Approximation of functions by de la Vallée-Poussin sums in weighted Orlicz spaces

We investigate problems of estimating the deviation of functions from their de la Vallée-Poussin sums in weighted Orlicz spaces LM (T, ω) in terms of the best approximation $${E_{n}(f)_{M, \, \omega }}$$En(f)M,ω.

or with the Luxemburg norm becomes a Banach space. This space is denoted by L M (T) and is called an Orlicz space [28, p.26]. The Orlicz spaces are known as the generalizations of the Lebesgue spaces L p (T), 1 < p < ∞. If M(x) = M(x, p) := x p , 1 < p < ∞, then Orlicz spaces L M (T) coincide with the usual Lebesgue spaces L p (T), 1 < p < ∞.
Note that the Orlicz spaces play an important role in many areas such as applied mathematics, mechanics, regularity theory, fluid dynamics and statistical physics. Therefore, the investigation into the approximation of the functions by means of Fourier trigonometric series in Orlicz spaces is also important in these areas of research.
The Luxemburg norm is equivalent to the Orlicz norm as holds true [28, p.80].
If we choose M(u) = u p / p, 1 < p < ∞, then the complementary function is N (u) = u q /q with 1/ p + 1/q = 1 and we have the relation where u L p (T) = T |u(x)| p dx 1/ p stands for the usual norm of the L p (T) space.
If N is complementary to M in Young's sense and f ∈ L M (T), g ∈ L N (T), then the so-called strong Hölder inequalities [28, p.80] first considered by Matuszewska and Orlicz [38], are called the Boyd indices of the Orlicz spaces L M (T ). It is known that the indices α M and β M satisfy 0 The detailed information about the Boyd indices can be found in [3,[5][6][7]39].
A measurable function ω : T → [0, ∞] is called a weight function if the set ω −1 ({0, ∞}) has Lebesgue measure zero. With any given weight ω we associate the ω-weighted Orlicz space L M (T, ω) consisting of all measurable functions f on T such that Let 1 < p < ∞, 1/ p + 1/ p = 1 and let ω be a weight function on T. ω is said to satisfy Muckenhoupt's By reference [18], Lemma 1.4, the shift operator ν h is a bounded linear operator on L M (T, ω): is called k-th modulus of smoothness of f ∈ L M (T, ω), where I is the identity operator. It can easily be shown that k M, ω (·, f ) is a continuous, nonnegative and nondecreasing function satisfying the conditions The function conjugate to a 2π−periodic summable function on [−π, π] given by . It is known that the conjugate series to Fourier series f ∈ L [0,2π ] will not always be the Fourier series (see, e.g., [53, p.155]).
The n-th partial sums, and de la Vallé e-Poussin sums [57] of series (1.1) are defined, respectively, as The best approximation to f ∈ L M (T, ω) in the class n of trigonometric polynomials of degree not exceeding n is defined by Note that the existence of T * n ∈ n such that and ψ stands for the inverse of ϕ. Let w = ϕ 1 (z) indicate a function that maps the domain G conformally onto the disk |w| < 1. The inverse mapping of ϕ 1 will be shown by ψ 1 . Let r be the image of the circle |ϕ 1 (z)| = r , 0 < r < 1 under the mapping z = ψ 1 (w).
Let us denote by E p , where p > 0, the class of all functions f (z) = 0 that are analytic in G and have the property that the integral r | f (z)| p |dz| is uniformly 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 (z) has definite limiting values f (z ) almost every where on , over all nontangential paths; f (z ) is summable on ; and 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 [12, pp.438-453], and [8, pp.168-185].
Let L M ( , ω) be a weighted Orlicz space defined on . We also define the ω-weighted Smirnov-Orlicz With every weight function ω on , we associate another weight ω 0 on T defined by Let h be a continuous function on [0, 2π]. Its modulus of continuity is defined by The curve is called Dini-smooth if it has a parameterization If is a Dini-smooth curve, then there exist [58] the constants c 1 and c 2 such that is an open disk with radius r and centered z. Let us denote by A p ( ) the set of all weight functions satisfying Muckenhoupt's A p -condition on . For a detailed discussion of Muckenhoupt weights on curves (see, e.g., [4]).
Let be a rectifiable Jordan curve and f ∈ L 1 ( ). Then, the function f + defined by Let ϕ k (z), k = 0, 1, 2, . . . be the Faber polynomials for G. The Faber polynomials ϕ k (z), associated with G ∪ , are defined through the expansion and the equalities hold [51, p.33-38].
for every z ∈ G. Considering this formula and expansion (1.3), we can associate with f the Faber series where Faber series expansion of f, and the coefficients a k ( f ), k = 0, 1, 2, . . . are said to be the Faber coefficients of f.

