Estimates of best approximations of functions with logarithmic smoothness in the Lorentz space with anisotropic norm Текст научной статьи по специальности «Математика»

URAL MATHEMATICAL JOURNAL, Vol. 6, No. 1, 2020, pp. 16-29

DOI: 10.15826/umj.2020.1.002

ESTIMATES OF BEST APPROXIMATIONS OF FUNCTIONS WITH LOGARITHMIC SMOOTHNESS IN THE LORENTZ SPACE WITH ANISOTROPIC NORM1

Gabdolla Akishev

L.N. Gumilyov Eurasian National University, 2 Pushkin str., Nur-Sultan, 010008, Kazakhstan

Ural Federal University, 19 Mira str., Ekaterinburg, 620002, Russia

[email protected]

Abstract: In this paper, we consider the anisotropic Lorentz space Lg g(Im) of periodic functions of m variables. The Besov space ' of functions with logarithmic smoothness is defined. The aim of the

paper is to find an exact order of the best approximation of functions from the class B^g'^' by trigonometric polynomials under different relations between the parameters p, $, and t.

The paper consists of an introduction and two sections. In the first section, we establish a sufficient condition for a function f € Lg (Im) to belong to the space Lg (Im) in the case 1< 92 < , j = 1,..., m, in terms of the best approximation and prove its unimprovability on the class E^g = {f € Lg g(Im): En(f)g g < An, n = 0,1,...}, where En(f )g g is the best approximation of the function f € Lg g(Im) by trigonometric polynomials of order n in each variable Xj, j = 1,..., m, and A = {An} is a sequence of positive numbers An 4 0 as n ^ In the second section, we establish order-exact estimates for the best approximation of functions

from the class B^^T' in the space Lg (Im).

Key words: Lorentz space, Nikol'skii—Besov class, Best approximation.

1. Introduction

Let X = (x1,...,xm) € Rm, Im = [0,2n]m, p = (p1,...,pm), and 0 = (01,...,dm), where Pj € (1, to) and 0j € [1, to) for j = 1,2,..., m. Denote by LP p(Im) the Lorentz space of real-valued functions f (x) that are 2n-periodic in each variable and

--1

c2n

f

^ 1

— — 1

(i!,...,im))Ölir dt 1

1/öm

dtm

< +TO,

where f*1,-.,*m is a nonincreasing rearrangement of the function |f(x1,...,xm)| in each of the variables Xj whereas the other variables are fixed (see [8, 18]).

In the case p1 = ■ ■ ■ = pm = 01 = ■ ■ ■ = 0m = p, the Lorentz space LP p (Im) coincides with the Lebesgue space Lp (Im) with the norm

p

/*2n f2n

/ . .. / |f (X1, ... ,xm)|pdx1 00

1/p

1This work was supported by the Competitiveness Enhancement Program of the Ural Federal University (Enactment of the Government of the Russian Federation of March 16, 2013 no. 211, agreement no. 02.A03. 21.0006 of August 27, 2013).

6

6

m

2

6

0

1

m

0

0

where p € [1, +to).

r * l

-p/- J, wc wm WLnciVpj

Instead of LP -(Im), we will write L* 0(Im) in the case pi = ■ ■ ■ = pm = p and 91 = ■ ■ ■ = 9m = 9

and L* di2)(Im) if p = (Pl,... ,pm) and d1 = • • • = 6m = 0(2). Given a natural number M, consider the set

□m = {k = € : |kj| < M, j = 1,...,m}.

Consider the multiple Dirichlet kernel

(x) = ^ ei(P 'X € fce^M

and its convolution with a function f € LP p(Im):

x) = Jf(y)(D^2s(x - y) - (x - y))dy,

where s € No = N U {0} and N is the set of positive integers.

Let M € N0, and let Tm(x) = ape^P 'x be a trigonometric polynomial of order M in each

fce^M

variable Xj, j = 1,..., m. Denote by 3bM the set of all such polynomials.

Let Em,... , m(f)p p = inf ||f — T||P e be the best approximation of a function f € LP p(Im) by

the set 3oM • Sometimes, we will use the notation EM(f )p p instead of EM, ... , M(f )p p. For a given class F c LP e(Im), let Em(FL p = sup Em(f )p p.

