  1−1/ p nd k f k p,µ = kL n (·, ξ )k p,µ   n (d+2µ)(1−1/ p) , Wµ (n; ξ ) which, combined with (2.8) leads to a contradiction if we take k sufficiently large. Lemma 3. Let µ > 0, β ∈ (0, 1/(2µ)) and let Λ, λξ be defined as in Lemma 2. Then X d(1+β) λ−β . ω n ω∈Λ

(2.9)

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Proof. We know that λξ  separable, we get I :=

X ω∈Λ

R

d(1+β) λ−β ω n

Bρ (ξ, nδ )

X

Wµ (x) dx  n −d Wµ (n; ξ ). Since Λ is maximal ( nδ , ρ)-

(Wµ (n; ω))−β−1

ω∈Λ

Z Bρ (ω, nδ )

Wµ (x) dx .

Note that if x ∈ Bρ (ω, nδ ), then Wµ (n; x)  Wµ (n; ω). Hence, XZ I  n d(1+β) (Wµ (n; x))−β−1 Wµ (x) dx δ ω∈Λ Bρ (ω, n )

 n

Z

d(1+β)

(Wµ (n; x))−β−1 Wµ (x) dx ! Z Z + √ √

Bd

 n d(1+β)

1−|x|2 ≤ n1

1−|x|2 ≥ n1

(1 − |x|2 )µ−1/2 p dx ( n1 + 1 − |x|2 )2(β+1)µ

:= n d(1+β) (I1 + I2 ).

(2.10)

In the first integral, using the fact that ( n1 + Z Bd

ϕ(|x|)dx = ωd−1

Z

1

p

1 − |x|2 )2(β+1)µ  n −2(β+1)µ and the formula

f (t)t d−1 dt,

0

where ωd−1 is the surface area of the unit sphere S d−1 , we get Z Z I1  n 2(β+1)µ √ Wµ (x) dx  n 2(β+1)µ q 1−|x|2 ≤ n1

= n 2(β+1)µ ωd−1

n

2(β+1)µ

Z

 1−

1

q

q

1− 12 n

|x|≥ 1−

1 n2

(1 − t 2 )µ−1/2 t d−1 dt  n 2(β+1)µ

1 − 1/n 2

µ+1/2

Wµ (x) dx 1

Z

q 1−

1 n2

(1 − t)µ−1/2 dt

 n 2(β+1)µ n −2(µ+1/2) = n 2βµ−1 .

(2.11)

Similarly, we have Z I2 

√ |y|≤

1−1/n 2

Z √1−1/n 2 

(1 − |y|2 )−βµ−1/2 dy = ωd−1

Z √1−1/n 2

(1 − t 2 )−βµ−1/2 t d−1 dt

0

(1 − t)−βµ−1/2 dt  1,

0

which, combined with (2.10), (2.11), gives (2.9). Lemma 3 is proved. Remark 4. We have the following more delicate estimate:  d(1+β) , if β ∈ (0, 1/(2µ)); n X d(1+β) λ−β  n ln n, if β = 1/(2µ); ω  d(1+β) 2βµ−1 ω∈Λ n n , if β > 1/(2µ).



H. Wang, H. Huang / Journal of Approximation Theory 154 (2008) 126–139

133

For a positive vector w = (w1 , . . . , wn ) ∈ Rm , wi > 0, 1 ≤ i ≤ n, let `np,w (1 ≤ p ≤ ∞) denote the space Rn equipped with the `np,w -norm defined by  !1 p n    X |x | p w , 1 ≤ p < ∞; i i kxk`np,w := i=1    max |xi |, p = ∞. 1≤i≤n

`np,w

The unit ball of is denoted by B`np,w . In the case w = (1, . . . , 1) we write as usual `np , n k · k`np , B` p instead of `np,w , k · k`np,w , B`np,w . Given 1 ≤ p ≤ ∞ and a positive integer n, we denote by B np,µ the set of all functions f ∈ Πnd on B d such that k f k p,µ ≤ 1. Lemma 4 (See [6, Lemma 2.2]). Let w = (w1 , . . . , wn ) ∈ Rn satisfy wi > 0, 1 ≤ i ≤ n and let n X −β w j ≤ n, for some β > 0. j=1

Then for 1 ≤ m ≤ n and 1 ≤ p ≤ q ≤ ∞, 1 1 1 1 1 1  n β ( p−q ) ( − ) n Sm (B`np,w , `q,w ) ≤ 2β p q S[ m2 ] (B`np , `qn ), m where Sm denotes either of the Kolmogorov m-width dm or the linear m-width δm . Let γ > 0 be the same as in Lemma 2 and let Λ = {ξ1 , . . . , ξu n } be a maximal ( nδ , ρ)separable subset of B d with δ = γ /10, where n is a positive integer, u n is the number of elements in Λ. It is easy to see that u n  n d . Lemma 5. Let Sm denote either of the symbols dm or δm and µ > 0. Then for 1 ≤ p ≤ q ≤ ∞, 1 ≤ m ≤ dim Πnd and any ε > 0, we have Sm (B np,µ , L q,µ )  n d(1/ p−1/q)



nd m+1

2(µ+ε)(1/ p−1/q)

S[m/2] (B`upn , `qu n ),

(2.12)

where u n is defined as above. Proof. By Lemma 2, there exists a sequence of positive numbers wi  n −d Wµ (n; ξi ), 1 ≤ i ≤ d, u n , for which the following quadrature formula holds for all f ∈ Π6n Z Bd

f (x)Wµ (x) dx =

un X

wi f (ξi ).

(2.13)

i=1

Moreover, for any 1 ≤ p ≤ ∞, f ∈ Πnd , we have !1/ p un X p n , k f k p,µ  wi | f (ξi )| = kUn ( f )k`up,w i=1

where w = (w1 , . . . , wu n ), Un : Πnd 7−→ Ru n is defined by Un ( f ) = ( f (ξ1 ), . . . , f (ξu n )).

(2.14)

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d by Next, we define the operator T : Ru n 7−→ Π2n

T v(x) :=

un X

vi wi L n (x, ξi ),

i=1

where v := (v1 , . . . , vu n ) ∈ Ru n . We will show that for 1 ≤ q ≤ ∞, un . kT vkq,µ  kvk`q,w

(2.15)

In fact, for q = 1, (2.15) follows directly from the inequality (see [8, (4.18)]) max kL n (·, y)k1,µ  1; y∈Bd

for q = ∞, by (2.14) with p = 1 and f (y) = L n (x, y), we have un kT vk∞ ≤ kvk`∞

un X

wi |L n (x, ξi )|  kvk`u∞n kL n (x, ·)k1,µ  kvk`u∞n ;

i=1

and for 1 < q < ∞, (2.15) follows from the Riesz–Thorin theorem. Now we can use (2.13) to obtain that for f ∈ Πnd , Z f (x) = Vn ( f )(x) = f (y)L n (x, y)Wµ (x, y) dx =

Bd u n X

wi f (ξi )L n (x, ξi ) = T U ( f )(x).

i=1

Hence, we can factor the identity I : Πnd U

i

T

d L p,µ 7−→ Π2n

T

L q,µ as follows:

T

un d n I : Πnd ∩ L p,µ −→ `up,w −→ `q,w −→ Π2n ∩ L q,µ ,

where i denotes the identity. It then follows by (2.14) and (2.15) that n , `u n ) n ) kT k u n S (B`up,w Sm (B np,µ , L q,µ ) ≤ kU k(Πnd ∩L p,µ ,`up,w d ∩L q,w (`q,w ,Π2n q,µ ) m n , `u n ).  Sm (B`up,w q,w

(2.16)

Finally, by Lemma 3, there exists a constant c > 0 such that un X (c n d wi )−β ≤ u n . i=1

Using Lemma 4 with β = 1/(2(µ + ε)) to the vector w˜ = (c n d w1 , . . . , c n d wu n ), we get n , `u n ) = (c n d )1/ p−1/q S (B`u n , `u n ) Sm (B`up,w m q,w p,w˜ q,w˜  d 2(µ+ε)(1/ p−1/q) n  n d(1/ p−1/q) S[m/2] (B`upn , `qu n ), m+1

which, combined with (2.16), gives (2.12) and therefore completes the proof of Lemma 5.



