On approximations by trigonometric polynomials of classes of functions defined by moduli of smoothness

In this paper, we give a characterization of Nikol'ski\u{\i}-Besov type classes of functions, given by integral representations of moduli of smoothness, in terms of series over the moduli of smoothness. Also, necessary and sufficient conditions in terms of monotone or lacunary Fourier coefficients for a function to belong to a such a class are given. In order to prove our results, we make use of certain recent reverse Copson- and Leindler-type inequalities.


Introduction
Let 1 f ∈ L p [0, 2π], 1 < p < ∞, be a 2π-periodic function. We say that the function f has monotone Fourier coefficients if it has a cosine Fourier series with f (x) ∼ ∞ n=0 a n cos nx, a n ↓ 0.
We say that the function f has lacunary Fourier coefficients if a µ cos 2 µ x, a µ ≥ 0.
By ω k (f, t) p we denote the modulus of smoothness of order k in L p metrics of a function f ∈ L p , 1 < p < ∞: is the k-th order shift operator.
By E n (f ) p we denote the best approximation in L p metrics of a function f ∈ L p , 1 < p < ∞, by means of trigonometric polynomials whose degree is not greater than n − 1, i.e.
A more detailed approach to the classes N (p, θ, r, λ, ϕ) is given in [6] and [12] (see also [2]). In the paper, we give a characterization of N (p, θ, r, λ, ϕ) classes of functions in terms of series over their moduli of smoothness. Then we give the necessary and sufficient conditions in terms of monotone or lacunary Fourier coefficients for a function f ∈ L p [0, 2π] to belong to a class N (p, θ, r, λ, ϕ). In the process of proving the results, we make use of certain recent reverse l p -type inequalities [10], closely related to Copson's and Leindler's inequalities.
Finally, by making use of our results, we construct an example of a function having a lacunary Fourier series, which shows that N (p, θ, r, λ, ϕ) classes are properly embedded between the appropriate Nikol'skiȋ classes and Besov classes.

Auxiliary statements
In order to establish our results, we use the following lemmas.  (2) for 0 < p ≤ 1 the following equalities hold where constants C 1 , C 2 , C 3 and C 4 depend only on numbers α, λ and p, and do not depend on m, n as well as on the sequence {a ν } ∞ ν=1 . Proof of the lemma is given in [4, p. 308]. Lemmas 3.3 and 3.4 that follow state certain l p -type inequalities which are reversed to the ones given in Lemma 3.2 and closely related to Copson's and Leindler's inequalities (see, e.g., [3,7,8,14]).
(2) for 0 < p ≤ 1, n ≥ 4m the following equalities hold where constants C 1 , C 2 , C 3 and C 4 depend only on numbers α, λ and p, and do not depend on m, n as well as on the sequence {a ν } ∞ ν=1 . Proof of the lemma is given in [10].
Lemma 3.4. Let a ν ↓, α > 0, λ a real number, m and n positive integers. For 0 < p < ∞ the following inequalities hold where constants C 1 , C 2 , C 3 and C 4 depend only on numbers α, λ and p, and do not depend on m, n as well as on the sequence {a ν } ∞ ν=1 . The lemma is also proved in [10].
The following inequalities hold where constants C 1 and C 2 do not depend on n and f .
The lemma is proved in [11].
where constant C does not depend on n.
Proof of the lemma is given in [6].
The following inequalities hold , where constants C 2 and C 1 do not depend on f .
Proof of the lemma is due to Zygmund [16, vol. I, p. 326].
Corollary 3.1. Lemma 3.7 yields the following estimate , where constants C 2 and C 1 do not depend on n and f .

Proofs
Now we prove our results.
Proof of Theorem 2.1. Put We have [4, p. 55] and, taking into account properties of modulus of smoothness [15], which proves inequality (1). Now we suppose that inequality (1) holds. For δ ∈ (0, 1) we choose the positive integer n satisfying 1 n+1 < δ ≤ 1 n . Then, taking into consideration the estimates from above for I 1 and I 2 we have Hence Proof of Theorem 2.1 is completed.
Proof of Theorem 2.2. Theorem 2.1 implies that the condition f ∈ N (p, θ, r, λ, ϕ) is equivalent to the condition where constant C 1 does not depend on n. Lemma 3.5 yields that the last estimate is equivalent to the estimate where constant C 2 does not depend on n. Hence, if we denote the terms on the left-hand side of the inequality by J 1 , J 2 , J 3 and J 4 respectively, then condition f ∈ N (p, θ, r, λ, ϕ) is equivalent to the condition Now we estimate the terms J 1 , J 2 , J 3 and J 4 from below and above by means of expression taking part in the condition of the theorem.
First we estimate J 1 and J 2 from below. We have For k − r > 0, making use of Lemmas 3.2 and 3.3 we obtain In an analogous way, for rθ > 0 we get We estimate the term J 2 from above: For J 1 we have , and applying once more Lemmas 3.2 and 3.3 we obtain Put Then for taking into account that (k + 1)p − 2 ≥ 0 and a ν ↓ 0 we get Since k − r − λ > 0, we have

From (8) it follows that
This way, inequalities (5), (6), (7) and (9) yield We estimate A 2 in an analogous way: We estimate the series Let θ p ≤ 1. For given n we choose the positive integer N such that 2 N ≤ n + 1 < 2 N +1 . Then we have

From (10) it follows that
Now, estimates (16) and (15) imply This way we proved that condition (1) is equivalent to the condition of the theorem. Since condition (1) is equivalent to the condition f ∈ N (p, θ, r, λ, ϕ), proof of Theorem 2.2 is completed.
Proof of Theorem 2.3. Considering Lemma 3.6, condition f ∈ N (p, θ, r, λ, ϕ) is equivalent to the condition where constant C does not depend on n.
where constant C 43 does not depend on n. Put , we estimate J 1 and J 2 from below and above.
Let 0 < θ 2 ≤ 1. Using Lemma 3.1, changing the order os summation we get Therefrom, taking into consideration that rθ > 0 while computing the second sum we obtain Let 1 ≤ θ 2 < ∞ and 0 < ε < r. Applying Hölder inequality we have Computing the second sum we obtain This way, for 0 < θ < ∞ we have where constant C 48 does not depend on n. Now we estimate J 1 from below. Let 1 ≤ θ 2 < ∞. Making use of Lemma 3.1 we get Computing the second sum we get Let 0 < θ 2 ≤ 1 and ε > 0. Applying Hölder inequality we have where is θ 2 + 1 θ ′ = 1. The last estimate implies Changing the order of summation and then computing the second sum we obtain where constant C 52 does not depend on n.
Now we estimate J 2 from above. Taking into consideration that (r + λ)θ > 0, we have Remark 4.1. Notice that another way of proving Theorems 2.2 and 2.3 is presented in [12]. Our approach here is similar to that used in [1].