MODIFIED MODULUS OF SMOOTHNESS AND APPROXIMATION IN WEIGHTED LORENTZ SPACES BY BOREL AND EULER MEANS Текст научной статьи по специальности «Математика»

Probl. Anal. Issues Anal. Vol. 10 (28), No 1, 2021, pp. 87-100 87

DOI: 10.15393/j3.art.2021.8950

UDC 517.518.832, 517.518.235

S. S. VOLOSIVETS

MODIFIED MODULUS OF SMOOTHNESS AND APPROXIMATION IN WEIGHTED LORENTZ SPACES BY BOREL AND EULER MEANS

Abstract. Using one-sided Steklov means, we introduce a new modulus of smoothness in weighted Lorentz spaces. The direct and inverse approximation theorem for this modulus of smoothness are proved. Also, we estimate the rate of approximation by the Borel and Euler means in weighted Lorentz spaces.

Key words: weighted Lorentz spaces, direct and inverse approximation theorems, Borel means, Euler means 2010 Mathematical Subject Classification: 42Á10, 42Á24

1. Introduction. Let f be a 2n-periodic continuous function (f E C2n), Tn be the space of trigonometric polynomilas of degree at most n, n E Z+ = {0,1,...}, |f 11^ = maxxe[o,2n] |f (x)|. Let us consider the best approximation En(f= inf{||f — ín||^> : tn E Tn}, n E Z+, and the modulus of continuity u(f,5) = sup0^h^¿ ||f (■ + h) — f Then the

classical Jackson theorem states that

En(f U ^ Cu(f, (n + 1)-1), n E Z+, while the inverse Salem-Stechkin inequality gives

n- 1

u(f, 1/n) ^ Cn-1 ^ Ek(f U n E N = {1, 2,...} k=0

(see [6, Ch. 7]). For a 2n-periodic locally integrable function f, we can consider two variants of Steklov means:

x+h x+h

Sh(f )(x) = h-1 J f (u) du, sh2)(f )(x) = (2h)-1 | f (u) du.

x x—h

© Petrozavodsk State University, 2021

Israfilov, Kokilashvili, and Samko [9] introduced a modulus of smoothness in a weighted Lebesgue space with variable exponent LW() of order r G N

= suP

0<hi<S

n(1 - sh?)(/)

i= 1

where I is the identical operator, and obtained a Jackson-type estimate En(/)Lw ^ Cn*(/, 1/n)Lw , n G N, and the inverse result. For another modulus of smoothness, direct and inverse approximation theorems were obtained by Ky [13]. Many mathematicians, such as Akgiin, Guven, Israfilov, Kokilashvili, Yildirir, studied the approximation by trigonometric polynomials in various weighted spaces. We note only the papers [17], [10] studying the moduli of smoothness defined with help of sh(/) and the papers connected with Lorentz spaces: [12], [20], [21], [1], [2].

2. Definitions. A Lebesgue measurable 2n-periodic function w : [0, 2n] ^ [0, ») is called a weight function if w-1({0}) has measure zero. If w(E) = J w(x) dx for a measurable subset E of [0, 2n], then

E

/w(t) = inf{A ^ 0 : w({x G [0, 2n] : |/(x)| > A}) ^ t}.

Let 1 < p,q < », w be a weight. A measurable function / on [0, 2n] belongs to the weighted Lorentz space LPwq, if

1/q

p,q,w = [j (/**(t))q tq/p-1 dt) q < », / ** (t) = t-1 y /W (u) du.

The classical Lorentz spaces were introduced by Lorentz (see [15]). If p = q, then LWq coincides with the weighted Lebesgue space LW.

A weight w belongs to the Muckenhoupt class Ap(T), 1 < p < », if

|w|Ap = sup ^|11 1 J w(x) dx^ 1 1 J w1 p (x) dx^ <

w Mx) dx < oo,

where p' = p/(p-1) and the supremum is taken with respect to all intervals I C R whose length |11 does not exceed 2n (see [16]).

