On approximation properties of functions by means of Fourier and Faber series in weighted Lebesgue spaces with variable exponent

. In this paper the approximation of functions by linear means of Fourier series in weighted variable exponent Lebesgue spaces was studied. This result was applied to the approximation of the functions by linear means of Faber series in Smirnov classes with variable exponent defined on simply connected domain of the complex plane.


Introduction and main results
Let T denote the interval [0, 2π] and L p (T), 1 ≤ p ≤ ∞, the Lebesgue space of measurable functions on T.
Let ℘ denote the class of Lebesgue measurable functions p : T −→ (1, ∞) such that 1 < p * := ess inf The spaces L p(.) (T) are called generalized Lebesgue spaces with variable exponent. It is known that for p(x) := p (0 < p ≤ ∞), the space L p(x) (T) coincides with the Lebesgue space L p (T). If p * < ∞ then the spaces L p(.) (T) represent a special case of the so-called Orlicz-Musielak spaces [37]. For the first time Lebesgue spaces with variable exponent were introduced by Orlicz [38]. Note that the generalized Lebesgue spaces with variable exponent are used in the theory of elasticity, in mechanics, especially in fluid dynamics for the modelling of electrorheological fluids, in the theory of differential operators, and in variational calculus [5,8,9,41,43]. The detailed information about properties of the Lebesque spaces with variable exponent can be found in [8,10,27,33,34,42,46]. Note that some of the fundamental problems of the approximation theory in the generalized Lebesgue spaces with variable exponent of periodic and non-periodic functions were studied and solved by Sharapudinov [47][48][49].
A function ω : T → [0, ∞] is called a weight function if ω is a measurable and almost everywhere (a.e.) positive.
Let ω be a 2π periodic weight function. We denote by L p ω (T) the weighted Lebesgue space of 2π periodic measurable functions f : . It is known (see [28]) that the set of trigonometric polynomials is dense in L p(.) Let B be the class of all intervals in T. For B ∈ B we set For given p ∈ ℘ the class of weights ω satisfying the condition will be denoted by A p(.) [1,15,23,30,32]. We say that the variable exponent p(x) satisfies Local log-Hölder continuity condition, if there is a positive constant c 1 such that A function p ∈ ℘ is said to belong to the class ℘ log , if the condition (1) is satisfied.
We denote by E n (f ) L p(.) ω (T) the best approximation of f ∈ L p(.) ω (T) by trigonometric polynomials of degree not exceeding n − 1, i.e., where Π n−1 denotes the class of trigonometric polynomials of degree at most n − 1.
Note that according to [53,54] Ω(f, δ) p(·),ω ≤ c(p)∥f ∥ L p(.) ω (T) . It can easily be shown that Ω(·, f ) p(·),ω is a continuous, nonnegative and nondecreasing function satisfying the conditions lim δ→0 Ω(f, δ) p(·),ω = 0, If the Fourier series of f is given by (2), then Zygmund-Rieszmeans of order k is defined as We denote by E n (f ) p(.),ω the best approximation of f ∈ L p(.) ω (T) by trigonometric polynomials of degree not exceeding n, i.e., where Π n denotes the class of trigonometric polynomials of degree at most n. Let The conjugate polynomial T n is defined by We will say that the method of summability by the matrix Λ satisfies condition b k,p(·) (respectively b * k,p(·) ) if for T n ∈ Π n the inequality holds and the norms In the present paper, the necessary and sufficient condition about the relationship between the approximation of functions by linear means of Fourier series and by Zygmund-Riesz means of order k was investigated in weighted Lebesgue spaces with variable exponent. Also, we investigate the approximation of functions by linear means of Fourier series in terms of the modulus of smoothness of these functions in weighted Lebesgue spaces with variable exponent. This result was applied to the approximation of the functions by linear means of Faber series in weighted Smirnov classes with variable exponent defined on simply connected domains of the complex plane. The similar problems in different spaces were investigated by several authors (see, for example, [1, 2, 4, 7, 12, 14, 16-26, 29-32, 35, 36, 39, 44, 50-57]).
The main results in the present work are the following theorems. ) . In order that it is sufficient and necessary that . If the summability method with the matrix Λ satisfies the condition (b k,M ) or (b * k,M ), then the inequality holds with a constant c 3 > 0 independent of n.
. If the summability method with the matrix Λ satisfies the condition (b k,p(·) ) or (b * k,p(·) ), then the estimate holds with a constant c 4 > 0 not depend on n, f and δ.