Remark 5. By Nikolskii inequalities (see [10, Proposition 2.4]), we obtain that for 1 ≤ p ≤ q ≤ ∞, S0 (B np,µ , L q,µ ) = sup k f kq,µ  n (d+2µ)(1/ p−1/q) . f ∈B np,µ

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H. Wang, H. Huang / Journal of Approximation Theory 154 (2008) 126–139

Lemma 6. Let Sn denote either of the symbols dn or δn and µ ≥ 0. Then for 1 ≤ p, q ≤ ∞, µ ≥ 0, there exists an N , N  n, N ≥ 2n such that r Sn (BW p,µ , L q,µ )  n −r/d+1/ p−1/q Sn (B` Np , `qN ).

Proof. We assume that m ≥ 6 and b1 m d ≤ n ≤ b2 m d with b1 , b2 > 0 being independent of n and m. We let {x j } Nj=1 ⊂ {x ∈ B d : |x| ≤ 2/3} such that N  m d and \ {x ∈ B d : |x − x j | ≤ 2/m} {x ∈ B d : |x − xi | ≤ 2/m} = ∅, if i 6= j. Obviously, such points x j exist. We may take b2 > 0 sufficiently small so that N ≥ 2n. Let ϕ be a nonnegative C ∞ -function on Rd supported in {x ∈ Rd | |x| ≤ 1} and be equal to 1 on {x ∈ Rd | |x| ≤ 1/2}. We define ϕi (x) = ϕ(m(x − xi )),

i = 1, . . . , N

and set ( A N := span {ϕ1 , . . . , ϕ N } =

f a (x) =

N X

) a j ϕ j (x) : a = (a1 , . . . , a N ) ∈ R

N

.

j=1

Clearly, supp ϕ j ⊂ {x ∈ B d : |x − x j | ≤ 1/m} ⊂ {x ∈ B d : |x| ≤ 5/6}, Z 1/ p Z 1/ p kϕ j k p,µ  |ϕ j (x)| p dx = |ϕ(mx)| p dx  m −d/ p , Bd

Bd

j = 1, . . . , N ,

and supp ϕ j

\

supp ϕi = ∅ (i 6= j).

(2.17)

Hence, if f a ∈ A N , a = (a1 , . . . , a N ) ∈ R N , then !1/ p N X −d p k f a k p,µ  m |a j | = m −d/ p kak` Np . j=1

Denote by PA N the orthogonal projector onto A N , that is, for any f ∈ L 1,µ and x ∈ B d , Z N X ϕ j (x) PA N ( f )(x) = f (y)ϕ j (y)Wµ (y)dy. 2 d j=1 kϕ j k2,µ B T Then PA N is the bounded projection operator from L p,µ to A N L p,µ . In fact, Z N X ϕ j (x)ϕ j (y) PA N ( f )(x) = f (y)K (x, y)Wµ (y)dy, where K (x, y) = . 2 d B j=1 kϕ j k2,µ Using (2.18), we get Z N X ϕ j (x) |K (x, y)|Wµ (y)dy  m −d 2 Bd j=1 kϕ j k2,µ 

N X j=1

|ϕ j (x)|  1

∀ x ∈ Bd .