If w G Ap(T), 1 < p,q < », then the Hardy-Littlewood maximal operator is bounded in LPwq (see [5]). As a consequence, the operators sh

Tp

L/1!

w

(2)

and sjl) are uniformly bounded in Lwq. Now, we define, for r E N, the following modulus of smoothness:

a (M

p,q,w SUp

0<hi<S, i=1,...,r

n(1 - Shi )(f)

i=l

p,q,w

It is clear that ar(f,5)p>q>w is finite for w E Ap (T) and f E LWq. In [12] and [1], the authors consider another modulus of smoothness H*(f, 8)p>q>w, where operators shi in (1) are substituted by s^. By definition, En(f )p,q,w = inft„eTn llf - tn\\p,q,w, n E Z+. For r E N, 1 <p,q< ro, and w E Ap (T) we denote by Wrqw the collection of all absolutely continuous on each period functions f (f E AC2n), such that f',..., f(r-1) E AC2n and f(r) E Lwq.

For a function f E L™ and r E N, we define the Peetre's K-functiomal

by

K(f,t,Lwq, Wpr^w)= inf {llf - g\\p,q,w + t\lg(r)l\p,q,w}.

If 1 < p,q < ro, w E Ap(T), then LPwlq C := L^, w0(x) = 1, (see the proof of Proposition 3.3 in [12]) and f E LPwq has the Fourier series

ac(f )/2 + Y,(ak(f )cos kx + bk(f) sin kx) =: Y Afc(f )(x). k=1 k=0

Let us consider partial sums Sn(f )(x) = Ak(f )(x), the Borel means

<x

Br(f )(x) = e-r Y rkSk(t)/k!, r > 0, k=0

and the Euler means

en(f)(x) = (1+ t)-n Y (nXn-kSk(f)(x), t> 0, n E N. k=0 k

More about these means can be found in the monograph by Hardy [7]. It is well known that for f E L2n, the following limit

f(x) = (2n)-1 ^ hmQ / (f (x - u) - f (x + u)) ctg(u/2) du

n

exists a. e. on R (see [3, Ch. VIII, §7]). The function /(x) is called the conjugate function to f. If f G L2n, then its Fourier series has the form

]C(«fc(f )sin kx - bfc(f) cos kx). fc=i

3. Auxiliary propositions.

Lemma 1. Let 1 < p, q < w, w G Ap(T). Then the conjugation operator is bounded in Lwq and the inequalities

< Cilf ll

llf - Sn(f)||

< (Ci + 1)En(f)

p,q,w> V 7

hold for n G Z+ and f G LW,q.

The statement concerning conjugation operator can be found in [11, Ch. 6, Theorem 6.6.2], while the inequalities (2) can be proved as in [3, Ch. VIII, §20].

Lemma 2 is stated in [20] for arbitrary r > 0 with a reference to the method of Ky [14]. We give another proof for r G N.

Lemma 2. Let 1 < p, q < w, w G Ap(T), tn G Tn, n G N, r G N. Then

llt(r)ll < CVIIt II (3)

holds.

Proof. It is sufficient to prove (3) in the case r = 1. Note that for ¿n(x) = Y1 n=0(ck cos kx + sin kx) we have

n n— 1

tn(x) = - k(Sfc(tn)(x) - Sfc—i(tn)(x)) = ^ Sfc(tn)(x) - ntn(x). fc=1 fc=1

Since the operators are uniformly bounded in Lw,q and 11 tn |p,q,w < < C1 |tn|p,q,w, we obtain

||4llp,q,w < C2(n - 1)||tn||p,q,w + n

< Ci(C2 + 1)n

□

Lemma 3 is proved in [1] also for r > 0. Lemma 3. Let 1 < p, q < w, w G Ap(T), r G N, f G Wpr,q,w. Then

En(f )p,q,w < C (n + 1)—r En(f (r))p,q,w < C (n + 1)—^ ||fW ||p,q,w , n G Z+.

Lemma 4 is proved in [12, Proposition 3.2].

Lemma 4. Let 1 < p, q < w G Ap(T) and <^(x,y) is a measurable 2n-periodic in each variable function. Then

2n

2n

<^(x, ■) dx

^ / ||^(x, -)||p,q,w dx.

p,q,w

Lemma 5. Let 1 < p, q < ro, w G Ap(T), r, k G N, f G Wp,q)W. Then

a+fc(f,5)M,w ^ C5k^(f (fc),£Ww, 5 G [0, 2n],

^ (f,5)p,q,w ^ C ||f (fc)|p,q,w , 5 G [0, 2n]. Proof. It is sufficient to prove the first inequality of the Lemma for k = 1.