' /eF

Let a > 0, y € (—to, +to), and 0 < t < to. Denote by App7'r) the space of all functions f € LPp(Im) such that the quasi-norm (see [9, 20])

i 1/t

En"1 (na(1 + log n)Y En(f )p-p)T]

n=l

ß,(af'T) =

is finite, where log a is the logarithm of the number a to the base 2. If t = to, then

|Aa,7,T = supna(1 + logn)7Era(f)pp < to.

n>1

It is known that APap7'T) is a quasi-Banach space (see [9, 10, 20]). It is called an approximate space (see [11]).

In the aniso f € LP p(Im) representable in the form of series

<x

EQ22" (f,X), Q22" (f) € fc22n (1.1)

In the anisotropic Lorentz space, we consider the space t < to, of all functions

* C[im\

P,<

n=0

and such that

for 1 < t < to and

n=0

[e(2na 11q2- (/)ii;,-) t]1/t < +to (1.2)

sup 2hiq2- (/)ii;-< to

raSNo

for t = to. The infimum of expression (1.2) over all representations (1.1) defines a quasi-norm in this space:

* _\ t

— "" | II y lip ,iJ

" " n=Q

,(0,alT) =in^(2nayg22n (/ )||p

1/T

The space B0^' T) is called the Besov space with logarithmic smoothness. In BpQia ' T\ we consider the unit ball

BpQfT) = {f G ) : 1/llB(0,«.T) < 1}•

It is known that / € iPQf+1/t' T) if and only if / € Ap^' T) (see [10]).

The main aim of the present paper is to obtain an exact order of the best approximation of the function classes ApQpYlT) and BPQpYi)~) in anisotropic Lorentz spaces.

In the one-dimensional case, sufficient conditions for a function / € Lp(11) to belong to the space Lq(I1) for 1 < p < q < to in terms of the best approximation and the modulus of continuity were established by P.L. Ul'ynov [30]. This study was continued by V.I. Kolyada [15], V.A. Andrienko [5], N. Temirgaliev [27, 28], E.A. Storozhenko [26], M.F. Timan, P. Oswald, L. Leindler, S.V. Lapin, B.V. Simonov, and others (see the references in [16]).

N. Temirgaliev established [28] a necessary and sufficient condition for a univariate function f € Lp(I1) to belong to the Lorentz space (I1) in terms of the modulus of continuity for 1 < 0 < p < to. L.A. Sherstneva studied [22] this problem in terms of the best approximation of a function. Such problems in the Lorentz space were investigated in [1, 4, 23].

Problems of estimating various approximative characteristics of function classes are well known and a survey of the results on this topic is given in [12, 29]. In particular, in the Lebesgue space Lp(Im), exact estimates of the best approximation of functions of the Besov class Bp p^)

were established by A.S. Romanyuk [21]. In the case 0(1) = pj = p, j = 1,...,m, estimates of approximative characteristics of the class bQ'were obtained by S.A. Stasyuk [24, 25]. In [13], the embedding and characterization problems of the Besov space with logarithmic smoothness in the Lebesgue space Lp(Im) were investigated.

Exact estimates of best approximations of functions from the Besov class in the Lorentz space with a mixed norm were obtained in [2, 6, 7].

The present paper consists of the introduction and two sections. In Section 1, we establish a sufficient condition for a function / € Lp p(Im) to belong to the space Lp e(2) (Im), 0(2) < ^j1), j = 1,..., m, and prove its accuracy on the class

Ep-'i = {/ € Lpi(Im) : Era(/)p-i < A„, n = 0,1,... },

where A = {An} is a sequence of positive numbers An 0 as n ^ +to.

In the case pj = = p, j = 1,...,m, V.I. Kolyada proved [15] a necessary and sufficient condition for the embedding of classes E- in the space Lq(I1), 1 < p < q.

In Section 2, we establish order-exact estimates of the value E^Bp0^)))qi(2) under various relations between coordinates of the parameters p, q-, -(2) , t (see Theorems and ).

The notation A (y) x B (y) means that there exists positive constants C1 and C2 such that C1A (y) < B (y) < C2A (y). If B(y) < C2A(y) or A(y) > C1B(y), then we write B(y) << A(y) and A(y) >> B(y), respectively.

