DIRECT THEOREMS OF TRIGONOMETRIC APPROXIMATION FOR VARIABLE EXPONENT LEBESGUE SPACES

Jackson type direct theorems are considered in variable exponent Lebesgue spaces Lp(x) with exponent p(x) satisfying 1 ≤ ess infx∈[0,2π] p(x), ess supx∈[0,2π] p(x) < ∞, and the Dini–Lipschitz condition. Jackson type direct inequalities of trigonometric approximation are obtained for the modulus of smoothness based on one sided Steklov averages Zvf(·) := 1 v ∫ v

The main difficulty related to L p (•) theory is that the spaces L p(•) are, in general, not translation invariant, see e.g.[10,Proposition 3.6.1].This inadequacy and the structure of L p(•) cause some additional problems.For example Young's Convolution inequality and Cavalieri's equality do not hold in the spaces L p (•) .Maximal, Poincáre, and Sobolev inequalities do not hold in a modular form in L p(•) either.Interpolation is not so useful in L p (•) .Solutions of the p (•)-Laplace equation are not scalable.
To obtain a Jackson type inequality in L p (•) we can not use the classical modulus of smoothness ω r (f, •) p because the classical translation t a f (x) := f (x + a) of 122 RAMAZAN AKG ÜN f (•), where a ∈ R, may not be in the class L p(•) even if f is in L p (•) , see e.g.[10,Proposition 3.6.1].So the classical modulus of smoothness ω r may not be suitable for functions f ∈ L p (•) .Instead of using the classical translation operator f → t a f , a ∈ R, we will consider the one sided Steklov average [22,23,2,24].Under some condition on p (x) Sharapudinov [22,23] obtained that the family of operators {Z v f } 0<v≤1 is uniformly bounded on L p(•) (see Theorem 1.1 below).
Let T := [0, 2π) and let E be the class of Lebesgue measurable functions p (x) : T → [1, ∞) such that 1 ≤ p ≤ P < ∞.The variable exponent p (x) is said to satisfy the Dini-Lipschitz property on T ( [22]) if there exists a (Dini-Lipschitz) constant A > 0, depending only on p (x), such that for all x, y ∈ T with x = y.We will denote by P the class of those exponents p ∈ E that satisfy the Dini-Lipschitz property (1.1) on T. We define the variable exponent Lebesgue space L p(•) as the collection of 2π-periodic Lebesgue measurable functions f : T → R having the norm where p ∈ E. The space L p(•) is a Banach space.If p ∈ P, p (x) := p (x) / (p (x) − 1) for p (x) > 1, and p (x) := ∞ for p (x) = 1, then Hölder's inequality holds when f ∈ L p(•) and g ∈ L p (•) .We know that where f p(•) is the Luxemburg norm (1.2) of f and Inequality (1.4) was obtained in [22,23]; (1.5) can be proved similarly.After this result one can define modulus of smoothness.For p ∈ P, f ∈ L p(•) , and 0  For r = 1 see [24].Note that modulus of smoothness is the main notion in the approximation theory and it is used in many of the approximation results such as Jackson type direct theorems, Salem-Stechkin type inverse theorems, Marchaud type inequalities, etc.First of all the set of trigonometric polynomials is a dense subset ([23, Theorems 6.1 and 6.2]) of L p(•) when p ∈ P.This allows us to consider approximation problems in L p (•) .Jackson type inequalities in L p(•) were investigated by several mathematicians.For example in [14] Israfilov and Testici obtained the following estimate: holds for n ∈ N, with constant depending only on p, where Π n is the class of trigonometric polynomials of degree not greater than n.
For p ∈ P := {p ∈ P : 1 < ess inf x∈T p (x)} see also [13].In [2] the author proved that holds for n ∈ N, with constant depending only on p, r, where Here, and in what follows, A = O (1) B means that A/B is less than or equal to some constant, depending on essential parameters only.
In [24,28] Sharapudinov and Volosivets considered the following type of modulus of smoothness of order r ∈ N: in the right hand side of (1.7): holds for n, r ∈ N, with constant depending only on p, r.
In the present work we will consider a more natural and smaller modulus of smoothness: •) and p ∈ P. Our result, given below, refines the Jackson type estimates (1.7) and (1.8).
Let X be a Banach space with norm • X .By X r we denote the class of functions f ∈ X such that f (r−1) is absolutely continuous and f (r) ∈ X.When r ∈ N, p ∈ P, and X = L p(•) we will denote W r p(•) := X r .The following second type Jackson inequality holds: Theorem 1.3.Suppose that r, k, n ∈ N, p ∈ P, and f ∈ W r p(•) .Then the following inequality holds: One of the results of this paper is the following theorem, which contains an equivalence of Ω r and Peetre's K-functional. and where z := max {y ∈ Z : y ≤ z}, with constants depending only on k, p.
The rest of the present work is organized as follows.In Section 2 we give upper estimates for the operator norm of Steklov and Jackson operators.Section 3 contains the main properties of the modulus of smoothness Ω r and some integral operators.In Section 4 we give a transference result.In Section 5 we give the proof of the equivalence of the modulus of smoothness Ω r and Peetre's K-functional K r .Finally, Section 6 contains the proof of the refined Jackson inequality and other proofs.
In what follows, letters c, p, A, B, C, . . .will stand for certain positive constants and these will not change in different places.