Let 0 ^ hi ^ 5, 1 ^ i ^ r + 1, f G WLw and g(x) = ]+[(/ - s^)(f)(x).

i=2

Then we have

r+1

hi t

n(1 - Shi)(f)(x) = -h"1^ J g'(x + s) dsdt.

0 0

¿=1

By Lemma 4 and the uniform boundedness of sh in Lw,q, we obtain

r+1

11(1 - Shi)(f)

¿=1

^ Clh-1 I t

hi t 1

p,q,w

0

t-1y g'(■ + s) ds

0

dt <

p,q,w

hi

^ C2h"1|g'|p,q,W tdt ^ 2"1 C2h1|g'|p,q,w ^ 2-1C2Î|g'|p,q,w. (5)

r+1

It is clear that g' = f! (1 — shi)(f ') a. e. on R. Taking the supremum

¿=2

in the left-hand side of (5) with respect to h G [0,5], 1 ^ i ^ r + 1,

we find that ^r+i(f,5)p>q>w ^ C35nr(f/,5)P)q,w. If we use the equality

h t

(sh — I)(f )(x) = h-1 / / f /(x + s) dsdt instead of (4) similarly to (5), we

00

obtain the second inequality of Lemma 5 in the case k = 1. The general case easily follows from this one. □

Lemma 6 is proved in [19].

Lemma 6. Let l E N, q > 0. Then there exists C = C(l, q) independent of n, such that

n /n\ qn-k (q + l)n

£ (n) (kW *c(THF' n E N

Lemma 7 was established by Iofina [8].

n

Lemma 7. Let Yn(t) = e-t E tk/k!, t > l. Then ^(t) ^ C > 0, where

k=0

[t] is the integer part of t.

4. Direct and inverse approximation theorems. As usually, A(t) x B(t), t E T, means that there exist Ci,C2 > 0, such that CiA(t) * B(t) * C2A(t), t E T.

Theorem 1. Let l <p,q< ro, w E Ap(T), r E N, f E Wprqw. Then

Qr(f,t)p,q,w x K(f,tr,Lwq, Wprq,w), t E [0, 2n].

Proof. By the uniform boundedness of the Steklov operators sh in Lwq and Lemma 5 for g E 2n, we have

^r (f ,t)p,q,w * ^r (f g,t)p,q,w + ^r (g,t)p,q,w *

* Ci||f - glUw + C2tr||g(r)|p>q,w. (6) Taking the infimum in the right-hand side of (6) over g E Wr , we obtain

(f,t)p,q,w * max(Ci,C2)K(f,tr,LwWw). For the converse inequality, we use the operator

z t

©z(f )(x) = Z2 J f f (x + s) dsdt.

00

In [18], it is proved that

2k

(©k (f ))(k)(x) = 3 [Sz - I ]k (f )(x).

Similar to the proof of Lemma 5, by Lemma 4 we have

z t

2

||Bz(/)IUw ^ ~2 I tilt-1 / f (x + s) ds||M;w dt ^

z2

00

z

< C1|f ||p,q,w Z2 J tdt = C1||f |p,q,w . (8) 0

For the operators AZr] = 1 — (1 — 6Z)r and Uj = by (7) we find

that

r1

j=0 j

r-1 / \ r-1 / \ or

M____, „sss^h X ^ / r\ 2'

£ . ||(©z(Uj(f)))(r)|Uw= £ . -||(1 -Sz)'[Uj(/)]|p>q,w.

x 1 / — V 1 / z

j=0 ^ j=0 VJ/

Since sh and 6Z commute, by (8) we obtain

r1

||(AZr](f ))(r)|Uw ^ 2r £ (jV- |Uj [(I - Sz )' (f )]||M>« ^

j=0 Vj/

j=0

^ C2z-r||(1 - Sz)r(f^ C2z-rQr(f,z)p>i>w. (9)

Using (8) and Lemma 4, we have for g G L^:

||(1 - ©)(g)|p>q,w ^ C3 sup |(1 - Sz)(g)|Uw. (10)

