On approximation of functions by rational functions in weighted generalized grand Smirnov classes

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hold. Note that the information about properties and applications of the grand Lebesgue spaces can be found in [11,13,26,33,35,36].
A Jordan curve is called Ahlfors 1-regular [37], if there exists a number c > 0 such that for every r > 0, sup {| ∩ D(z, r )| : z ∈ } ≤ cr, , where D(z, r ) is an open disk with radius r and centered at z and | ∩ D(z, r )| is the length of the set ∩ D(z, r ).
Let ω be a weight function on . ω is said to satisfy Muckenhoupt's A p -condition on if Let us further assume that B is a simply connected domain with a rectifiable Jordan boundary and B − := ext . Without loss of generality we assume that 0 ∈ B. Let The inverse mapping of φ * 1 will be denoted by ψ * 1 . Note that the mappings ψ * and ψ * 1 have in some deleted neighborhood of ∞ representations: are analytic in the domain D − . The following expansions hold: where k, p−ε (z) and F k, p−ε ( 1 z ) are the p − ε Faber polynomials of degree k with respect to z and 1 z for the continuums B and B\B, respectively (see also [5,20,23] and ( [34], pp. 255-257).
Let E 1 (B) be a classical Smirnov class of analytic functions in B. The set E p),θ (B, ω) := f ∈ E 1 (B) : f ∈ L p),θ ( , ω) is called the ω-weighted generalized grand Smirnov class in B.
Let ω ∈ A p (T). For f ∈ L p),θ ( , ω) we define the operator If ω ∈ A p (T) and f ∈ L p (T, ω), then the operator ν h is a bounded on L p),θ (T, ω) [24]: It can be easily shown that r p),θ,ω ( f, ·) is a continuous, non-negative and nondecreasing function satisfying the conditions: Let G be a doubly connected domain in the complex plane C, bounded by the rectifiable Jordan curves 1 and 2 (the closed curve 2 is in the closed curve 1 ). Without loss of generality we assume 0 ∈ int 2 . Let G 0 1 : = int 1 , G ∞ 1 : =ext 1 , G 0 2 : =int 2 , G ∞ 2 :=ext 2 . We denote by w = φ (z) the conformal mapping of G ∞ 1 onto domain D − normalized by the conditions: and let ψ be the inverse mapping of φ. We denote by w = φ 1 (z) the conformal mapping of G 0 2 onto domain D − normalized by the conditions: and let ψ 1 be the inverse mapping of φ 1 .
Let us take For k, p−ε (z) and F k, p−ε 1 z the following integral representations hold [5,20,23,34], pp. 255-257: If a function f (z) is analytic in the doubly connected domain bounded by the curves C ρ 0 and r 0 , then the following series expansion holds: The series (1.5) is called the p − ε Faber-Laurent series of f, and the coefficients a k and b k are said to be the p − ε Faber-Laurent coefficients of f . For z ∈ G by Cauchy's integral formulae we have If z ∈ int 2 and z ∈ ext 1 , then Let us consider The function I 1 (z) determines the functions I + 1 (z) and I − 1 (z), while the function I 2 (z) determines the functions I + 2 (z) and I − 2 (z). The functions I + 1 (z) and I − 1 (z) are analytic in int 1 and ext 1 , respectively. The functions I + 2 (z) and I − 2 (z) are analytic in int 2 and ext 2 , respectively.
Let B be a finite domain in the complex plane bounded by a rectifiable Jordan curve and f ∈ L 1 ( ). Then the functions f + and f − defined by are analytic in B and B − respectively, and f − (∞) = 0. Thus the limit exists and is finite for almost all z ∈ .
The quantity S ( f )(z) is called the Cauchy singular integral of f at z ∈ . According to the Privalov theorem ( [12], p. 431), if one of the functions f + or f − has the non-tangential limits a.e. on , then S ( f )(z) exists a.e. on and also the other one has the non-tangential limits a.e. on . Conversely, if S ( f )(z) exists a.e. on , then the functions f + (z) and f − (z) have non-tangential limits a.e. on . In both cases, the formulae and hence holds a.e. on . From the results given in [33], it follows that if is an Ahlfors 1-regular curve, then S is bounded on L p),θ ( , ω).
a.e. on T. Now, in the doubly connected domain we define the ω-weighted generalized grand Smirnov class . Let E 1 (G) be a classical Smirnov class of analytic functions in G. The set E p),θ (G, ω) := f ∈ E 1 (G) : f ∈ L p),θ ( , ω) is called the ω-weighted generalized grand Smirnov class in G. We denote by E p),θ (G, ω) the closure of Smirnov class E p (G, ω) in the space E p),θ (G, ω). Lemma 1.1 [23,24]. Let g ∈ E p),θ (D, ω) , ω ∈ A p (T), 1 < p < ∞ and θ > 0. If n k=0 d k (g)w k is the nth partial sum of the Taylor series of g at the origin, then there exists a constant c 2 > 0 such that , r ∈ N for every natural number n.
We set The rational function R n ( f, z) is called the p − ε Faber-Laurent rational function of degree n of f. Since series of Faber polynomials are a generalization of Taylor series to the case of a simply connected domain, it is natural to consider the construction of a similar generalization of Laurent series to the case of a doubly-connected domain.

Theorem 1.2 Let G be a finite doubly connected domain with the Ahlfors 1-regular boundary
where R n (., f ) is the p − ε Faber-Laurent rational function of degree n of f.
For z ∈ int 1 , using (1.1) and (2.1) we obtain n k=1 a k k (z) Now, by virtue of (2.6) and (2.7) for z ∈ int 2 , we conclude that n k=0 a k k (z) Taking the limit as z → z * ∈ 2 along all non-tangential paths inside 2 , we reach f z * − n k=0 a k k, p z * − n k=1 b k F k, p a.e. on 2 .