2. Conditions for embedding classes in the Lorentz space

Theorem 1 [19, Theorem 10]. Let 1 < pj < +to and 1 < < qj < +to for j = 1,..., m, let p = (pi,... ,pm) and q = (qi,..., qm), and let 0 = ..., 0m). Then a trigonometric polynomial

n-l nm

eî(x,fc)

Tn(x)= ... £ b

k1 = n1 km = nm

satsfies the following inequality :

||Tn||Pp < C(p, q, 0) J] (ln(1 + nj))1/öj-1/qj ||T j=i

Lemma 1. Let 1 < pj < to and 1 < q2 < qj1^ < +to for j = 1,..., m. Let {un} be a sequence of non-negative measurable functions on the cube = [0,2n]m such that (1)

llunll^a) < ^n+i < ß^n, ß € (0,1); (2) there exists a sequence of positive numbers {An} such that

£(1/öj -1/j )

UnHp,0 < CAn £n, n — 1,2,3,...,

any € (0, q(1)), j = 1,..., m. Then the inequality

Hf up,® < C{E at1 42}

n=1

holds for every function of the form f (x) = ^^=1 un (x).

This lemma is proved by V.I. Kolyada's method (see [15, Proof of Lemma 4]) as in [3].

Remark 1. Lemma 1 was proved by L.A. Sherstneva [22, Lemma 13] in the one-dimensional case and by the author [3] in the multi-dimensional case for q(1) = ■ ■ ■ = q^.

Now, let us consider a condition for a function f € LP (Im) to belong to the space LP 0(2) (Im), 1 < 0(2) < j < +TO,j = 1,...,m.

Theorem 2. Let 1 < 0(2) < 0(1) < +to and 1 < pj < to for j = 1,... ,m, and let 0(1) =

j

?11),..., 01(m)). Assume that f € LPp(1)(Im) and

0(2) £ (1/0(2)-1/0(1) )-1

j = 1 j

V(l1"" -<:,„(/w, < +=0. (2.1)

n=2

Then f € LP,Ö(2) (Im) and

m

ö(2) £ (1/e(2) -1/0(1))-1

* ^/llfll* I rv(ln(fc + l)) ^ ' ,(3) iV^S (2.2)

p,0(2) « {11/lip,0(D + [2^ -1-^fc,...,fc(/)p,0(1) j, V '

k=2

m

En,...,n(/}p,e(2) << (ln(n + 1))

£(1/0(2)-1/0<1) )

)j=l

En,...,ra(/)pi0(l) +

+ [E

fc=ra+1

(ln(k +1))

0(2) E (1/0(2)-i/0(1))-i

j=i

Efc(..).,fc(/)p,0(1)

1/0(2)

(2.3)

Proof. Since En'...,n(/)pi(i) = ^ 0 as n ^ +to for every function / € Lp ^(Im), 1 < pj, 0(1) < +to, j = 1,..., m, there exists a numerical sequence {nv} such that (see [15, Sect. 2])

n-v + l

< ~:£nv, £nv+1-i > ~£nv, v — 1,2

2

2

Let Tn(f, x) be a trigonometric polynomial of the best approximation of a function / € Lp i(1)(Im), 1 < pj, 0j1) < +to, j = 1,..., m. Consider the series

Tni (/, x) ^(T„v+1 (/, x) - Tnv (/, x))

(2.4)

V=1

Let us prove the convergence of this series in the norm of the space Lp ^2) (Im). Suppose that

Uv (x) = |T„v+1 (/, x) - Tnv (/, x)|, v = 0,1,...

