Approximation by rational functions in Smirnov classes with variable exponent

In this article, we investigate the direct problem of approximation theory in the variable exponent Smirnov classes of analytic functions, defined on a doubly connected domain bounded by two Dini-smooth curves.

We denote by w = φ(t) (w = φ 1 (t)) the conformal mapping of G ∞ 1 (G 0 2 ) onto domain D − which satisfies the conditions and let ψ and ψ 1 be the inverse mappings of φ and φ 1 , respectively. Throughout this paper, we assume that the letters c 1 ,c 2 ,. . . always remain to denote positive constants that may different at each occurrence.  Let F be some Jordan rectifiable curve Γ ⊂ C or the segment [0, 2π] and let |F | denote the Lebesgue measure of F . We define the classes of functions P(F ), P log (F ) and P Detailed information on variable exponent Lebesgue space can be found in the books [1,2].

Definition 2.2
Let a finite simply connected domain U with the rectifiable Jordan curve boundary Γ in the complex plane C be given, and let Γ r be the image of circle {w ∈ C : |w| = r, 0 < r < 1} under some conformal mapping of D onto U . By E 1 (U ), we denote the class of analytic functions f in U which satisfy It is known that every function of class E 1 (U ) has nontangential boundary values almost everywhere on Γ and the boundary function belongs to L 1 (Γ ) [3, pp. 438-453].

Definition 2.3
Let a finite simply connected domain U with the rectifiable Jordan curve boundary Γ in the complex plane C be given, and let p(.) ∈ P log 0 (Γ ). The variable exponent Smirnov class of analytic functions is defined as:

Definition 2.4
Let L = L 1 ∪ L − 2 and p(.) ∈ P log 0 (L). The variable exponent Smirnov class with respect to the doubly connected domain G is defined as: For f ∈ E p(.) (G), the norm E p(.) (G) can be defined as: Definition 2. 5 We define the modulus of continuity of a function g ∈ L p(.) (γ 0 ) by the relation Definition 2.6 Let Γ be a rectifiable Jordan curve in the complex plane C. For a given t ∈ Γ and f ∈ L 1 (Γ ), the operator defined by is called the Cauchy singular operator.
where σ (s) 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 Ω(σ, s).
Kokilashvili and Samko proved in [11] that, if Γ is a Dini-smooth curve, then the operator To prove our main theorem, we need the following lemma. It can be found in [3, p. 431].
The level lines of the domains G 0 1 and G 0 2 are defined for r, R > 1 by The Faber polynomials Φ k (t) of degree k are defined by the relation and have the following integral representations [12]: And for t ∈ extC r , we have Similarly, the Faber polynomials Φ k (1/t) of degree k with respect to 1/z are defined by the relation and satisfy the following relations: And in case t ∈ extC R , we obtain has the following formula [13]: where In case if G is an annulus domain, then the series Eq. (6) becomes the Laurent series for the function f (t).
Taking the first n terms of the series Eq. (6), we obtain the rational function For large values of n and if f ∈ E p(.) (G), we will prove that such a rational function R n ( f, t) approximated the function f (t) arbitrarily closely.
If L 1 and L 2 are Dini-smooth, then by [15, pp. 321-456], it follows that where c 2 , c 3 , c 4 and c 5 are positive constants. Let L i (i = 1, 2) be a Dini-smooth curve, we define the following functions From [7], it follows that f 0 ∈ L p 0 (.) (γ 0 ) with p 0 ∈ P log 0 (γ 0 ) and = 0 and the following relations hold a. e. on γ 0 The following lemma was proved in [7].

Lemma 2.9 Let g ∈ E p(.) (D) with p ∈ P
log 0 (γ 0 ). If n k=0 a k w k is the n th partial sum of the Taylor series of g at the origin, then the following estimate holds, where c 6 is a positive constant.
In the literature, there are sufficiently wide investigations relating to the approximation problems in the simply connected domains. For example, the problems of approximation theory for Smirnov classes with variable exponent, weighted Smirnov classes, weighted Smirnov Orlicz classes and weighted rearrangement invariant Smirnov classes were studied in [4][5][6][7][8]. But the approximation problems in the doubly connected domains were not investigated sufficiently wide.
In this work, we study the direct theorem of approximation theory in the variable exponent Smirnov classes, defined in the doubly connected domains bounded by two Dini-smooth curves.

The main result
Our main result is given in the following theorem.

Theorem 3.1 Let G be a finite doubly connected domain with the Dini-smooth boundary, L
holds, where c 7 is a positive constant and R n ( f, .) is the rational function defined by Eq. (8).
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