Let G be a finite domain in the complex plane C, bounded by a rectifiable Jordan curve Γ, and let G − := extΓ. Further let and let ψ denote the inverse of ϕ. Let w = ϕ 1 (z) denote a function that maps the domain G conformally onto the disk |w| < 1. The inverse mapping of ϕ 1 will be denoted by ψ 1 . Let Γ r denote circular images in the domain G, that is, curves in G corresponding to circle |ϕ 1 (z)| = r under the mapping z = ψ 1 (w).
Let us denote by E p , where p > 0, the class of all functions f (z) ̸ = 0 which are analytic in G and have the property that the integral Γr |f (z)| p |dz| is bounded for 0 < r < 1. We shall call the E p -class the Smirnov class. If the function f (z) belongs to E p , then f (x) has definite limiting values f (z ′ ) almost everywhere on Γ, over all nontangential paths; |f (z ′ )| is summable on Γ; and lim r→1 Γr It is known that φ ′ = E 1 (G − ) and ψ ′ ∈ E 1 (D − ). Note that the general information about Smirnov classes can be found in the books [13, pp. 438-453] and [11, pp. 168-185].
Let L M (T, ω) is a weighted Orlicz space defined on Γ. We define also the ω-weighted Smirnov class of variable exponent E p(·) (G, ω) as Let h be a continuous function on [0, 2π]. Its modulus of continuity is defined by The curve Γ is called Dini-smooth if it has a parameterization If Γ is Dini-smooth curve, then there exist (see [58]) the constants c 5 and c 6 such that Note that if Γ is a Dini-smooth curve, then by (7) we have f 0 ∈ L p(·) ω 0 (T) for f ∈ L p(·) ω (Γ). It is known (see [20]) that, if Γ is a Dini-smooth curve, then p 0 ∈ ℘ log (T) if and only if p ∈ ℘ log (Γ).
Let 1 < p < ∞, 1 p + 1 p ′ and let ω be a weight function on Γ. ω is said to satisfy Muckenhoupt's A p -condition on Γ, if r) is an open disk with radius r and centered z.
Let us denote by A p (Γ) the set of all weight functions satisfying Muckenhoupt's A p -condition on Γ. For a detailed discussion of Muckenhoupt weights on curves, see, e.g., [3].
Let ϕ k (z), k = 0, 1, 2, . . . , be the Faber polynomials for G. The Faber polynomials ϕ k (z), associated with G ∪ Γ, are defined through the expansion and the equalities for every z ∈ G. Considering this formula and expansion (8), we can associate with f the formal series This series is called the Faber series expansion of f , and the coefficients a i (f ) are said to be the Faber coefficients of f . Let (10) be the Faber series of the function f ∈ E p(·) (G, ω). For the function f we define the summability method by the tringular matrix Λ = λ ij j,∞ i,j=0 by the linear means The n-the partial sums and Zygmund means of order k of the series (10) are defined, respectively, as Let Γ be a Dini-smooth curve. Using the nontangential boundary values of f + 0 on T we define the modulus of smoothness of f ∈ L p(·) ω (Γ) as Ω(f, δ) p(·),Γ,ω := Ω(f + 0 , δ) p 0 (·),ω 0 , δ > 0. The following theorem holds.
Let P be the set of all algebraic polynomials (with no restriction on the degree), and let P(D) be the set of traces of members of P on D. We define the operator T : P(D) −→ E p(·) (G, ω) as Then, from (9) we have The following result holds for the linear operator T [53].

Proof of the main results
Proof of Theorem 1. It is clear that the inequality (4) follows from the inequality (3).
holds. Taking into account the relations (12), (14) and (15), we obtain If the summability method with the matrix Λ satisfies condition (b * k,p(·) ), the proof is made anologously to the above.
The inequality (20) holds for Zygmund-Riesz means of order k. Note that in the Lebesgue spaces L p (T), 1 < p ≤ ∞, the inequality (20) was proved in [50].

Conclusion
In Theorem 1 of this work, the relationship between the linear means of Fourier and Zygmund means of Fourier series in weighted variable exponent Lebesgue spaces has been investigated. The necessary and sufficient condition has been found for this relationship.
In Theorem 2, the approximation of the function by the linear means of Fourier series in weighted variable exponent Lebesgue spaces was studied in terms of modulus of smoothness.
In Theorem 3, the modulus of smoothness of the linear means of Fourier series of the function has been estimated.
In Theorem 4, the result obtained in Theorem 2 was applied to the approximation of the functions by linear means of Faber series in Smirnov classes with variable exponent defined in the domains with a Dini-smooth boundary of the complex plane.
In Remark 1, the approximation of the function by linear means of Fourier series has been obtained in terms of the best approximation of the function.