Then

and, by Theorem 1,

|uv lip,0(1) < 2tv

< 2ev, v = 0,1,

llu

v lip , r

E (1/Tj -1/0(1) ) << (ln nv+1)J=1

for any Tj € (0,0j1)), j = 1,..., m. Hence, by Lemma 1, we obtain

E (Tnv+1 (/) - Tnv (/))

v=fc+1

I

<< { S (ln nv+1) J=1

v=fc+1

p,0(2)

<

£ uv v=fc+1

p,0(2)

<<

0(2) ^E1(1/0(2)-1/0(1)) 0(2) -,1/0(2)

^v J

(2.5)

Condition (2.1) implies that

£(ln nv+1)

0(2) E1(1/0(2) -1/0« ) 0(2)

v=1

(2.6)

It follows from (2.5) and (2.6) that series (2.4) converges to a function g € Lp ^2) (Im) in the norm.

p,<

It is easy to see that g(x) = /(x) almost everywhere on Im. Hence, / € Lp ^2)(Im). Setting k = 0 in (2.5), we get

lTm+1 (/)||P 0(2) <<

+ £(ln nv+1)

m

0(2) E1(1/0(2)-1/0(1)) 0(2)

v=1

1/0(2)

<<

oo

k

*

v

г те

<< < II/ ГрЖ1) +

0(2) Е (1/0(2)-1/0(1))-1 (ln(n +1)) ^ J ,(2)

---Аг,...,пШр,0(1>

Lra=2

By tending l to +то in this inequality, we obtain

i/0(2)

p,0(2) << II/Hp,0(i) +

0(2) E (1/0(2)-1/0(1))-1

\ - (ln(n + 1)) ' = 1_ gi2)

Lra=2

1/0(2)

Thus, inequality (2.2) is proved.

Applying inequality (2.2) to the function f — Tn(f) € Lp0(2)(Im), it is easy to prove inequality (2.3). The proof of Theorem 2 is complete. □

Let us prove that condition (2.1) is exact on the classes Ep, .

Theorem 3. Let 1 < pj < to and 1 < 0(2) < 0(1) < +to for j = 1,..., m. The following condition is necessary and sufficient for the inclusion EA ^ c LP 0(2) (Im):

0(2) E (1/0(2)-1/0(1))-1 E -—Af1 < +=o. (2.7)

n=2

Proof. The sufficiency of condition (2.7) follows from Theorem 2. Let us prove the necessity.

iA f— T * 'p,0« C Lp, (

Let Ea 0(1) с LP 0(2)(Im). Assume that condition (2.7) is violated, i.e.,

m ,.,4

0(2) E (i/0(2)-i/e(1))-i

E"11'" "„-(2-8)

n=2

We choose a sequence of numbers } with the following properties (see [15]):

-AVk, \Uk+1-i > —

Since the function (ln x)e/x with в € R decreases to 0 as x ^ +то, we have

V,+1 0(2) E(1/0(2)-1/0( )-1 vfc+1 0(2) E(1/0(2)-1/0(1))-1

y^1 (Inn) ^ < ^ Мп-щ + l)) ^

11 ~ 11 — Vk

ra=vfc+1 ra=Vfc+1

<<

0(2) E (1/0(2)-1/0(1) ) << (ln(vfc+1 - Vfc + 1)) j=1 •

Thus, (2.8) implies that

m

~ 0(2) E (1/0(2)-1/0(1) ) 0(2)

]T(ln(vfc+1 - vfc + 1)) AV, = +то. (2.10)

fc=1

Let us consider the function

/o(x) = Avk (ln(vfc+1 - vfc + 1)) j=1 3 Tfc(x),

m

- E 1/0(4

ln(

V k \

fc=0

where

m z'fc+l i

"/■'•'•) II £ {n3-vk)pj :

j=1 raj=vfc+1

It is known that (see [22])

m

llTfcllp'^d) x (ln(vfc+1 - vfc + 1))j=1" , 1 <pj< +to, j = 1,..., m. (2.11)

Using this relation and (2.9), we can verify that

m

~ - E 1/e(1) ~

ll/Qllp'i-(i) <£ Avfc(ln(vfc+1 - Vfc + 1)) llTkllp p(i) < ^Avfc < to.

fc=Q fc=Q

Hence, /q € Lp,^ (Im), 1 < p;, j < to, j = 1,...,m.

Let a positive integer n satisfy the inequalities v < n < v1+1. Then, by the best approximation property and according to relation (2.11) and inequality (2.9), we have

m

~ - E 1/i(1)

E 1/i(1)

En(fQ)j-'-(1) < Ev (/q)j-'-(1) < £ Avfc (ln(vfc+1 - vfc + 1)) j=1 j ^Tfc yp-'-d) <<

fc=z

<x

<< £Avfc << Av << 2Av1+1-1 < CQAn.

