Научная статья на тему 'MAXIMAL CONVERGENCE OF FABER SERIES IN WEIGHTED REARRANGEMENT INVARIANT SMIRNOV CLASSES'

MAXIMAL CONVERGENCE OF FABER SERIES IN WEIGHTED REARRANGEMENT INVARIANT SMIRNOV CLASSES Текст научной статьи по специальности «Математика»

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MAXIMAL CONVERGENCE / BANACH FUNCTION SPACE / FABER SERIES / WEIGHTED REARRANGEMENT INVARIANT SPACE

Аннотация научной статьи по математике, автор научной работы — Testici Ahmet

Let be a bounded set on the complex plane C with a connected complement 𝐾- := C∖𝐾. Let D := {𝑤 ∈ C : |𝑤| < 1} and D- := C∖D. By we denote the conformal mapping of 𝐾-onto {𝑤 ∈ C : |𝑤| > 1} normalized by the conditions (∞) = ∞ and lim𝑧→∞ (𝑧) /𝑧 > 0. Let := {𝑧 ∈ 𝐾- : |𝜙 (𝑧)| = > 1} and := Int 􀀀𝑅. Let also (𝑧), = 0, 1, 2, . . . be the Faber polynomials for constructed via conformal mapping 𝜙. As it is well known, if is an analytic function in 𝐺𝑅, then the representation (𝑧) = ∞Σ 𝑘=0 (𝑓) (𝑧), ∈ holds. The partial sums of Faber series play an important role in constructing approximations in complex plane and investigating properties of Faber series is one of the essential issue. In this work the maximal convergence of the partial sums of the partial sums of the Faber series of in weighted rearrangement invariant Smirnov class (𝐺𝑅, 𝜔) of analytic functions in is studied. Here the weight satisfies the Muckenhoupt condition on 􀀀𝑅. The estimates are given in the uniform norm on 𝐾. The right sides of obtained inequalities involve the powers of the parameter and (𝑓,𝐺)𝑋.𝜔 called the best approximation number of in (𝐺𝑅, 𝜔), defined as (𝑓,𝐺)𝑋.𝜔 := inf { ‖𝑓 - 𝑃𝑛‖𝑋(Γ,𝜔) : ∈ } . Here is the class of algebraic polynomials of degree not exceeding 𝑛. These results given in this paper is a kind of generalisation of similar results obtained in the classical Smirnov classes.

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Текст научной работы на тему «MAXIMAL CONVERGENCE OF FABER SERIES IN WEIGHTED REARRANGEMENT INVARIANT SMIRNOV CLASSES»

ISSN 2074-1871 Уфимский математический журнал. Том 14. № 3 (2022). С. 121-130.

MAXIMAL CONVERGENCE OF FABER SERIES IN WEIGHTED REARRANGEMENT INVARIANT SMIRNOV CLASSES

A. TESTICI

Abstract. Let K be a bounded set on the complex plane C with a connected complement K- := C\K. Let D := {w e C : H < 1} and D- := C\D. By <p we denote the conformal mapping of K-onto {w e C : |w| > 1} normalized by the conditions <p (to) = to and lim^^ tp (z) /z > 0 Let r^ := {z e K- : (z)| = R> 1} and Gr := Intr^. Let also (z), k = 0,1, 2,... be the Faber polynomials for K constructed via conformal mapping As it is well known, if f is an analytic function in Gr, then the representation

f (z) = S ak (f) ^fc (z), z e Gr holds. The partial sums of Faber series play an impor-k=0

tant role in constructing approximations in complex plane and investigating properties of Faber series is one of the essential issue. In this work the maximal convergence of the partial sums of the partial sums of the Faber series of f in weighted rearrangement invariant Smirnov class Ex (Gr, w) of analytic functions in Gr is studied. Here the weight w satisfies the Muckenhoupt condition on Tr. The estimates are given in the uniform norm on K. The right sides of obtained inequalities involve the powers of the parameter R and En (f,G)x w called the best approximation number of / in Ex (Gr,w), defined as

En (f, G)x w := in^ ||/ — Pn\\x(rw) : Pn e n^^. Here nn is the class of algebraic polynomials of degree not exceeding n. These results given in this paper is a kind of generalisation of similar results obtained in the classical Smirnov classes.