(2.18)

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By the H¨older inequality, it follows that Z Z p |PA N ( f )(x)| Wµ (x) dx =

p f (y)K (x, y)Wµ (y)dy Wµ (x) dx Bd Bd Bd  Z  p0 Z Z p |K (x, y)|Wµ (y)dy Wµ (x) dx ≤ | f (y)| p |K (x, y)|Wµ (y)dy · Bd Bd Bd  Z Z  | f (y)| p |K (x, y)|Wµ (y)dy Wµ (x) dx Bd Bd  Z Z = |K (x, y)|Wµ (x) dx | f (y)| p Wµ (y)dy d Bd ZB  | f (y)| p Wµ (y)dy, Z

Bd

which implies that kPA N ( f )k p,µ  k f k p,µ . Hence, we have   \ r r Sn (BW p,µ , L q,µ ) ≥ Sn BW p,µ A N , L q,µ   \ \ r  Sn BW p,µ A N , L q,µ AN .

(2.19)

For a positive integer v > r and f a ∈ A N , a = (a1 , . . . , a N ), by the definition of −Dµd and (2.17), it is easy to verify that k(−Dµd )v (ϕ j )k p,µ  m 2v−d/ p

and

k(−Dµd )v ( f a )k p,µ  m 2v−d/ p kak` Np .

It then follows by the Kolmogorov type inequality (see [11, Theorem 8.1]) that r

2v−r

2v 2v k f a k p,µ  m r −d/ p kak` Np . k(−Dµd )r/2 ( f a )k p,µ  k(−Dµd )v ( f a )k p,µ T r Hence, each function from BW p,µ A N is associated with an element of the ball of radius of c · m −r +d/ p of the space ` Np , where c is a positive constant independent of N . By (2.19) and (2.18), we obtain that

r Sn (BW p,µ , L q,µ )  m −r +d/ p−d/q Sn (B` Np , `qN )  n −r/d+1/ p−1/q Sn (B` Np , `qN ),

which completes the proof of Lemma 6.



3. Proof of Theorem 1 Proof of Theorem 1. We first prove the lower estimates of the widths. The lower estimates for r ,L dn (BW p,µ q,µ ) follow from Lemma 6 with Sn = dn and the following estimates (see [3, p. 236]):  1/q−1/ p n , 1 ≤ q ≤ p ≤ ∞;    1/q−1/ p n , 2 ≤ p ≤ q ≤ ∞; N N dn (B` p , `q )  (3.1)  n 1/q−1/2 , 1 ≤ p ≤ 2 ≤ q ≤ ∞;   1, 1 ≤ p ≤ q ≤ 2. Using Lemma 6 with Sn = δn , (3.1) and the following relations

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δn (B` Np , `qN ) = δn (B`qN0 , ` Np0 ) ≥ max{dn (B` Np , `qN ), dn (B`qN0 , ` Np0 )}, 1/ p + 1/ p 0 = 1/q + 1/q 0 = 1, r ,L we get the desired lower estimates for δn (BW p,µ q,µ ). Now we prove the upper estimates. In the case µ = 0, all weights λξ , ξ ∈ Λ in (2.5) are equivalent, so the proof runs along the same lines as that of the sphere case, see [12]. We omit it. We just consider the case µ > 0. In region I: 1 ≤ q ≤ p ≤ ∞, by (2.2), (2.3) and the inequality

k f k p,µ  k f kq,µ , r , r > 0, we have for f ∈ BW p,µ

k f − Vm ( f )kq,µ  k f − Vm ( f )k p,µ  E m ( f ) p,µ  m −r k(−Dµd )r/2 ( f )k p,µ  m −r , where the operator Vm is defined by (2.1), m  n 1/d . Since the rank of the linear operator Vm is d  m d , we get equivalent to dim Π2m r r dn (BW p,µ , L q,µ ) ≤ δn (BW p,µ , L q,µ )  k f − Vm ( f )kq,µ  m −r  n −r/d .

For f ∈ L p,µ , we define A0 ( f ) = V1 ( f ), Clearly for f ∈

A j ( f ) = V2 j ( f ) − V2 j−1 ( f ) for j ≥ 1. P f = ∞ j=0 A j ( f ), and by (2.2) and (2.3), we have

r , BW p,µ

kA j ( f )k p,µ  2− jr k(−Dµd )r/2 ( f )k p,µ  2− jr ,

j ≥ 0.