Hence, /1 = Cq-1/q € e* .

fc=z

-1

e(1).

e(2)

Let us show that f1 € L* (2)(Im), 1 < 0(2) < to. To this end, we consider the function

go(x) = £(ln(vfc+1 - vfc + 1))j= fl(1) AVf-1^(x)

fc=Q

where

5 lyfc+1 ^--1

= n £ ("j - Vk)P'j sin HjXj, p'j = —J—r, 3 = 1,..., m.

pj - 1

j=1 raj=vfc+1

It is clear that (see (2.11))

m

E 1/lj

p, p x (ln(vfc+1 - vfc + 1))j=1 , 1 <pj < +to, 1 <0j < to, j = 1,..., m. Further, in view of the orthogonality of the trigonometric system, for any number N, we have

n £

Bn = J /1(x)£ AVk2)-1(ln(vfc+1 - vfc + 1))j= j £fc(x)dx =

Jm k—0

N

-1(2) E 1/0(1) _ m Vk+1 i c - + D] * ' C n E ^"hr » <2-12»

k=Q j=1 nj=vk + 1

W i(2) E (1/i(2) -1/e(i))

>>£[ln(vfc+1 - vfc+1)] -1 av;) .

fc=Q

Using the integral Holder inequality, we obtain

N £ i^Ei

bn << n/iii;-ö(2)£(in(vfc+i -vfc + 1))-1 j Av;2)-iefc p , (2.13)

k=Q

where

0(2) - 1 '

We set uk(x) = (ln(i/fe+1 - vk + 1))J=1 ^ A^'"1^)!. Then (see (2.11))

J

0(1) « Kk 1 = ßk,

* ^ \0(2)-1 _

A (T.i „(1) I

llp,;r << [ln(vfc+1 - vfc + 1)]j E1 J j &, k = 0, 1,.... Thus, all the conditions of Lemma 1 hold for the sequence of functions (uk(x)}. Therefore,

N y _

lEdnK+i-^ + Dr1 V Af-^l*

k=Q

{]T(ln(vfc+i - Vk + 1)) j=1 A^} .

<<

p ,e(2)'

(2.14)

k=Q

Now, it follows from inequalities (2.12), (2.13), and (2.14) that

{]T(ln(vk+i - Vk + 1)) j=1 A^} << ||/iHp,ö(2).

k=Q

By (2.10), we find that /1 / LP0(2)(Im), 1 < 0(2) < j < +to, j = 1,... ,m. This contradicts the inclusion EAr(1) C LP0(2)(Im). The proof of Theorem 3 is complete. □

Remark 2. The results of L.A. Sherstneva [22] follow from Theorems 2 and 3 in the case m = 1.

3. Estimates of best approximations of functions with logarithmic smoothness

Now, let us prove estimates of the value EM (F)p 0(2) for the classes F = Bp^a T) and F = Ap^T).

Theorem 4. Let 1 < pj < to and 1 < 0(2) < < to for j = 1,..., m, and let 1 < t < to. If f ), then ßP^T) C L;>ö(2)(

a > £ (1/0(2) - 1/0(i)), then Äf C LP,(2)(Im) and

j=i

Pi0(2) << ||f Ho(0,a,T ) .

B(

P,f

Proof. Let f € B^0e(1')rThen, by the definition of the class, this function can be represented in the form of the series

ro

E Q22v (f, X) , Q22v (f, X) € F^22n ,

v=0

in the sense of convergence in the quasi-norm of the space LP (Im) and

E(2valQ22v (f )iyr

v=0

l/T

< +TO.