Keywords: Maximal convergence, Banach function space, Faber series, weighted rearrangement invariant space.

Mathematics Subject Classification:30E10, 41A10, 41A30

1. Introduction

Banach function spaces include many important particular cases including Lebesgue and Orlicz spaces (see, [1, 2]). Earlier some theorems of approximation theory in the rearrangement of invariant Banach function spaces and Smirnov classes were proved in [3, 5, 6, 7, 8, 9]. The partial sums of Faber series are used in constructing approximation aggregates on complex plane generally. Faber series are used for solving many problems in mechanical science, such as the problems on the stress analysis on the piezoelectric plane in [10, 11]. As described below we investigate the maximal convergence property of the Faber series in the rearrangement invariant Smirnov classes. Some classical results of the series of Faber polynomials and their applications were considered comprehensively in [12] and [13]. Moreover, the distribution of zeros of Faber polynomials were investigated in [14]. The Faber series is defined as follows.

Let K be a bounded continuum with the connected complement K- := C\K. Let D := {w e C : |u>| < 1} T := SO and D- := C\0, Let also be conformal mappings of K-

A. Testici, Maximal convergence of Faber series in weighted rearrangement invariant Smirnov classes.

© A. Testici 2022.

Submitted January 25, 2021.

onto D normalized by the conditions

p (to) = to, lim p (z) ¡z > 0.

and let be the inverse mappings of <p. We set

rR := {z E K- : (z)l = fl^d GR :=IntrR ,R > 1.

As it is known from [12], if a function is analytic on continuum K, then it has the Faber series expansion

f (z) = ^ ak (f)$fc (z), z E K, (1.1)

k=0

which converges absolutely and uniformly on K. Here $fc (z), k = 0,1, 2,..., are Faber polynomials for K, which can be defined by the series representations

^(t) r (z)

^ (f) -z tk+l

z E K, \t\ > 1,

where the Faber coefficients (f ), k = 0,1, 2,..., are defined as

kyj J 2ni J tk+l In view of (1,1) we use the notation

T

Rn (f, z) := f (z) - ^ ak (f) $fc (z) = ^ ak (f) $fc (z), z E K. (1.2)

k=0 k=n+\

The maximal convergence theorem estimates the rate of the convergence of Rn (f, z) to zero in uniform norm on K in terms of parameter R and the best approximation number of analytic function f belonging to a given space. The results on maximal convergence properties of orthogonal polynomials can be found in [13]. The maximal convergence properties of the Faber series in the Smirnov-Orlicz classes were investigated in [16], Later these results were extended to Smirnov classes with variable exponent by Israfilov et al, in [17, 18], In this work we investigate the maximal convergence properties of the Faber series in the weighted rearrangement invariant Smirnov classes of analytic functions.

This work is organized as four sections. Necessary definitions and notations are given in second section. In the third section, some auxiliary results proved previously are formulated according to our notation used in this work. Finally, we state and prove the main results in the last section.

2. Preliminaries

Let M be the set of all measurable complex-valued functions on reetifiable Jordan curve r with respect to Lebesgue length measure |dr | and let m+ be the subset of functions from m whose values lie in [0, to]. The characteristic function of a Lebesgue measurable set E c r is denoted by xe-

A mapping p : m+ ^ [0, to] is called a function norm if it satisfies the following properties for all measurable functions f,g,fn (n = 1, 2,...), for all const ants a ^ 0 and for all measurable sets E c r :

1. p (f ) = 0 ^ f = 0 a.e., p (af) = ap (f), p (f + g) ^ p (f) + p (g) ,

2. If 0 ^ g ^ f a.e., then p (g) ^ p (f) ,

3. If 0 ^ fn S f a.e., then p (fn) / p (f),

4. If £ h^ a finite Lebesgue measure, then p (xe) < to,

5, If E has a finite Lebesgue measure, then

i f (r) Idr| ^ CEp (f) Je

where Ce is a positive constant depending on E and p does not depend on f.

