# Approximation by Rational Functions in Variable Exponent Morrey–Smirnov Classes

Authors: Ahmed Kinj
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this work, the direct theorem of approximation theory in variable exponent Morrey–Smirnov classes of analytic functions, defined on a doubly connected domain of the complex plane bounded by two sufficiently smooth curves, is investigated.


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## Body

## 1. Introduction

The classical Morrey spaces were introduced by Morrey in [1] in order to investigate the local behavior of the solutions of elliptic differential equations. Recently, many researchers have investigated function spaces with variable exponents due to their use in several fields of applied mathematics. In particular, function spaces with variable exponents have many applications in areas involving the modeling of electrorheological fluids [2] and image restoration [3]. The variable exponent Morrey spaces introduced in [4] have been studied intensively by various authors (see, for example, [5–7]). The fundamental problem in approximation theory is to express complicated functions by simple functions such as polynomials, wavelets, or rational functions with more useful structures. The theory of approximation is strongly related with the operators and has a considerable number of applications in areas including general marginal distributions such as sampling and machine learning (see [8–10]). Also, the approximation problems in the variable Morrey–Smirnov classes of analytic functions defined on a simply connected domain with a Dini-smooth boundary are proved in [11]. The direct and converse theorems of approximation theory in the classical Morrey–Smirnov classes defined on a simply connected domain with a Dini-smooth boundary were obtained in [12, 13]. Similar results in the variable exponent Smirnov classes were studied in [14, 15].On a doubly connected domain, the rate of approximation byp−Faber–Laurent rational function in Smirnov classes was studied in [16]. Also, the rate of approximation by Faber–Laurent rational function in Smirnov–Orlicz classes and Smirnov classes with variable exponent was obtained in [17]. The approximation property of p−ϵ−Faber–Laurent rational functions in the weighted generalized grand Smirnov classes on doubly connected domains is proved in [18].In the current paper, approximation one direct theorem of approximation theory in variable exponent Morrey–Smirnov classes of analytic functions, defined on a doubly connected domain bounded by two Dini-smooth curves, is obtained.


## 2. Preliminaries

LetJ denote the interval 0,2π or a Jordan rectifiable curve Γ, and let ℘ denote the class of all Lebesgue measurable functions p.:Γ⟶1,∞ such that(1)1<p−=essinfz∈Jpz≤p+=esssupz∈Jpz<∞.We denote byJ to the Lebesgue measure of J. We say p.∈℘log if there is a constant c such that(2)pz1−pz2lnJz1−z2≤c,for all z1,z2∈J.Forp.∈℘logΓ, we define Lp.Γ the set of all measurable functions f such that(3)∫Γfzpzdz<∞.Lp.Γ is a Banach space with respect to the norm(4)fLp.Γ=infλ>0,∫Γfzλpzdz≤1.LetU be a finite simply connected domain with a rectifiable Jordan curve boundary Γ. Denote U−=extΓ, γ0=w∈ℂ:w=1, D=intγ0, and D−=extγ0. Let Γr be the image of circle w∈ℂ:w=r,0<r<1 under some conformal mapping of D onto U.For given1≤p<∞, we denote by EpU the class of analytic functions f in U which satisfies(5)∫Γrftpdt<∞,uniformly in r.It is known that every function of classEpU has nontangential boundary values almost everywhere on Γ and the boundary function belongs to LpΓ ([19], pp. 438–453).Also, suppose thatϕ∗ is the conformal mapping of U− onto D− normalized by(6)ϕ∗∞=∞,limz⟶∞ϕ∗zz>0,and let ψ∗ be the inverse of ϕ∗. Let ϕ1∗ be the conformal mapping of U on to D−, normalized by(7)ϕ1∗0=∞,limz⟶0zϕ1∗z>0.The inverse mapping ofϕ1∗ will be denoted by ψ1∗.The functionsψ∗ and ψ1∗ have in some deleted neighborhood of ∞ representations(8)ψ∗w=αw+α0+α1w+α2w2+⋯+αkwk+⋯,α>0,ψ1∗w=β1w+β2w2+⋯+βkwk+⋯,β1>0.The functions(9)ψ∗′wψ∗w−z,z∈U,ψ1∗′wψ1∗w−z,z∈U−,are analytic in the domain D−, and the following expansions hold [20–23]:(10)ψ∗′wψ∗w−z=∑k=0∞Fkzwk+1,z∈U,w∈D−,ψ1∗′wψ1∗w−z=∑k=1∞−F˜k1/zwk+1,z∈U−,w∈D−,where Fkz and F˜k1/z are the Faber polynomials of degree k with respect to z and 1/z for the continuums U¯ and ℂ¯\U, respectively.LetΓ be a rectifiable Jordan curve in the complex plane with length ℓ and let Γt,r=Γ∩Bt,r, t∈Γ, r>0, where Bt,r=z∈ℂ:z−t<r. The classical Morrey spacesLp,λΓ for given 0≤λ≤1 and 1≤p<∞ are defined as the set of functions f∈LlocpΓ such that(11)fLp,λΓ=supz∈Γ0<r<ℓr−λ/pfLpΓt,r<∞.LetU=intΓ, we define the classical Morrey–Smirnov classesEp,λU for 0<λ≤1 and 1<p<∞ as(12)Ep,λU=f∈E1U,f∈Lp,λΓ.We definefEp,λU≔fLp,λΓ.Definition 1.
Letp.:Γ⟶1,∞ be a Lebesgue measurable function satisfying the condition (1), and let λ.:Γ⟶0,1 be a measurable function. We define the variable exponent Morrey spacesLp.,λ.Γ as the set of Lebesgue measurable functions f defined on Γ, such that(13)μp.,λ.f=supt∈Γ,0<r≤ℓr−λt∫Γt,rfspsds<∞.
Lp.,λ.Γ becomes a Banach space with respect to the norm:(14)fLp.,λ.Γ=infu>0:μp.,λ.fu≤1.
We define the variable exponent Morrey–Smirnov classEp.,λ.U as(15)Ep.,λ.U≔f∈E1U,f∈Lp.,λ.Γ.
If we definefEp.,λ.U≔fLp.,λ.Γ, the class Ep.,λ.U becomes a Banach space.Definition 2.
A smooth curveΓ is called Dini-smooth if(16)∫0δΩσ,ssds<∞,δ>0,where σs is the angle, between the tangent line of Γ and the positive real axis expressed as a function of arclength s with the modulus of continuity Ωσ,s, where(17)Ωσ,s≔sups1−s2≤sσs1−σs2,s>0.Definition 3.
Letp.:γ0⟶1,∞ and λ.:γ0⟶0,1 be measurable functions such that(18)0≤λ−≔essinft∈γ0λt≤λ+≔esssupt∈γ0pt<1.
Also, assume thatp.∈℘log. For f∈Lp.,λ.γ0, we define the operator(19)vhfw=1h∫0hfweitdt,w∈γ0,0<h<π.
The operatorvh is bounded linear operator on Lp.,λ.γ0 [24]. Hence, we can define the modulus of smoothness of f∈Lp.,λ.γ0 as(20)Ωf,δp.,λ.=sup0<h≤δf−vhfLp.,λ.γ0,δ>0.
The functionΩf,δp.,λ. is a continuous, nonnegative, and nondecreasing on 0,∞ satisfying the following properties for any f,g∈Lp.,λ.γ0:(21)limδ⟶0Ωf,δp.,λ.=0,Ωf+g,δp.,λ.≤Ωf,δp.,λ.+Ωg,δp.,λ.,δ>0,Ωf,nδp.,λ.≤nΩf,δp.,λ.,n=1,2,…,δ>0.
Suppose thatG is an arbitrary doubly connected domain in the complex plane ℂ, bounded by two rectifiable Jordan curves L1 and L2. Without loss of generality, we may assume that the closed curve L2 is inside the closed curve L1 and 0∈intL2. Let G10≔intL1,G1∞≔extL1,G20≔intL2,and G2∞≔extL2.
We denote byw=ϕtw=ϕ1t the conformal mapping of G1∞G20 onto domain D− normalized by the conditions:(22)ϕ∞=∞,limt⟶∞ϕtt>0,ϕ10=∞,limt⟶0tϕ1t>0.
Moreover,ψ and ψ1 will denote the inverse mappings of ϕ and ϕ1, respectively.
The level lines of the domainsG10 and G20 are defined for r,R>1, by(23)Cr≔t:ϕt=r,CR≔t:ϕ1t=R.
The Faber polynomialsFkt and F˜k1/z have the following integral representations [22].
Ift∈intCr, then(24)Fkt=12πi∫Crϕξkξ−tdξ=12πi∫w=rψ′wwkψw−tdw.
Ift∈extCr, then(25)Fkt=ϕtk+12πi∫Crϕξkξ−tdξ.
Ift∈intCR, then(26)F˜k1t=ϕ1tk−12πi∫CRϕ1ξkξ−tdξ.
Ift∈extCR, then(27)F˜k1t=−12πi∫CRϕ1ξkξ−tdξ=−12πi∫w=Rψ1′wwkψ1w−tdw.
Iffz is an analytic function in the doubly connected domain bounded by the curves Cr and CR, then(28)ft=∑k=0∞akFkt+∑k=1∞bkFk˜1t,where(29)ak=12πi∫w=r1fψwwk+1dw,1<r1<r,k=0,1,2,…,bk=12πi∫w=R1fψ1wwk+1dw,1<R1<R,k=1,2,….
LetL=L1∪L2− and let G be a doubly connected domain bounded by L1 and L2, where L2 is in L1. We define the variable exponent Morrey–Smirnov classes Ep.,λ.G as(30)Ep.,λ.G=f∈E1G,f∈Lp.,λ.L.
Forf∈Ep.,λ.G, the norm is defined by(31)fEp.,λ.G=fLp.,λ.L.
LetU be a simply connected domain in the complex plane ℂ, bounded by a rectifiable Jordan curve Γ, and let U− be the exterior of Γ. Then, for f∈L1Γ, the functions f+ and f− defined by(32)f+t≔12πi∫Γfξξ−tdξ,t∈U,f−t≔12πi∫Γfξξ−tdξ,t∈U−.are analytic in U and U−, respectively, f−∞=0.
For a givent∈Γ, the operator SΓ defined by(33)SΓft≔limε⟶0+12πi∫Γ∩ξ:ξ−t>εfξξ−tdξ,is called the Cauchy singular operator.
According to the Privalov theorem [19] if one of the functions f+ or f− has the nontangential limits a.e. on Γ, then SΓft exists a.e. on Γ and also the other one has the nontangential limits a.e. on Γ. Conversely, if SΓft exists a.e. on Γ, then the functions f+ and f− have nontangential limits a.e. on Γ. In both cases, the following formulae:(34)f+t=SΓft+12ft,f−t=SΓft−12ft,ft=f+t−f−t,hold a.e. on Γ.
In Kokilashvili and Meskhi [25], it is proved that, if Γ is a Dini-smooth curve, then the operator SΓ is bounded on Lp.,λ.Γ, i.e., there exists a positive constant c1 such the following inequality holds for any f∈Lp.,λ.Γ:(35)SΓfLp.,λ.Γ≤c1fLp.,λ.Γ.
IfL1 and L2 are Dini-smooth curves, then by [26] there are positive constants c2,c3,c4, and c5 such(36)c2<ψ′w<c3,c4<ψ1′w<c5.
LetLii=1,2 be a Dini-smooth curve, we define the following functions f0≔f∘ψ for f∈Lp.,λ.L1, p0=p∘ψ and f1≔f∘ψ1 for f∈Lp.,λ.L2, p1=p∘ψ1.
Iff∈Lp.,λ.L1 and f∈Lp.,λ.L2, then by (36), f0∈Lp0.,λ.γ0 and f1∈Lp1.,λ.γ0. From (34), we get f0−∞=0,f1−∞=0 and the following relations hold a.e. on γ0:(37)f0w=f0+w−f0−w,f1w=f1+w−f1−w.
Using Theorem 6.1 from [24] and taking into account the proof of a similar result in [20], we deduce the following lemma.Lemma 1.
Letp.:γ0⟶1,∞ and λ.:γ0⟶0,1 be measurable functions. Let g∈Ep.,λ.D with p.∈℘logγ0 and 0≤λ−≤λ+<1. If ∑k=0nakgwk is the n−th partial sum of the Taylor series of g at the origin, then for any n=1,2,…, there is a constant c6 such the following estimate:(38)gw−∑k=0nakgwkLp.,λ.γ0≤c6Ωg,1np.,λ.,holds.


