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.
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 by 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 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
Let denote the interval or a Jordan rectifiable curve , and let denote the class of all Lebesgue measurable functions such that
We denote by to the Lebesgue measure of . We say if there is a constant such thatfor all .
For , we define the set of all measurable functions such that
is a Banach space with respect to the norm
Let be a finite simply connected domain with a rectifiable Jordan curve boundary . Denote , , , and . Let be the image of circle under some conformal mapping of onto .
For given , we denote by the class of analytic functions in which satisfiesuniformly in .
It is known that every function of class has nontangential boundary values almost everywhere on and the boundary function belongs to ([19], pp. 438–453).
Also, suppose that is the conformal mapping of onto normalized byand let be the inverse of . Let be the conformal mapping of on to , normalized by
The inverse mapping of will be denoted by .
The functions and have in some deleted neighborhood of representations
The functionsare analytic in the domain , and the following expansions hold [20–23]:where and are the Faber polynomials of degree with respect to and for the continuums and , respectively.
Let be a rectifiable Jordan curve in the complex plane with length and let , , where . The classical Morrey spaces for given and are defined as the set of functions such that
Let , we define the classical Morrey–Smirnov classes for and as
We define .
Definition 1. Let be a Lebesgue measurable function satisfying the condition (1), and let be a measurable function. We define the variable exponent Morrey spaces as the set of Lebesgue measurable functions defined on , such that becomes a Banach space with respect to the norm:We define the variable exponent Morrey–Smirnov class asIf we define , the class becomes a Banach space.
Definition 2. A smooth curve is called Dini-smooth ifwhere is the angle, between the tangent line of and the positive real axis expressed as a function of arclength with the modulus of continuity , where
Definition 3. Let and be measurable functions such thatAlso, assume that . For , we define the operatorThe operator is bounded linear operator on [24]. Hence, we can define the modulus of smoothness of asThe function is a continuous, nonnegative, and nondecreasing on satisfying the following properties for any :Suppose that is an arbitrary doubly connected domain in the complex plane , bounded by two rectifiable Jordan curves and . Without loss of generality, we may assume that the closed curve is inside the closed curve and . Let and .
We denote by the conformal mapping of onto domain normalized by the conditions:Moreover, and will denote the inverse mappings of and , respectively.
The level lines of the domains and are defined for , byThe Faber polynomials and have the following integral representations [22].
If , thenIf , thenIf , thenIf , thenIf is an analytic function in the doubly connected domain bounded by the curves and , thenwhereLet and let be a doubly connected domain bounded by and , where is in . We define the variable exponent Morrey–Smirnov classes asFor , the norm is defined byLet be a simply connected domain in the complex plane , bounded by a rectifiable Jordan curve , and let be the exterior of . Then, for , the functions and defined byare analytic in and , respectively, .
For a given , the operator defined byis called the Cauchy singular operator.
According to the Privalov theorem [19] if one of the functions or has the nontangential limits a.e. on , then exists a.e. on and also the other one has the nontangential limits a.e. on . Conversely, if exists a.e. on , then the functions and have nontangential limits a.e. on . In both cases, the following formulae:hold a.e. on .
In Kokilashvili and Meskhi [25], it is proved that, if is a Dini-smooth curve, then the operator is bounded on , i.e., there exists a positive constant such the following inequality holds for any :If and are Dini-smooth curves, then by [26] there are positive constants , and suchLet be a Dini-smooth curve, we define the following functions for , and for , .
If and , then by (36), and . From (34), we get and the following relations hold a.e. on :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. Let and be measurable functions. Let with and . If is the partial sum of the Taylor series of at the origin, then for any , there is a constant such the following estimate:holds.
3. Main Result
From now on, we will assume that the set of rational functions is dense in the space . Our main result is the following.
Theorem 1. Let be a finite doubly connected domain with the Dini-smooth boundary, , and let , and . be the variable Morrey–Smirnov space with and . If , then for every , there are a rational function and a constant such that
Proof. Let , then and .
Putting and in place of in (37), we obtainLet , then from (24), we haveand using (40), we getSince ,Thus,Now, for from (27) and (41), we haveFor any , we haveBecause the relations (45), (46), and (47) are valid for any and this givesTaking the limit as along nontangential path outside for almost every , we getThe rational function Rn(f,z) is defined asBy (51), Minkowski’s inequality, and (35), we getAnd from Lemma 1, we obtainLet . From (26) and (41), we getand for any , from (24) and (40), we haveSince , relations (19) and (20) are valid for , and this gives Taking the limit as along nontangential path inside for almost every , we getUsing Minkowski’s inequality and (35), we getBy Lemma 1, we obtainFrom (52) and (58), we obtain
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest.