On the Jackson-Type Inequality for the Best S-Approximations of Functions by Trigonometric Polynomials

We find the sharp constant in the Jackson-type inequality between the value of the best approximation of functions by trigonometric polynomials and moduli of continuity of m-th order in the spaces S, 1 6 p < ∞. In the particular case we obtain one result which in a certain sense generalizes the result obtained by L.V. Taykov for m = 1 in the space L2 for the arbitrary moduli of continuity ofm-th order (m ∈ N). Introduction Trigonometric polynomials are the object of the study for a long time. The significant results in the approximation theory were obtained by Jackson. He proved that for an arbitrary 2π-periodic continuous function the following inequality holds en−1(f)C 6 Kω(f ; 1 n ) , where en−1(f)C = inf { ∥f − Tn−1∥C : Tn−1 ∈ Tn−1 } is the value of the best approximation of function f by the subspace Tn−1 of trigonometric polynomials of degree n− 1 in the continuous metric; ω(f ; t) = sup { ∥f(·+ h)− f(·)∥C : |h| 6 t } is the modulus of continuity of function f , and K is a constant which doesn’t depend on n and f . This inequality and analogous relations are known in the approximation theory as the Jackson-type inequalities. In approximation theory it is of importance to find the smallest constant from all possible ones in the Jackson-type inequalities. Such constants are called the sharp constants. The questions of the obtaining the Jackson-type inequalities in case of approximation by trigonometric polynomials in the uniform and integral metrics were studied by many mathematicians, see for example the articles [1]-[25]. A.I. Stepanets in [26] introduced the normed spaces S (1 6 p < ∞) of the integrable functions f(x) having the period 2π for which ∥f∥Sp df = {∑ k∈Z |f̂(k)| }1/p <∞ , where f̂(k) = (2π)−1/2 π ∫


Introduction
Trigonometric polynomials are the object of the study for a long time. The significant results in the approximation theory were obtained by Jackson. He proved that for an arbitrary 2π-periodic continuous function the following inequality holds is the value of the best approximation of function f by the subspace T n−1 of trigonometric polynomials of degree n − 1 in the continuous metric; is the modulus of continuity of function f , and K is a constant which doesn't depend on n and f . This inequality and analogous relations are known in the approximation theory as the Jackson-type inequalities. In approximation theory it is of importance to find the smallest constant from all possible ones in the Jackson-type inequalities. Such constants are called the sharp constants. The questions of the obtaining the Jackson-type inequalities in case of approximation by trigonometric polynomials in the uniform and integral metrics were studied by many mathematicians, see for example the articles [1]- [25].
A.I. Stepanets in [26] introduced the normed spaces S p (1 p < ∞) of the integrable functions f (x) having the period 2π for which are the Fourier coefficients of the function f (x) on the trigonometric system (2π) −1/2 e ikx , k ∈ Z. It was proved that the spaces S p (1 p < ∞) have the substantial properties of the Hilbert spaces, i.e. the minimal property of the partial Fourier sums. If is the value of the best approximation of function f (x) ∈ S p by the subspace T n−1 of trigonometric polynomials of degree n − 1 in the metric of the space S p then where is the partial sum of the Fourier series A.I. Stepanets stated in [26] that for p = 2 it is hold the equality Let is a modulus of continuity of order m of the function is a finite difference of order m of the function f (x) at the point x with the step h. If X = L p (1 p < ∞) then the value ω m (f, t) Lp is the known integral modulus of continuity [27]. In case of X = S p the modulus of continuity ω m (f, t) S p was introduced in the article [28]. Let Ψ(k) and β(k) df = β k (k ∈ N) are the constrictions on N of the arbitrary functions Ψ(x) and β(x) defined on the half-segment [1, ∞). Let's suppose that the series is the Fourier series of some summable function which we denote by f Ψ β (x) according to [29]. The and β(k) = r then we use notation L r (S p ); L r 2 ≡ L r (S 2 ). A lot of articles are devoted to solving problems of approximation theory in the spaces S p (1 p < ∞). For example, in the articles [30]- [36] were studied the approximation properties of trigonometric system and were solved several problems on obtaining the Jackson-type inequalities and finding the sharp constants for the fixed values of m, n, t and p, that is the values 28 IJARM Volume 8 We assume that the ratio 0/0 is equal to zero. Let's define the following notation In the spaces S p the values of the type (4) were studied by A.I. Stepanets, A.S. Serduk [28] , S.B. Vakarchuk [33] ( . The analogous to (4) values were considered by B.P. Voycehivskiy [34], S.B.Vakarchuk and A.N.Shchitov [35].
In the article [36] were Sharp constant in the Jackson-type inequality for the best S p -approximation of functions by trigonometric polynomials is found in the next theorem. Theorem 1. For the arbitrary numbers n, m ∈ N, 0 < τ 3π 4n and 1 p < ∞ the following equality holds Proof. Using following we can write the Fourier coefficients (1) in the form

International Journal of Advanced Research in Mathematics Vol. 8 29
Then the relation (2) can be written in the next form where It is known [29] that Fourier coefficients of the functions f (x) and f Ψ β (x) are connected by the formula From (6) and (8) we have In the article [28] it was shown that for an arbitrary function Using (6) and (10) we write From the (9) it immediately follows the equation Then using the last equation from the (11) we have Using (7) we can write Applying the Holder's inequality to the right part of the (13), using (2), (12), definition of the modulus of continuity of the m-th order and the decreasing character of the function Ψ(x), from the (13) Integrating the relation (14) by the variable h over the limits from 0 to τ we have In the [3] it was obtained the relation Dividing the inequality (15) by τ and taking into account (7) and (16) Therefore from (17) we get From (18) for an arbitrary 0 < τ 3π 4n we have the upper bound To obtain the lower bound we consider the function which belongs to the class L Ψ β (S p ). Based on the (7) we have International Journal of Advanced Research in Mathematics Vol. 8 For (Ψ, β)-derivative of the function f f Ψ β (x) = √ 2/πΨ −1 (n) cos(nx + β n π/2) due to (11) and definition of the modulus of continuity of order m for 0 < t π n we can write From the (21) for 0 < t π n we obtain Then taking into account (20) and (22) we get From the upper bound (19) and lower bound (23) it follows the equality (5). Theorem 1 is proved.
If Ψ(n) = n −r , r ∈ Z + , then from the theorem 1 it follows the next result. The result of the theorem 2 in a certain sense generalizes for the arbitrary modulus of continuity of m-th order (m ∈ N) one result obtained by L.V. Taykov for the case m = 1 in the article [3].

Conclusions
For the functions from the class L Ψ β (S p ) (1 p < ∞) the sharp constant in the Jackson-type inequality between the value of the best approximation e n−1 (f ) S p of functions by trigonometric polynomials and moduli of continuity of m-th order ω m (f Ψ β , t) S p in the spaces S p has been found. From the obtained result it follows the statement which in a certain sense generalizes for the arbitrary modulus of continuity of m-th order ω m (f (r) , t) L 2 (m ∈ N) the result obtained by L.V. Taykov for m = 1 in the space L 2 .