## 3. Main Result

From now on, we will assume that the set of rational functions is dense in the spaceEp.,λ.L. Our main result is the following.Theorem 1.
LetG be a finite doubly connected domain with the Dini-smooth boundary, L=L1∪L2−, and let p.:L⟶1,∞ , and λ.:L⟶0,1. Ep.,λ.L be the variable Morrey–Smirnov space with p.∈℘logL and 0≤λ−<λ+<1. If f∈Ep.,λ.G, then for every n∈ℕ, there are a rational function Rnf,. and a constant c7 such that(39)f−Rnf, .Ep.,λ.G≤c7Ωf0,1np0.,λ.+Ωf1,1np1.,λ..Proof.
Letf∈Ep.,λ.L, then f0∈Ep0.,λ.γ0 and f1∈Ep1.,λ.γ0.
Puttingϕξ and ϕ1ξ in place of w in (37), we obtain(40)fξ=f0+ϕξ−f0−ϕξ,ξ∈L1,(41)fξ=f1+ϕ1ξ−f1−ϕ1ξ,ξ∈L2.
Lett∈extL1, then from (24), we have(42)∑k=0nakFkt=∑k=0nakϕkt+12πi∫L1∑k=0nakϕkξξ−tdξ,and using (40), we get(43)∑k=0nakFkt=∑k=0nakϕkt+12πi∫L1∑k=0nakϕkξ−f0+ϕξξ−tdξ+12πi∫L1fξξ−tdξ+12πi∫L1f0−ϕξξ−tdξ.
Sincef0−ϕξ∈Ep0.,λ.G1∞,(44)12πi∫L1f0−ϕξξ−tdξ=−f0−ϕt.
Thus,(45)∑k=0nakFkt=∑k=0nakϕkt+12πi∫L1∑k=0nakϕkξ−f0+ϕξξ−tdξ+12πi∫L1fξξ−tdξ−f0−ϕt.
Now, fort∈extL2, from (27) and (41), we have(46)∑k=1nbkF˜k1t=−12πi∫L2∑k=1nbkϕ1kξξ−tdξ=12πi∫L2f1+ϕ1ξ−∑k=1nbkϕ1kξξ−tdξ−12πi∫L2fξξ−tdξ.
For anyt∈extL1, we have(47)12πi∫L1fξξ−tdξ=12πi∫L2fξξ−tdξ.
BecauseextL1⊂extL2, the relations (45), (46), and (47) are valid for any t∈extL1, and this gives(48)∑k=0nakFkt+∑k=1nbkF˜k1/t==∑k=0nakϕkt−f0−ϕt−12πi∫L1f0+ϕξ−∑k=0nakϕkξξ−tdξ+12πi∫L2f1+ϕ1ξ−∑k=1nbkϕ1kξξ−tdξ.
Taking the limit ast⟶z∈L1 along nontangential path outside L1 for almost every z∈L1, we get(49)fz−∑k=0nakFkz+∑k=1nbkF˜k1z=f0+ϕz−∑k=0nakϕkz+12f0+ϕz−∑k=0nakϕkz+SL1f0+ϕz−∑k=0nakϕkz−12πi∫L2f1+ϕ1ξ−∑k=1nbkϕ1kξξ−tdξ.
The rational functionRn(f,z) is defined as(50)Rnf,z=∑k=0nakFkz+∑k=1nbkF˜k1z.
By (51), Minkowski’s inequality, and (35), we get(51)f−Rnf,.Lp.,λ.L1≤c8f0+w−∑k=0nakwkLp0.,λ.γ0+c9f1+w−∑k=0nbkwkLp1.,λ.γ0.
And from Lemma1, we obtain(52)f−Rnf,.Lp.,λ.L1≤c10Ωf0,1np0.,λ.+Ωf1,1np1.,λ..
Lett′∈intL2. From (26) and (41), we get(53)∑k=1nbkF˜k1t′=∑k=1nbkϕ1kt′−12πi∫L2∑k=1nbkϕ1kξξ−t′dξ=∑k=1nbkϕ1kt′−12πi∫L2∑k=1nbkϕ1kξ−f1+ϕ1ξξ−t′dξ−12πi∫L2fξξ−t′dξ−f1−ϕ1t′.and for any t′∈intL1, from (24) and (40), we have(54)∑k=1nakFkt′=12πi∫L1∑k=0nakϕkξξ−t′dξ=12πi∫L1∑k=0nakϕkξ−f0+ϕξξ−t′dξ+12πi∫L1fξξ−t′dξ.
SinceintL2⊂intL1, relations (19) and (20) are valid for t′∈intL2, and this gives (55)∑k=0nakFkt′+∑k=1nbkF˜k1/t′=12πi∫L1∑k=0nakϕkξ−f0+ϕξξ−t′dξ−12πi∫L2∑k=1nbkϕ1kξ−f1+ϕ1ξξ−t′dξ−f1−ϕ1t′+∑k=1nbkϕ1kt′.
Taking the limit ast′⟶z∈L2 along nontangential path inside L2 for almost every z∈L2, we get(56)fz−∑k=0nakFkz+∑k=1nbkF˜k1z=f1+ϕ1z−12∑k=1nbkϕ1kz−f1+ϕ1z−SL2∑k=1nbkϕ1kz−f1+ϕ1z−12πi∫L1∑k=0nakϕkξ−f0+ϕξξ−zdξ.
Using Minkowski’s inequality and (35), we get(57)f−Rnf,.Lp.,λ.L2≤c11f1+w−∑k=1nbkwkLp0.,λ.γ0+c12f0+w−∑k=0nakwkLp1.,λ.γ0.
By Lemma1, we obtain(58)f−Rnf,.Lp.,λ.≤c13Ωf0,1np0.,λ.+Ωf1,1np1.,λ..
From (52) and (58), we obtain(59)f−Rnf,.Ep.,λ.G≤c7Ωf0,1np0.,λ.+Ωf1,1np1.,λ..



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