Let p be a function norm. The set

X (r) = {f e M : p (|/1) < to}

is called a Banach function space generated bv p and X (r) becomes a Banach space equip with the norm \\f := p (|/|).

If p is a function norm, then associate norm of p is defined as

p' := sup | J f (t) g (t) |dr| : f eM+,p (f) ^ 1

for g e M+ and p' is also itself a function norm. The Banach function space determined by p' is called the associate space of X (r) and the associate space of X (r) is denoted bv X' (r) in [1]-

If f e X (r^d g e X' (r), then as it is known from Theorem 2,4 in [1], the Holder inequality

J If (r) g (T )||dr | ^ ||/ |X(r) MxT) r

holds, where

x (r) := sup if If (t ) g (r )| |dr | : g e X' (r) , -(r) ^ 1

||y|x/(r) := sup { j If (t) g (r)| |dr| : f e X (r) , ||/(^ ^ 1

Let M0 and M+ ^e classes of a.e. ^^^^e functions in M and M+ , respectively. The distribution function of f defined as

pf (A) := mes {z e r: If (z)| > A} for A ^ 0. The pair of functions f,g e M0 is called equimeasurable if pf (A) = pg (A) for all A ^ 0.

Definition 2.1. [1] If p (f) = p (g) for every pair of equimeasurable functions f,g e M+ then the function norm p is called a rearrangement invariant function norm and the Banach function space generated by p is called a rearrangement invariant spaces.

The function f * (a) := inf {A : pf (A) ^ a} , a ^ 0, is called the decreasing rearrangement of the function f e M0.

Let |r| be the Lebesgue measure of r. We use the notation ([0, |r|] ,m) to indicate Lebesgue measure spaces over the interval [0, |r|]_ By Luxemburg representation theorem [1] we obtain that there is a (not necessarily unique) rearrangement invariant function norm p over ([0, |r|] ,m) such that p (f) = ~p (f *) for f e M+. The rearrangement invariant space over ([0, |r|] ,m) generated by p is called Luxemburg representation of X (r) and it is denoted by X. We define the operator Hx on ([0, | r | ] ,m) for each x > 0 as

(f (xt), xt e [0, |r|]

{H'S)(t)-{ 0. / [0. n- 1 e [0-|r|]' ' eM0.

By [1] the operator H1/x is bounded on X with the operator norm

hx (x) := ||#i/ik||b(x)

where B (X) is the Banach algebra of bounded linear operators on X, The limits defined as

log hx (x) log hx (x) ax := lim —--, px := lim —--

log x x^x log x

are called lower and upper Boyd indices of X (r), respectively [1], The Bovd indices satisfy 0 ^ ax ^ Px ^ 1- The Boyd indices are said to be nontrivial if they satisfy 0 < ax ^ Px < 1-A function u : r ^ [0, to] is called a weight if u is measurable and the preimage u-1 ({0, to}) has the zero measure. The weighted rearrangement invariant space is defined as

X (r,u) = {f EM : fu E X (r)}

which is equipped with the norm ||/||x:= \\fw\\X(r) where X (r) is rearrangement invariant spaces.

Definition 2.2. [20] Let 1 < p < to and 1/p +1/q = 1. Let u be weight function on r such that u E Lploc (r)and u E Lqloc (T). We say that u satisfies the Muckenhoupt eondition on r if

sup sup

ter £>q

/ \1/v ( \ 1/q 1 J u (t)p \dr| 1 J u (t)-q \dr| V r(i>e) / V r(i>e)

< TO

where r (t, e) := (t g r : \t — t\ < e] and £ > 0.

Let us we denote by Ap (r) the set of all weight functions satisfying Muckenhoupt condition on r.

Let G c C be a Jordan domain bounded by reetifiable curve r. We denote by ¿1(r), 1 ^ p < to, the set off all measurable complex-valued functions / defined on r such that \f \ is Lebesgue integrable with respect to arc length on r. If there exists a sequence (Gvc G of domains G^, the boundary of which is a rectifiable Jordan curve (rsuch that the domain Gv contains each compact subset G* of G for v ^ uq for some uq g N and

lim sup / \f (z)\ \dz\ < to,

V^-X J

r

then we say that / belongs to the Smirnov class E 1(G). Each function f G E1 (G) has the nontangential boundary value almost everywhere (a.e.) on r and the boundary function belongs toL1(r)[15],

Definition 2.3. Let u be weight function on r. The class of analytic functions

Ex (G, u¡) := {/ g E1 (G): f G X (r,w)} is called a rearrangement invariant Smirnov class.

The best approximation number of / in Ex (G,u) is defined as

En (f, G)Xm := inf {|| / — Pn\\x(r,W) : Pn G n„} where nn is the class of algebraic polynomials of degree not exceeding n.

3. Auxiliary results

The direct theorems of approximation theory in the weighted rearrangement invariant Smirnov class were proved in [4]. For the sake of clarity, in this section we formulate these results and some notations in our terms.

We recall that rR := {z G K- : [p (z)| = fl^d GR := Int R> 1, where K is a bounded continuum with connected complement K- = C \ K. If u G X(rB) and w-1 G X'(rR) for R > 1, then bv Holder inequality we have

L™ (rR) C X(rR,u) C L1 (rR). Since rR with R > 1 is a analytic curve, by [19] there are positive constants such that

0 <ci ^

0 <cs ^

«) ^ C2 < to, K| = R, <p (z) ^ c4 < to, z g rß. (3,1)

Let us we define the functions f0 (w) := f o ^ (Rw) and (w) := u o ^ (Rw) for w G T,

If f G X(rR,u) for R > 1, then by (3.1) we have f0 G X(T, w0). The Cauchy type integral for a given f0 G L1 (T) is defined as

/o+ H := 2-. i ^dr, w G D, 2m J t — w

T

D

Lemma 3.1. [4, Lm. 1] If Boyd indices ax and fix are nontrivial and

^o G Ai/ax (T) n Ai/px (T), then f + G Ex (D,u0) for each f G X(T,w0).

Given f G X(T,^0), we define the mean value operator as

h

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1

ah (f ) (w) := — J f (welt) dt, 0 < h < n andw G T. -h

If the Bovd indices a^d fix are nontrivial and G A1/ax (T) n A1/^x (T), then for each f G X (T, w0) the inequality

K (f (T>W0) ^ C HZHx(T>W0) follows from Lemma 2,2 proved in [3],

If R > 1, then by (3,1) we have the conditions u G A1/ax (rR) n A1/px (rR) and G A1/ax (T) n A1/px (T) are equivalent. Lemma 3,1 implies that the nontangential boundary value of /+ belongs to X (T, w0). Consequently, we can give the following definition.

Let X be a Banach space and X* be its dual space. We define the dual of X* by setting X** := (X*)*. Let J (X) be the image of X in the canonical mapping J : X ^ X**, A Banach space X is said to be reflexive if J (X) = X** (see, [2, p. 21]), If the rearrangement invariant space X is reflexive, then X is called reflexive rearrangement invariant space.

Definition 3.1. Let Boyd indices ax and fix are nontrivial and u G A1/ax (Fr) n A1/px (Fr) with R > 1. Let X (T) be a reflexive rearrangement invariant space. The function

(f, ^ := suP

i=1,2,...,r 0<hi^S

n (J - ^ ) (/o+)

i=1

, v = 1, 2,..., andS > 0,

X (T,wo)

is called v a modulus of smoothness of f G Ex (Gr,u).

Since X (T) is a reflexive rearrangement invariant space, the Boyd indices ax and fix of which are nontrivial, where u0 e A1/ax (T) nA1/^x (T) the set of continuous functions is dense in X (T,u0), see [3, Lm. 2], Hence, it is guaranteed that

lim (f, S) = 0.

