Ukrainian Mathematical Journal, Vol. 53, No. 7, 2001

ON ESTIMATES OF THE KOLMOGOROV WIDTHS OF THE CLASSES B rp, q IN THE SPACE Lq A. S. Romanyuk

UDC 517.5

We obtain an order estimate for the Kolmogorov width of the Besov classes Bpr, θ of periodic functions of many variables in the space Lq for 2 < p < q < ∞, which complements the result obtained earlier by the author.

dM

(

In the present work, we complement one result concerning estimates of the Kolmogorov widths Bpr, θ , Lq , 2 ≤ p < q < ∞, obtained in [1]. First, recall necessary notation and definitions.

)

Let R be the Euclidean space with elements x = ( x 1 , … , x d) and let L p( π d) , π d = d

∏ j = 1[−π; π] , d

be

the space of functions 2 π -periodic in each argument and such that

f

p

  =  (2π)− d ∫ f ( x ) p dx   πd

1/ p

< ∞,

1 ≤ p < ∞,

and f

= ess sup f ( x ) < ∞,

∞

p = ∞.

In what follows, we assume that π

∫ f ( x) dx j

= 0,

j = 1, d ,

−π

for f ( x ) ∈ L p ( π d ). For vectors k = ( k 1 , … , k d ) , k j ∈ Z, and s = ( s 1 , … , s d ) , s j ∈
, j = 1, d , we set ρ (s ) = δ s ( f, x ) =

∑

{k : 2

s j −1

≤ kj < 2

fˆ (k ) ei(k , x ) ,

k ∈ρ( s)

sj

},

f ( x ) ∈ Lp ( π d ) ,

where Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 51, No. 7, pp. 996–1001, July, 2001. Original article submitted January 21, 2000; revision submitted November 23, 2000. 0041–5995/01/5307–1189 $25.00

© 2001 Plenum Publishing Corporation

1189

1190

A. S. ROMANYUK

fˆ (k ) = (2π)− d

∫ f (t) e

– i( k , t )

dt

πd

are the Fourier coefficients of f ( x ) . Let 1 < p < ∞ and r = ( r1 , … , rd ) , rj > 0, j = 1, d . Then the Besov classes Bpr, θ are defined as follows: Bpr, θ

 =  f ( x) 

f

Bpr, θ

  =  ∑ 2(s, r )θ δ s ( f , x ) θp   s 

1/ θ

 ≤ 1 

for 1 ≤ θ < ∞

and Bpr, ∞ =  f ( x ) 

f

Bpr, ∞

= sup 2(s, r ) δ s ( f , x ) s

p

≤ 1  . 

Without loss of generality, we can assume that the coordinates of the vector r = ( r1 , … , rd ) are ordered as follows: 0 < r1 = r2 … = rν < rν + 1 ≤ … ≤ r d . We associate the vector γ = ( γ 1 , … , γ d) , γ j = rj / r1 , j = 1, d , with the vector r = ( r1 , … , rd ). Denote by l pd the space R d with the norm

x

l pd

1/ p  d p   ∑ xi  ,   =  i =1   max xi ,  1≤i≤d

1 ≤ p < ∞, p = ∞,

and by Bpd the unit ball in l pd . If A is a certain subset of the integer lattice Z d, then | A | denote the number of elements of A. For convenience, we recall the definition of Kolmogorov width. Let F be a certain centrally-symmetric set of a Banach space . Then the quantity d M ( F,  ) = inf sup inf

LM f ∈ F a ∈ LM

f −a



is called the M-dimensional Kolmogorov width of F. Here, L M is a subspace of dimension M of the space . The following statement was proved in [1]: 1 1 Theorem A. Suppose that 1 ≤ θ ≤ ∞, 2 ≤ p < q < ∞, and r1 > β =  −   p q

(

)

(

d M Bpr, θ , Lq ⱱ M − r1 log ν −1 M where a + = max {a, 0 }.