0<t<z

(see similar arguments in [18, (4.6)]). Using the equality 1—AZ] = (1—6r) and applying (10) r times, we obtain

||f - 4r](f )|| ^ C3Qr (f,z) (11)

From (9) and (11) we deduce

||f - Azr](f )|p,q,w + zr ||(Azr](f ))(r) | |p,q,w ^ C4Qr (f,Z )p,q,w ,

where Ar](f) G . Thus, K(f, zr, , Wp^) ^ C4Qr(f,z)p(0, and

the proof of Theorem 1 is complete. □

Now we compare our modulus of smoothness with Q*(f,5)p,q,w used in [12] and [1].

Corollary 1. Let l <p,q < ro, w E Ap(T), r E N. Then

^2r (f,S)p,q,w x n*r(f,S)Piq,w, S E [0, 2n]. (12)

Proof. In [1] it is proved that, under conditions of the Corollary,

nr(f,S)p,q;w x K(f,Sr,Lwq, Wpi^w), s E [0, 2n].

Combining this result with Theorem 1, we obtain (12). □ Theorem 2. Let l < p,q < ro, w E AP(T), r E N. Then

En (f )p,q,w * C Q (f, (n + l)-1)p,q,w , n E Z+.

Proof. For n E Z+ we choose a function g E WpqJ,w, such that

If - gll)p,q,w + (n + l)-r||g(r) llp,q,w * 2K(f, (n + l)-r, LwqWpr^w). By Lemma 3 and Theorem 1, we have, for n E Z+:

En(f %)p,q,w * En(f g%)p,q,w + En (g)p,q,w *

* Ci (llf - gllpqw + lg(r)lp,q;w(n +l)-r) * * 2CiK(f, (n + l)-r, Iff, WT^w) * C2^r(f, (n + l)-1)p,q;w.

□

Theorem 3. Let 1 <p,q < <x, w G Ap(T), r G N. Then

n

Mr(f,n-1)pq,w ^ Cn-r Y kr-lEk-l(f )pq,w, n G N. (13)

k=1

Proof. Let tk G Tk be the polynomial of the best approximation for f G Lwq, k G Z+. Using Lemma 5 and Lemma 2, we obtain

(f,n 1)p,q,w ^ (f — t2m ,n 1)p,q,w + (t2m ,n 1)p,q,w ^ ^ C1(\\f — t2m \\p,q,w + n r ||t2") \\p,q,w) ^

^ C2

/ m— 1

E (f) + n—r[\|t(r) - t(r)ll + Y^ llt(r) -t(r)ll

E2m (f )p,q,w + n I WH b0 \\p,q,w + / j Hb2i+1 b2i \p,q,w ^ i=0

^ C3

m— 1

^ C

E2m (/) + n-r ||ti - to||

+ ||t2 i+i — ^ ^ i=0

m- 1

+ n r I Eo(/)

+ (/) ^ i=0

2m

^ C5 £ kr-1 Efc-i(/)p,q,w, n G N.

|p,q,w

p,q,w

k=i

If n G N is fixed and 2m ^ n < 2m+1, m G Z+, then (13) easily follows from the last inequality. □

If u is increasing and continuous on [0;2n], u(0) = 0, then u G $. A function u G $ belongs to the Bary-Stechkin class Ba, a > 0, if ka-1u(k-1) = O(nau(n-1)), n G N (see [4]).

Corollary 1. Let 1 < p, q < œ, w G Ap(T), r G N and u G Br. Then, the conditions En(/)p,q,w = O(u((n + l)-1)), n G Z+, and (/, &)p,q,w = O(u(i)), # G [0, 2n], are equivalent.

Remark 1. The converse inequality from Theorem 3 has the same form as the classical converse approximation theorem (see [6, Ch. 7, Theorem 3.1]), while (see Corollary 1)

' V- k2r-1Ek-1(/)p,q,w, n G N.