Lemma 3.2. [4, Cor, 1] Let X (T) be a reflexive rearrangement invariant space. Let Boyd indices ax and ßx be nontrivial and u E Al/ax (r#) nAl/ßx (r#) with R> 1. If f E Ex (Gr,u),

Wf-Pn (•, f) ||*(raiW) ^ ^^ (A n+y) , * = 1 2,...,

holds for each, n = 1, 2,..., where Pn (•, f) is the nth partial sum of the Faber series of f. It is known that [12]

Ek (0)=2^ / rkF (r> Odr, K| ^ r > 1 (3.2)

|r l=r

and Lebedev's results

i J \F (^ 0\\dr\ ^ ^ ln ^, id > 1 (3.3)

|r l=r

where

F (r, <):= . ( V'(T). (n - \r\ >1, \C\ >1.

4. Main results

Our main results are as follows.

Theorem 4.1. Let the Boyd indices ax and fix be nontrivial and

u e Ai/ax (rR) n Ai/px (rR) with R> 1. If f e Ex (Gr ,u), then

\Rn (^ z)\ ^ Rn+1 (R - 1) E (^ Gr)x.uj ^ Z e K, holds for each, n = 1, 2,..., where c is a positive constant independent of n. Combining Theorem 4.1 and Lemma 3.2, we have the following corollary. X (T)

axand fix be nontrivial and u e A1/ax (Fr) n A1/px (Fr) with R > 1. If f e Ex (Gr, u), then

R (A *)\ ^ Rra+1 R - 1) ^^ z e K,

n = 1, 2, . . . , = 1, 2, . . . , n

Proof of Theorem 4-1 ■ Let z e r and 1 < r < R. Let the Bovd indices ax and fix be nontrivial and u e A1/ax (rR) n A1/px (r^). Let Pn be the best approximating polynomial of degree at most n to f e Ex (Gr,u). Since f is analytic function on Gr, then we have the Faber coefficients

ltl=R

Taking into account Pn G nn and applying the Cauchy integral formula for derivatives, we have

2S / P" «'(Î))

^ $k (z) ¿fc+i

,k=n+1

dt = 0.

(4.1)

Using (1.2) and (4.1) respectively we obtain (see, [12]

1

Rn (f, Z)=2^i (f))

^ $k (z) ¿k+1

,k=n+1

dt

and

E

k= n+1

(z)

tk+1

dt.

(4.2)

The $k (z) ,k = 1, 2,..., are the polynomial part of Laurent series expansion of [p (z)] such that

$k (z) = [ ip (z)]k + Ek (z) , zgK -, where Ek is an analytic function on KTherefore we get

L

$k (z)

^ ^ (z)]k ^ Ek (z)

Z^ + k+1 + Z^

k+1 k+ k=n+1 k=n+1

k=n+1

tk+1

and by (4.2) we obtain

k (/, z)i ^ -j if & (t)) -Pn & m

\t\=R

E

k=n+1

k+1

Denoting

and

we get

+ ^ J if ^ (f)) -Pn № (*))i

\t\=R

h J if № (t)) -Pn & (t))|

\t\=R

E Ek w (0)

k= n+1

k+1

E

k= n+1

tk+1

h J if & (t)) -Pn & (t))|

\t\=R

£ Ek w (0)

k= n+1

k+1

idt i

iRn (f, z)l ^ h + h. (4.3)

Since z G rr and $ G r R for 1 < r < fl then (z)| = r and [p (c)| = R. Thus I R — r| ^ [p (q) — p (z)| implies that

1

1

< -. (4.4)