Choose a positive integer J such that n  2 J d and C1 2 J d ≤ n with C1 > 0 to be specified later. We may take sufficiently small positive numbers ρ, ε > 0 and define ( dim Π2dj+1 , i if 0 ≤ j ≤ J, n j = h J d(1+ρ)−dρ j 2 , if j > J. Then X

nj 

X

2 jd +

j≤J

j≥0

X

2 J d(1+ρ)−dρ j  2 J d .

j>J

P J d ≤ n. After discretization, it Hence, we can take C1 sufficiently large so that ∞ j=0 n j ≤ C 1 2 follows by Lemma 5 and Remark 5 that ∞   X \ r Π2dj+1 , L q,µ Sn (BW p,µ , L q,µ )  2− jr Sn j B L p,µ j=0

X



2

− j (r −d/ p+d/q)



J < j≤J (1+ρ −1 )

+

2 jd nj + 1

2(µ+ε)(1/ p−1/q)

u

u

S[n j /2] (B` p j , `q j )

2− jr 2 j (d+2µ)(1/ p−1/q)

X j>J (1+ρ −1 )



2− j (r −d/ p+d/q) 22( j−J )d(1+ρ)(µ+ε)(1/ p−1/q) S[n j /2] (B` p j , `q j ) u

X

u

J < j≤J (1+ρ −1 )

+ 2−J (1+ρ

−1 )(r −(d+2µ)(1/ p−1/q))

,

(3.2)

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where Sm denotes either of the symbols dm or δm , u j  2 jd , 1 ≤ p ≤ q ≤ ∞, and r > (d + 2µ)(1/ p − 1/q). In region IV: 1 ≤ p ≤ 2 ≤ q ≤ ∞, if r > (1 + 2(1 + ρ)(u + ε))d/ p + dρ/2, then applying (3.2) with Sn = dn and the following improved Kashin’s inequality (see [4, p. 465]), M M M −1/2 dm (B` M (log2 (eM/m))1/2 , p , `∞ ) ≤ dm (B`2 , `∞ )  m

1 ≤ m ≤ M,

we have r r dn (BW p,µ , L q,µ )  dn (BW p,µ , L ∞,µ ) X − j (r −d/ p) 2( j−J )d(1+ρ)(µ+ε)(1/ p) −(J d(1+ρ)− jdρ)/2  2 2 2 J < j≤J (1+ρ −1 )

× (( j − J )d(1 + ρ))1/2 + 2−J (1+ρ  2−J (r −d/ p+d/2) + 2−J (1+ρ

−1 )(r −(d+2µ)/ p)

−1 )(r −(d+2µ)/ p)

 n −r/d+1/ p−1/2 .

(3.3)

For Sn = δn , if p = 1, q = p 0 = ∞, r > (1 + 2(1 + ρ)(u + ε))d + dρ/2, then the upper r ,L estimate for δn (BW1,µ ∞,µ ) follows from (3.2) and the following equality (see [4, p. 412]): M M δm (B`1M , `∞ ) = dm (B`1M , `∞ ),

1 ≤ m ≤ M.

If 1 < p ≤ 2, q = p 0 , r > (1 + 2(1 + ρ)(u + ε))(d/ p − d/ p 0 ) + d(ρ/2 + 1 − 1/ p), then using (3.2) with Sn = δn and the following Gluskin’s estimate (see [4, p. 473]), M 1−1/ p −1/2 δm (B` M m , p , ` p0 )  M

1 ≤ m ≤ M,

we get X

r δn (BW p,µ , L p0 ,µ ) 

2− j (r −d/ p+d/ p ) 22( j−J )d(1+ρ)(µ+ε)(1/ p−1/ p ) 0

0

J < j≤J (1+ρ −1 )