This series is called the
The n-th partial sums and de la Vallée-Poussin sums of the series (1.6) are defined, respectively, as Let be a Dini-smooth curve. Using the nontangential boundary values of f + 0 on T we define the r-th modulus of smoothness of f ∈ L M ( , ω) as Let P :={all 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 as follows: Then, taking into account (1.4) and (1.5) we have We use the constants c, c 1 , c 2 , . . . (in general, different in different relations) which depend only on the quantities that are not important for the questions of interest We need the following results.
holds with a constant c 3 > 0 independent of n.

Then, for r ∈ N and f
holds with a constant c 6 > 0 independent of n.

If f (r ) ∈ L M (T, ω), then the estimate
holds with a constant c 7 > 0 independent of n.
Proof The function f (r ) can be written in following form: Then, by (1.9) we obtain (1.10) By [18] the following inequality holds: Then, from the inequality (1.11) we conclude that Using the Bernstein inequality for weighted Orlicz spaces [18], we have Theorem 1.6 [20] If is a Dini-smooth curve, 0 < α M ≤ β M < 1, and ω ∈ A 1/α M ( ) ∩ A 1/β M ( ), then the operator is one-to-one and onto.
Note that the approximation problems by trigonometric polynomials in weighted Lebesgue spaces with weights belonging to the Muckenhoupt class A p (T) were studied in [13,36,37]. Detailed information on the weighted polynomial approximation can be found in the books [9,41].
In the present paper, we investigate the problems of estimating the deviation of functions from their de la Vallée-Poussin sums in weighted Orlicz spaces L M (T, ω). This result is applied to estimate of approximation of de la Vallé e-Poussin sums of Faber series in weighted Smirnov-Orlicz classes defined on simply connected domains of the complex plane in terms of the modulus of smoothness. We also study the approximation of conjugate function by de la Vallée-Poussin sums of the Fourier series of the conjugate function in weighted Orlicz spaces L M (T, ω). Note that the estimates obtained in this work depend on sequence of the best approximation E n ( f ) M, ω . Similar problems in different spaces have been investigated by several researchers (see, e.g., [1,2,11,27,42,43,[48][49][50][52][53][54][55][56][57]59,60]).
Our main results are as follows.

Theorem 1.7 Let L M (T ) be a reflexive Orlicz space and
holds with a constant c 11 > 0 independent of n.
Note that this result in the spaces of continuous functions and Lebesgue space L p (1 < p < ∞) have been investigated in [49,50,60].
holds with a constant c 12 > 0 independent of n.
Similar result for the other modulus of smoothness in the spaces of continuous functions has been obtained in [49]. Also, similar results for the Cesaro means, Zygmund means of order 2 and Abel-Poisson means in weighted Orlicz spaces can be found in [15].
is satisfied for f ∈ L M (T , ω), then the estimate holds with a constant c 13 > 0 independent of n.
This result for the spaces of continuous functions has been obtained in [1]. Similar results for the other means of Fourier trigonometric series in the Smirnov classes E p (G) (1 < p < ∞) and weighted Orlicz spaces E M (G, ω) can be found in [15,29].

Proofs of the main results
Proof of Theorem 1. 7 We take the integer j such that the inequality 2 j ≤ m + 1 < 2 j+1 is satisfied. The following identity holds: Taking into account of (2.1), we have Consideration of (1.11) and (2.1) gives us On the other hand, the following inequality holds: Thus, the proof of Theorem 1.7 is completed.
the inequality (2.6) yields Then, the last inequality yields From Theorem 1.7 we have Since 0 ≤ 2m ≤ n, this implies (n + 1)/(n − m + 1) ≤ 3. Then, using the last inequality we reach 2. Suppose that the inequality n < 2m ≤ 2n is satisfied. From Theorem 1.3 we have Using the last inequality and Theorem 1.7, we find that Hence, the proof of Theorem 1.9 is completed. Since ω 0 ∈ A 1/α M (T )∩ A 1/β M (T ), considering Lemma 1.4 given in Ref. [20] we get f + 0 ∈ E M (D, ω 0 ). Then, function f + 0 has the following Taylor expansion a k ( f )w k .
Note that f + 0 ∈ E 1 (D) and boundary function f + 0 ∈ L M (T, ω 0 ). Then, using Theorem 3.4 [8, p.38] for the function f + 0 (w) we get Fourier expansion Using the boundedness of the operator T, Theorems 1.7 and 1. Thus, the theorem is proved.