(c) — <p (z)I " R — r { }

We know that 1/u G X' (r) bv Theorem 2.1 in [21]. Hence, bv (3.1), Holder inequality and (4.4) we have

h =1 j if № (t)) -Pn № (t))i

\t\=R

E

k=n+1

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tk+1

idt i

k

1

k

1

if \ /(0 -Pn (0\

rr

E

k=n+1 [V 1

fc+1

\v' m \ \ <k\

« /(0 -P" Wu (0 u w

E

fc=ra+1

[ V (z)f

[ V 00]

fc+1

\ <k \

^— ||(f -pn)u|x(r)

V [v (*)]*

u 00 fc=b, [v (^)]fc+1

X'(r)

2^ h/ - P«|x(r,

2^ Pra|x(r,

^ 2^Era (^ Gr)x-u

W)

[ v( )]

ra+1

u (0 [v (0]ra+1 (V (0 -V (*))

X' (r)

u ( )

\ v( )\

ra+1

\V OOr1 \ V (0 -V (*)\

X' (r)

X'(r)

1 ra+1

u (<0 Rra+1 (R - r)

..ra+1

= —Era (f, GR)X,UJ Rn+1 (R - r) H 1/u (<0 ^'(P)

~ra+1

^ 2^En (^ Gr)x» Rra+1 (R - r).

Therefore,

~ra+1

Rra+1 (R - r)'

(4.5)

On the other hand by (3.2) and applying Fubini's theorem we have

h — J \f (V (t)) -Pra (V (t))\

ltl=R

= ^ j \f (V (t)) -Pra (V (t))\

ltl=R

E Efc (V «)) ¿fc+r

fc=ra+1

E 1 ! ^F (r, 0 \dr\

fc=ra+1

= ^ / \/(V W) -Pra (V «))\j ^ /

2i j tfc+1 | |=

a

E

^2lJ \f(v(*))-Pra(Vw)'1iI

ltl=R I |r|=r

a fc tfc

¡fcT1

fc=ra+1

rra+1

ira+1 (i - r)

\F (r, C)\\dr\

\F (r, C)\\dr^\di|

< ¿A\ra+1\ f e. o\) ¿/ ;ra+-*; (t))\

|r|=r I |t|=.R

1

1

\r\=r I \t\ = Ä

idri .

Changing the variables and by (3,1), Holder inequality, (3,3), (4,4), we get

h C

2^ / |f• j b / ^« - P" ^ dS-fW№i >^

\r\=r I \t\=Ä

~n+1

Cc

i tf (0 i

|f(r,OU U^^-PnM ^ (Ç) -,(z)i

\r\=r I \t\=Ä

idçi } idri

n+1

c^^nrr j i f(^0 iii(/-Pnmi*(p)

\ \=

1

i tf (0 i

C

n+1

— Pnl

n+1

11

4^2Än+1 (R - r)

„n+1

En ( f, Gr)

u (ç) R - r 1

i i F ( r, 0 ii dr i

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i dr i

X'(r)

X '(r)

\ \=

X.ÜJ

i F (r, 0 i i dr i

X '(r)

\ \=

C

■4n2Rn+1 (R - r)

„n+1

En (f,GR)XM iF (r, 0i Mri

\ \=

C

: 2ttRn+1 ( ft - r)

En ( f, GR)X,

X.Üj \ 4

ln-

4 1 2 1

Thus we get

2C

n+1

En ( f, Gr)

2tt Rn+1 ( fl — r) and combining (4,3), (4,5) and (4,6) we have

X.u^ r4

ln

4 1 2 1

n+1

i Rn ( f, z) i C 2^En ( I,Gr)x.u Rn+1 (ft - r)

+

n+1

;En ( f, Gr)

ln

2kRn+1 ( fl — r ) ~r4 — 1 r2 — 1 • Finally, setting z g K and r := 1 + n, we obtain the desired inequality

i Rn ( f, Z) i C

Rn+1 (R - 1)

En (f, GR)XujVn\nn.

Acknowledgements

(4.6)

The author would like to thank the referees for their valuable comments and suggestions which helped to improve the manuscript.

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Ahmet Testici,

Balikesir University,

Department of Mathematics,

10145, Balikesir, Turkey

E-mail: testiciahmet@hotmail.com

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