)

1 1 r1 +  −  2 θ +,

1 − 2  . Then  q (1)

ON E STIMATES OF THE KOLMOGOROV WIDTHS OF THE CLASSES B rp ,q

IN THE S PACE

Lq

1191

In fact, this statement was proved only for r1 > 1 / 2 because, in the proof of the upper bound in (1), we assumed that p = 2, whereas, in this case, we have β = 1 / 2. The objective of the present paper is to extend Theorem A to the case β < r1 ≤ 1 / 2. The following statement is true: 1 1 Theorem 1. Suppose that 2 < p < q < ∞,  −   p q

(

)

1 − 2  = β < r ≤ 1 2, and θ ≥ p. Then / 1  q

(

d M Bpr, θ , Lq ⱱ M − r1 log ν −1 M

1

1

)r + 2 − θ . 1

(2)

Proof. The lower bound in (2) can be established by analogy with [1]. To obtain the upper bound, we use a certain modification of the arguments presented in [1]. Thus, for natural numbers k and l, we set Sl, k =

{s ∈ N d : l − 1 ≤ (s, γ ) < l, (s, 1) = k}

and note that k ≥ d and Sl, k = ∅ for k ≥ l. Then, denoting

Sl, k =

U ρ(s)

Sl, k = 2 k Sl, k .

, we can write

s ∈ Sl, k

Further, on the basis of the number M, we select µ according to the condition M ⱱ 2 µ µ ν −1 and define the numbers

Ml, k

 Sl, k ,  =   S 2 µ + αµ − 2αl + αk ,  l, k

d ≤ k ≤ l, l ≤ µ ; d ≤ k ≤ l, l > µ ,

where α is the number that will be selected in the course of establishing the required estimate. Let us show that l

∑ ∑ Ml, k

<< M.

(3)

l≥d k=d

We have µ

l

∑ ∑ Ml, k

=

l≥d k=d

l

l=d k=d

ⱱ

Let us estimate each term obtained.

=

l>µ k=d

∑ 2(s, 1) + ∑

d ≤ ( s, γ ) ≤ µ

µ

l

∑ ∑ Ml, k + ∑ ∑ Ml, k

l

∑ ∑

l=d k=d

Sl, k +

∑ 2µ + αµ − 2αl + α(s, 1)

l > µ l −1 ≤ ( s, γ ) < l

l

∑ ∑

l>µ k=d

= I1 + I2 .

Sl, k 2 µ + αµ − 2αl + αk

(4)

1192

A. S. ROMANYUK

To estimate I 1, we use the known relation [2]

∑ 2(s, δ)

<< 2 n n ν−1 ,

(5)

( s, γ ) ≤ n

where 1 = γ 1 = δ 1 = … = δ ν , 1 ≤ δ j < γ j , j = ν + 1, … , d. As a result, we get I1 ≤

∑ 2(s, 1)

<< 2 µ µ ν −1 ⱱ M.

( s, γ ) ≤ µ

(6)

Applying estimate (5) to I 2 , we obtain I2 ≤

∑ 2µ + αµ − 2αl

l>µ

∑ 2α(s, 1)

<<

l −1 ≤ ( s, γ ) < l

∑ 2µ + αµ −αl l ν−1

l>µ

= 2 µ + αµ ∑ 2 −αl l ν −1 << 2 µ + αµ 2 −αµ µ ν −1 ⱱ M.

(7)

l>µ

Substituting (6) and (7) in (4), we obtain the required estimate (3). Let l, k be the subspace of trigonometric polynomials with the “numbers” of harmonics from the set Q l, k = U ρ(s) . Then, for f ∈ l, k and θ ≥ p, by virtue of the Hölder inequality, we have s ∈ Sl, k

  p δ ( f , x ) s  ∑ p   s ∈ Sl, k

1/ p

 ≤  ∑ δ s ( f , x)   s ∈ Sl, k

1

   ∑ 1  s ∈ Sl, k 

θ θ p  

θ− p θp

1

θ  =  ∑ δ s ( f , x ) θp  Sl, k   s ∈ Sl, k

We need the following auxiliary statement: Theorem B [3]. Between the space of trigonometric polynomials of the form f (t ) =

and the space R2

( s, 1)

∑ ck ei(k, t)

k ∈ρ( s)

, there is an isomorphism that associates a function f ( ⋅ ) with the vector δs f fn(t) =

(

∑

j

=

ck ei (k, t ) ,

sgn kl = sgn nl

{ fn (τ j )} ∈ R2

( s, 1)

,

l = 1, d , n = (± 1, … , ± 1) ∈ R d,

)

τ j = π2 2 − s1 j1, … , π2 2 − sd jd ,

j i = 1, … , 2 si −1 ,

i = 1, d .