H*(/, n ) ^ Cn-

k=1

5. Approximation by the Borel and Euler means.

Theorem 4. Let 1 < p, q < w, w G Ap(T), r G N, t > 0 and f G • Then

If - en(f )IU,w ^ C^(f, n-1)p,q;w, n G N, (14)

en(/)lp,q,w ^ Cn-^ kr-1Ek-1(/)p,q,w, n G N

:15)

k=1

Proof. By the definition of the Euler means, Lemma 1, and Theorem 2, we obtain

— en(/)||p,q,w = (1+ t)-

è (nV-i (/—Sj (/))

j=o j

<

p,q,w

* (ir^£ jtn-JE(f)p,q,w *

* £ q(f, j + l)-i)p,q;w.

Due to Theorem 1, we have the property

Qr (f, AS)p,q,w * C (A + l)r Qr (f, S)p,q,w , A> 0. By this property and Lemma 6, we find that

llf - en(f)lp,q,w * ^^ £ jtn-j(^ + ^rQr(f, ^Ww *

* UTtjn(n+l)rQr^nrr^£ (n) jrly * CQr(f,^+l)p,q,w

and (14) is proved. The inequality (15) follows from (14) and Theorem 3. □

Theorem 5. Let l <p,q < ro, w E Ap(T), r E N, t ^ l and f E Lpwq. Then

M tk

Bt(f )\\p,q,w ^ C Ykl e—tEk (f )p,q,w , k=0 '

where [t] is the integer part of t.

Proof. Let Tn G Tn be such that ||f - Tn||p,q,w = En(f)p,q,w. Then, by Lemma 1,

||f - Bt(f)|p,q,w ^ ||ßt(Tn) - Bt(f)||p,q,w + ||Bt(Tn) - TnHP,q,w + ^ tk

+ ||тn - f b,q,w ^ Y k! ||Sk(f - ^L^w + ||Bt(Tn) - ^¡p^w + k=0 '

+ ||Tn - f !p,q,w ^ C1En(f) + ||Bt(Tn) -

Now,

\\Bt(Tn) - T

n) 'n\\p,q,w

n—1 tk e—t

z2~kT(Sk (Tn) - Tn) k=0 k'

p,q,w

n-1tfce-t S C2 k! (rra)P,q,w•

fc=o !

By the definition of Tn,

Efc(Tra)p,q,w S Efc(f Tra)p,q,w + Efc(f )p,q,w S 2Efc(f )p,q,w•

Using the previous inequaities and Lemma 7 and taking n = [t], we obtain

tfce-t

Bt(f )|p,q,w S C1E[t](f )p,q,w + 2C2 £ —kpEfc(f )p,q,w S

fc=o !

A tke-t

S C3 £ —kpEfc(f )p,q,w• fc=o !

□

Let us show that the estimate of Theorem 5 gives a clear result on some subclasses of L^f.

Corollary 1. Let 1 < p, q < w, w G Ap(T), r G N, t ^ 1 and f G Lwq• If En(f )p,q,w = O((n + 1)-a), n G Z+, or n(f,5)p,,,w = O(5a), 5 G [0, 2n], then

Ilf - Bt(f)|p,q,w S Ct-", t ^ 1.

Proof. Under conditions of Corollary 1, we have, by Theorem 5 and Theorem 2:

JtL tfce-t JtL tfce-t / t +1' "

If - w)»p.,.w S Ci E = ^+r"E ^ kTT

fc=0 V ' fc=0 v

For m = [a] + 1 and 0 S k S [t], we have

t + 1 \ a /1 + 1N m < 2mtm

k + 1/ \k + V (k + 1)m and

[t] j.fc+m

t

- Bt(f) IUw S C2(t + 1)-°e-^

--t \ ~ s

m

< C2 ^ tk+m (k + 1)... (k + m) <

< (t + 1)aet ^ (k + m)! (k + 1)m <

k=0

< C2m! ^ tk+m < C3

< (t + 1)aet (k + m)! < (t + 1)a.

k=0

□

Acknowledgment. Supported by the Ministry of science and education of the Russian Federation in the framework of the basic part of the scientific research state task, project FSRR-2020-0006

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Received August 17, 2020. In revised form,, November 30, 2020. Accepted December 09, 2020. Published online December 22, 2020.

Saratov State University

83 Astrakhanskaya St., Saratov 410012, Russia

E-mail: VolosivetsSS@mail.ru