× 2−(J d(1+ρ)−dρ j)/2 2 jd(1−1/ p) + 2−J (1+ρ  2−J (r −d/ p+d/2) + 2−J (1+ρ

−1 )(r −(d+2µ)(1/ p−1/ p 0 ))

−1 )(r −(d+2µ)(2/ p−1))

 n −r/d+1/ p−1/2 . (3.4)

r ,L Hence, in region IV: 1 ≤ p ≤ 2 ≤ q ≤ ∞, the upper estimates for δn (BW p,µ q,µ ) follow from (3.4) and the following relation  δ (BW r , L 0 ), if p 0 = p ≥ q; n p ,µ p,µ r p−1 , L q,µ )  δn (BW p,µ δn (BW r0 , L q,µ ), if p 0 ≤ q(equivalently, q 0 ≤ p). q ,µ

In regions II and III: 1 ≤ p ≤ q ≤ 2 and 2 ≤ p ≤ q ≤ ∞, for the linear widths, if r > (1 + 2(1 + ρ)(u + ε))(d/ p − d/q) (ρ, ε sufficiently small), then using the inequality M δm (B` M p , `q ) ≤ 1,

1≤m≤M

and (3.2) with Sn = δn , we get r δn (BW p,µ , L q,µ )  2−J (r −d/ p+d/q)  n −r/d+1/ p−1/q .

(3.5)

For Kolmogorov widths, in region II: 1 ≤ p ≤ q ≤ 2, the upper estimates follow from (3.5) and r ,L r the relation dn (BW p,µ q,µ ) ≤ δn (BW p,µ , L q,µ ). In region III: 2 ≤ p ≤ q ≤ ∞, the upper

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139

estimates follow from (3.3) and the fact r r dn (BW p,µ , L q,µ )  dn (BW2,µ , L ∞,µ )  2−Jr  n −r/d .

Theorem 1 is now proved.



Acknowledgments The authors are very grateful to the anonymous referees for many valuable comments and suggestions which helped in improving the drafts. References [1] C.F. Dunkl, Y. Xu, Orthogonal Polynomials of Several Variables, Cambridge Univ. Press, 2001. [2] Y. Xu, Summability of Fourier orthogonal series for Jacobi weight on a ball in Rd , Trans. Amer. Math. Soc. 351 (1999) 2439–2458. [3] A. Pinkus, n-Widths in Approximation Theory, Springer-Verlag, 1985. [4] G.G. Lorentz, M.V. Golitschek, Y. Makovoz, Constructive Approximation, Advanced Problems, Springer-Verlag, New York, 1996. [5] V.N. Temlykov, Approximation of Periodic Functions, Nova Science Publishers, Inc., New York, 1993. [6] F. Dai, Kolmogorov and linear widths of weighted Besov classes, J. Math. Anal. Appl. 315 (2) (2006) 711–724. [7] Y. Xu, Weighted approximation of functions on the unit sphere, Constr. Approx. 21 (2005) 1–28. [8] P. Petrushev, Y. Xu, Localized polynomial frames on the ball, Constr. Approx. 27 (2008) 121–148. [9] F. Dai, Multivariate polynomial inequalities with respect to doubling weights and A∞ weights, J. Funct. Anal. 235 (2006) 137–170. [10] G. Kyriazis, P. Petrushev, Y. Xu, Decomposition of weighted Triebel-Lizorkin and Besov spaces on the ball, Proc. London Math. Soc. 97 (2008) 477–513. [11] Z. Ditzian, Fractional derivatives and best approximation, Acta Math. Hungar. 81 (4) (1998) 323–348. [12] G. Brown, F. Dai, Approximation of smooth functions on compact two-point homogeneous spaces, J. Funct. Anal. 220 (2005) 401–423. [13] G. Szeg¨o, Orthogonal Polynomials, fourth ed., in: American Mathematical Society Colloquium Publications, vol. 23, American Mathematical Society, Providence, RI, 1975.