If 0(2) < t < to, then, using the Holder inequality and taking into account that

m

(

m

a > E (1/^(2) — 1/j), we obtain

where

f ~ v E(i/e(2)-i/e(1))e(2) wi/e(2)

{E2j=1 j (IQ22V(fe } <

v=0

<{532—(\\Q2,Hf)\\;^y} {E2 }■9 ,3 ^ (3-1}

v=0 v=0

ro i ,

< ^{53 2vTa (1Q22V (f )i;,p(1^r1

v=0

0(2)' K £ — 1' If t = to, then

f ~ v E (i/e(2)"i/e(1) )e(2) w i/e(2)

{E2j=1 j (IQ22V(f)ii>))e } <

v=0

ro ve(2)(£ (i/e(2)-i/e(1))-«) ^ i/e(2)

< sup 2va IQ22V (f )li; *(!){£ 2 j=1 j .

veN0 ^ v=0 J

(3.2)

If t < 0(2), then, using the Jensen inequality (see [17, p. 125]), we obtain

f ~ v E (i/e(2)-i/e(1))e(2) w i/e(2) , ~ i/t

{E2j=1 (HQ**(f)i;,p(10e} <{E2—(IQ22V(f)i;,p(10T} . (3.3)

v=0 v=0

Thus, (3.1)-(3.3) imply that the series

v E(i/e(2)-i/e(1))e(2) ^

E2 j=1 (IQ22V(fe (3.4)

v=0

is convergent for every function f € B(0ea1')r).

Taking into account the monotonicity of the best approximation and the properties of the norm, it is easy to verify that

ro

m

V (1/0(2) _1/0(1))_1 m

y(ln n)hu (3) < (2,

-n-En,...,n(/)p,0< 1)<<Z^2 V.....2^1/Wl11 «

n=2 v=0

m

~ V E (1/0(2) _1/0(1) )0(2) /11 ~ * N0(2)

<<E2 j=1 ' (||EQ221 (/)

v=0 1=v

p,^1)/ <<

(3.5)

~ v£ (1/0(2)_1/0(1))0(2) ~ ,0(2)

<<E2 j (Ell^ (/

v=0 1=v

Since 0(2) < , j = 1,..., m, we have

mm

n V E (1/0(2)_1/0(1))0(2) n E (1/0(2)_1/0(1))0(2)

^2 j=1 << 2 j=1 , n € N0.

V=0

Therefore, according to [14, Lemma 2.2], we find from (3.5) that

m

E (1/0(2)-1/0(1) )0(2)-1 m m

A (ln n)j=1 3 0(2) A v E (1/0(2)-1/0(1))0(2) , 0(2)

E—-^-<:!.n(/w><<£2 (WQ^un;^) • o-e)

n=2 v=Q

Since the series (3.4) converges, it follows from (3.6) that

m

E (1/0(2) -1/0(1) )0(2)-1

v- (1dw)j-1_P0(2) (f\ / _

/ ^---< OO■

n=2

Hence, by Theorem 3, we have / € Lp0(2) (Im).

Let us estimate the quasi-norm ll/||p 0(1). By the quasi-norm property and the Holder inequality, we obtain

p,0(1) <<E »Q22V (/)y;,0(1) <<

V=0

V =(1/0(2) _1/0(1) )0(2) ^ * (2) N1/0(2)

(3.7)

<<(E2 j=1 " j (1Q22V(/)ii;;0-(1))02)

j=1

V=0

Therefore, according to relations (2.2), (3.6), and (3.7), we have

0j(1))

„>) <<J^ 2 j=1 j

m

r ~ v e(1/0(2)_1/0(1))0(2) w^1/0(2)

<<{E 2 j=1 (IQ22V (/)i;,0-(1^0 } . (3.8)

V=0

Taking into account (3.1)-(3.3) and (3.8), we obtain

" ,t,1/T

;,0(2) << {E2vr(7+1/r)(IQ22V(/)iT} T (3.9)

V=0

for every function / € B^aT). The proof of Theorem 4 is complete. □

Theorem 5. Let 1 < pj < to and 1 < 0(2) < 0(1) < to for j = 1,..., m, and let 1 < t < to. If

j

m

a > £ (1/0(2) — 1/j), then

j=i j

m

(0aT) -(«- E (i/e(2)-i/e(1)))

P0eiT1T))p ,e(2) ~ (log(M + 1)) j=1 j , M € N.