1 1 − p θ.

(8)

ON E STIMATES OF THE KOLMOGOROV WIDTHS OF THE CLASSES B rp ,q

IN THE S PACE

Lq

1193

Furthermore, the following relation is true: 2  ⱱ  2 − (s, 1) ∑ δ s f  j =1 ( s, 1)

δ s ( f , x)

p

 j p

1/ p

p ∈ ( 1, ∞ ).

,

 

Thus, on the one hand, by virtue of inequality (8) and Theorem B, for f ∈ Bpr, θ I l, k we have 1

1 ≥

f

Bpr, θ I l, k

 θ =  ∑ 2(s, r )θ δ s ( f , x ) θp   s ∈ Sl, k  1

θ  ⱱ 2 r1l  ∑ δ s ( f , x ) θp  ≥ 2 r1l Sl, k   s ∈ Sl, k

  ∑ δ s ( f , x)  s ∈ Sl, k

1

p p p 

1

 1 1 − r l − k/ p Sl, k θ p 2 1 

≥

1 1 − θ p

2k

∑ ∑

 s ∈ Sl, k j = 1

δs f

j

p p  . 

This relation implies that if f ∈ Bpr, θ I l, k , then 2   ∑ ∑ δs f  s ∈ Sl, k j = 1 k

1 p  p j

 

<< 2 − r1l + k / p Sl, k

1/ p −1/θ

.

(9)

On the other hand, for g ∈ Lq I l, k , q ≥ 2, we have 1

g

q

2  <<  ∑ δ s (g, x ) 2q    s ∈ Sl, k

and, by virtue of the Hölder inequality with exponent q / 2, we get 1

g

q

1

2 q   <<  ∑ δ s (g, x ) qq   ∑ 1   s ∈ Sl, k   s ∈ Sl, k

–

1 q

ⱱ

1 1 − Sl, k 2 q 

1

∑

 s ∈ Sl, k

q δ s (g, x ) qq  . 

(10)

Applying Theorem B to the last sum in (10), we establish the estimate

g

q

<<

1 1 − 2 − k / q Sl, k 2 q

2k  j  ∑ ∑ δ s g  s ∈ Sl, k j = 1

1

q q  . 

(11)

1194

A. S. ROMANYUK

Thus, by using relations (9) and (11), we can establish the correspondence between the classes Bpr, θ I l, k and the finite-dimensional spaces l p nite-dimensional spaces lq mate

(

Sl, k

)

d M Bpr, θ , Lq <<

and also between the subspaces Lq I l, k

and the fi-

. Consequently, carrying out the corresponding discretization, we obtain the esti-

∑ dM ( Bpr, θ I l, k , Lq I l, k ) l, k

l, k

l

<<

Sl, k

∑ ∑

1/ 2 − 1/ q + 1/ p − 1/ θ

2 − r1l + k / p − k / q Sl, k

l>µ k=d

S

S S d Ml , k  Bp l , k , lq l , k  ,

(12)

S

where Bp l, k is the unit ball in the space l p l, k . Note that, in the course of the proof of the second inequality, we have used the relation

(

d Ml, k Bpr, θ I l, k , Lq I l, k

)

d ≤ k ≤ l,

= 0,

l < µ.

To continue estimate (12), we need the following auxiliary statement: Lemma A [4]. Suppose that M < n, 2 ≤ p < q < ∞, and β =

(

)

{

1/ p − 1/ q . Then 1 − 2 /q

}

d M Bpn , lqn ⱱ min 1, n 2β / q M −β .

(13)

Under our conditions, according to (13) we have S S d Ml, k  Bp l, k , lq l, k  <<

Sl, k

Further, substituting (14) in (12) and taking into account that β =

(

)

d M Bpr, θ , Lq <<

<<

∑ 2 − r1l

l>µ

l

∑ 2 k / p − k / q Sl, k

(14)

1 1 2β – + , we get p q q

1/ 2 − 1/ q + 1/ p − 1/ θ

Sl, k

2β / q

Ml−, kβ

k=d

∑ 2 − r1l − µβ − βαµ + 2α lβ

l>µ

2β −β q Ml, k .

l

∑ 2 k β − βα k

Sl, k

1/ 2 − 1/ θ

.