Upper estimates for the operator norm of Steklov and Jackson operators
Theorem 2.1 We point out that (2.1) was obtained in [23] and (2.2) can be obtained in the same way. Let Let n ∈ N and be the Jackson operator (polynomial) where J 2,n is the Jackson kernel The Jackson kernel J 2,n satisfies the relations 3) The next theorem was obtained in [24].
Theorem 2.2.If p ∈ P and f ∈ L p(•) , then the sequence of Jackson operators where

Modulus of smoothness
By Theorem 1.1 we have For k ∈ N we define the modulus of smoothness of f ∈ L p(•) , p ∈ P, as From [28, (3.2) and Corollary 2] we have Lemma 3.1.Let p ∈ P, k ∈ N, and 0 ≤ v ≤ 1.Then and hold.

Transference result
Here we state a variant of the transference result obtained in [5, Theorem 14].Let p ∈ P, f ∈ L p(•) , G ∈ L p (•) , and G p (•) = 1.Note that the dual of L p(•) is L p (•) .For any ε > 0 there exists an h 1 ≤ 1 such that (see [5, (18)]).Everywhere in this work, this h 1 will be fixed.We define We define Peetre's K-functional , the class of functions continuous on T, then

RAMAZAN AKG ÜN
Note that for 0 < v ≤ 1, p ∈ P we know that Lemma 5.3.Let 0 < v, p ∈ P and f ∈ L p (•) .Then ) , we can use the generalized Minkowski integral inequality and Lemma 4.4 to obtain Remark 5.4.Note that the function R v f is absolutely continuous ( [24]) and differentiable a.e. on T.
Lemma 5.5.Let 0 < v ≤ 1, p ∈ P, and Proof.The proof is the same as the proof of Lemma 24 of [5].
Rev. Un.Mat.Argentina, Vol.60, No. 1 (2019) One can find by Lemma 4.4 and the generalized Minkowski integral inequality: Proof.The proof is the same as the proof of Lemma 26 of [5].
Proof of Theorem 1.4.Let r = 1, p ∈ P, and f ∈ L p(•) .Since , from Lemma 5.3 and (5.3) we find and taking infimum on g ∈ W 1 p(•) in the last inequality we get and the equivalence of ( Now we will consider the case r > 1.For r = 2, 3, . . .we consider the operator (see [2]) From the identity On the other hand, using Lemmas 5.7 and 5.3 recursively, For the reverse of the last inequality, when g ∈ W r p(•) , from Lemma 3.1: and hence (5.5) 6. Proofs of the results 6.1.Proof of the transference result.
Proof of Theorem 4.2.Let p ∈ P and f ∈ L p (•) .In this case, by Theorem 2.1.
On the other hand, for any ε, η > 0 and appropriately chosen Since ε, η > 0 are arbitrary we have This gives the required result.
≤ 72 max On the other hand, for any ε, η > 0 and appropriately chosen Since ε, η > 0 are arbitrary we obtain max and hence 1 3

Proof of Jackson's inequalities.
Proof of Theorem 1.2.Let n, r ∈ N.For g ∈ W r p(•) we have by Theorem 1.3 that .
Taking infimum in the last inequality with respect to g ∈ W r p(•) , one can get, by (5.5), At this stage, we will use the method given by Natanson and Timan [21] to obtain (1.10).By Corollary 1.5 we have, for ρ ≤ n, and the result (1.10) follows.
Proof of Theorem 1.3.Suppose that Θ n ∈ Π n , E n (f ) p(•) = f − Θ n p(•) , and β/2 is the constant term of Θ n , namely, We get We set u n ∈ Π n so that u n = Θ n − β/2.Then dt, r := max {n ∈ N : n ≤ x}, {r} := r− r , I is the identity operator, and (I − Φ t ) {r} is the fractional binomial series expansion of I − Φ t .