Proof. Let f € bP0^. We have a > E(1/0(2) — 1/^jl)); therefore, f € LP e(2)(Im)

by Theorem 4. Take a positive integer l such that 22 < M < 22l+1. Then, using the best approximation property and inequality (3.9), we have

f ~ v E (i/e(2)-i/e(1))e(2) „(2), i/e(2)

EM(f)p-,e(2) < E22i (f)p-*(2) << { £ 2 j=1 (HQ22" (f)NP.edO } • (3.10)

v=l

m (i)

If 0(2) < t, then by the Holder inequality and in view of the fact that a > (1/0(2) — 1/0,- )),

j=i

(3.10) implies that (see formula (3.1))

ro 1 /

Em (f )p,e(2) < 2vTa (HQ** (f )| T} 'x

v=0

^2 j=1 ^ << 2 j=1 v=l ^

for every function f € bp^T) in the case 0(2) < t.

If t < 0(2), then, arguing as in the proof of formula (3.3), by means of the Jensen inequality, we find from (3.10) that

m

r ~ , 1 i/T -i(a- E (i/e(2)-i/e«))

EM(f)p-e(2) ^^2vTa(HQ** (f^-pa))T} 2 -1 . (3.12)

v=0

Now, taking into account that 22 < M < 22l+1, by formulas (3.11) and (3.12), we obtain

-(«-E (i/e(2)-i/e( ))

Em(f)p,e(2) << (log(M + 1))

for every function f € B^e"')"). Thus, the upper estimates are proved. Let us prove the lower estimates. Consider the function

-(n+l)(a+E l/dW) 2"+2 m _

i=i V V 1 iL-j T 1 • 1) ei{k^

f2(X) = 2 ^ E E L[(kj — 2s-1 + 1)

s=2n+1+i fce^2s \^2S-1 j=i

where x € Im and n € N0. It is well known that

s

= 2 i=1 x x 1 r

P,e(1)

£ ^(/2) =2 J £ £ 11 i/ / ' ' 1)

=2n+1 s=2n+1+1 fce^2s \^2S-1 j=1

*

e (1)

<<

m (1) m (1)

-(n+1)(a+ £ 1/Ö«) 2n+2 2n+1 £ V^ ( +1)

<< 2 j=1 j (log(22n+2 - 22n+1 ))j=1 j << 2-(n+1)a.

Thus,

2V+1

v=0

E2VTi II E ( f2 )

s=2v

* \ t ") 1/t

2n+2

—2

(n+1)a

E ^s(f2)

s=2n+1

>,(P(1)

< C1.

Hence, C-1f2 € bP0^ for 1 < 0(2) < to and 1 < t < to. Next, by the definition of the best approximation and the estimate

2n+2

E E Il(kj - 2s-1 + 1)

s=2n+1 + 1 fceÜ2S\d2S_1 j=1

__i - _

Pj ei{k,x)

>,0(2)

» 2'\(2) ,

we have

E22n (f2)p,0(2) — C-

-1

p,0(2)

— C-12

-(n+1)(«+£ 1/0((1) )

j=1

(1) 2n+2

E E n(kj - 2s-1 + 1)

s=2n+1+1 fceD2s \Ü2S_1 J=1

Pj 1 ei(k,x)

>,0(2)

>>

m

-(n+1)(«-£ (1/0(2)-1/e(.1))) >> 2 j=1 .

Taking into account that 22" < M < 22"+1, we obtain

Em(f2ke(2) >> (log(M + 1))

(«-£ (1/ö(2)-1/ö(1)))

=1

for 1 < 0(2) < to and 1 < t < to. Thus, the proof of Theorem 5 is compete.

□

Theorem 6. Let 1 < pj < to and 1 < 0(2) < ) < to for j — 1,..., m, and let 1 < t < to.

m

If Y > E (1/0(2) - 1/0(1)) - 1/t, then j=1 j

Em(A^k^) x (log(M + 1))

-(y+1/T-£ (1/Ö(2)-1/ö(1)))

=1

Proof. Since ap0'?,')) and bp0'?1+ 1/t't) coincide, the statement of Theorem 6 follows from

Theorem 5.

.,e(1)

□

*

m

*

m

m

■i

4. Conclusion

The best approximations of functions of the classes bp^t) and AP0e(l)) in the space LP e(2) (Im)

Jp

have logarithmic order.

Note that estimates of the quantities EM(bP0^)))pp(2) and EM(aP0^))))pP(2) are unknown in

the case j — 0(2), j — 1,...,

m.

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