(15)

k=d

To continue (15), we estimate the inner sum in this relation. For this purpose, we represent this sum in the form

ON E STIMATES OF THE KOLMOGOROV WIDTHS OF THE CLASSES B rp ,q l

∑

k=d

2 k β − βα k Sl, k

1 1 − 2 θ

IN THE S PACE

1195

 l  l =  ∑ ′ + ∑ ′′  2 k β − βα k Sl, k   k = d k = d 

l

∑ ′ is carried out over k for which

where the summation in

Lq

k=d

1/ 2 − 1/ θ

,

(16)

Sl, k ≤ l ν – 1 and the summation in

l

∑ ′′

is

k=d

carried out over k for which Sl, k > l ν – 1. Therefore, for arbitrary 0 < α < 1, we have l

∑ ′ 2 k β(1 − α ) Sl, k

1/ 2 − 1/ θ

≤ l ( ν −1)(1/ 2 −1/ θ)

l

∑ ′ 2 k β(1 − α)

<< 2l β(1− α ) l ( ν −1)(1/ 2 −1/ θ) .

(17)

k=d

k=d

If Sl, k > l ν – 1, then l

∑ ′′ 2 k β(1−α ) Sl, k

1/ 2 − 1/ θ

k=d

l

∑ ′′ 2 k β(1−α ) Sl, k

=

Sl, k

−1/ 2 − 1/ θ

k=d

< l − ( ν −1)(1/ 2 +1/ θ)

l

∑ ′′ 2 k β(1−α) Sl, k

≤ l − ( ν −1)(1/ 2 +1/ θ)

∑

2β(1− α )(s, 1)

( s, γ ) ≤ l

k=d

<< l − ( ν −1)(1/ 2 +1/ θ) 2βl(1− α ) l ( ν −1) = 2βl(1− α ) l ( ν −1)(1/ 2 −1/ θ) .

(18)

Thus, substituting (17) and (18) in (16), we obtain

∑ 2 k β − βα k

Sl, k

1/ 2 − 1/ θ

<< 2βl(1− α ) l ( ν −1)(1/ 2 −1/ θ) .

k=d

Returning to (15), we get

(

)

d M Bpr, θ , Lq << 2 −µβ + αβµ ∑ 2 − r1l + l β + α l β l ( ν −1)(1/ 2 −1/ θ) .

(19)

l>µ

Finally, we choose 0 < α < 1 so that the condition r1 > ( 1 + α ) β is satisfied (since r1 > β according to the conditions of the theorem, such a value of α always exists). By using (19), we establish the required estimate

(

)

(

d M Bpr, θ , Lq << 2 −µ r1 µ ( ν −1)(1/ 2 −1/ θ) ⱱ M − r1 log ν −1 M The upper bound is proved, which completes the proof of the theorem.

)r + 1/ 2 − 1/ θ . 1

1196

A. S. ROMANYUK

(

)

Remark. The upper bound of the width d M H pr , Lq , 2 ≤ p < q < ∞, β < r1 was obtained earlier by Galeev [3]. The author is grateful to Prof. Sun Yongsheng and Prof. Wang Heping for drawing his attention to an inaccuracy in the formulation of the condition imposed on r1 in Theorem A in [5]. REFERENCES 1. A. S. Romanyuk, “On the best trigonometric approximations and Kolmogorov widths of the Besov classes of functions of many variables,” Ukr. Mat. Zh., 45, No. 5, 663–675 (1993). 2. V. N. Temlyakov, “Approximation of functions with bounded mixed derivative,” Tr. Mat. Inst. Akad. Nauk SSSR, 178, 1–112 (1986). 3. É. M. Galeev, “Kolmogorov widths of classes of periodic functions of many variables W˜ pα and H˜ pα in the space Lq ,” Izv. Akad. Nauk SSSR, Ser. Mat., 49, No. 5, 916–934 (1985). 4. B. S. Kashin, “Widths of some finite-dimensional sets and classes of smooth functions,” Izv. Akad. Nauk SSSR, Ser. Mat., 41, No. 2, 334–351 (1977). 5. Sun Yongsheng and Wang Heping, “Representations and approximation of multivariate periodic functions with bounded mixed moduli of smoothness,” Tr. Mat. Inst. Akad. Nauk SSSR, 219, 356–377 (1997).