Special Moduli of Continuity and the Constant in the Jackson–Stechkin Theorem

We consider a special 2k-order modulus of continuity W2k(f,h) of 2π-periodic continuous functions and prove an analog of the Bernstein–Nikolsky–Stechkin inequality for trigonometric polynomials in terms of W2k. We simplify the main construction from the paper by Foucart et al. (Constr. Approx. 29(2), 157–179, 2009) and give new upper estimates of the Jackson–Stechkin constants. The inequality W2k(f,h)≤3∥f∥∞ and the Bernstein–Nikolsky–Stechkin type estimate imply the Jackson–Stechkin theorem with nearly optimal constant for approximation by periodic splines.


Introduction
The paper is devoted to the question of equivalence of two types of direct theorems in approximation theory: (a) the case of smooth functions (Favard estimates); (b) the case of arbitrary continuous functions (Jackson-Stechkin estimates).
We show that the Jackson-Stechkin inequality with the optimal constants follows from the Favard inequality (see the proof of Theorem 2 and (5.3)). The main tool in the proof of this statement is the function W 2k , measuring the smoothness of an integrable periodic function. Modulus W 2k is the special case of the generalized modulus of smoothness introduced by H. Shapiro [2,15]. This characteristic is more delicate than the standard modulus of continuity of order 2k. The function W 2k allows us to obtain asymptotically sharp results for the approximation by Favard-type operators. For example, we obtain the Jackson-Stechkin inequality for the periodic splines with constants close to optimal.
The following two facts play a key role here: 1. Uniform (in k) boundedness of operators W 2k : 2. The Bernstein-Nikolsky-Stechkin inequality in terms of W 2k .
The paper is organized as follows. In the second section, we introduce notation. In the third section, we consider the smooth characteristic W 2k and prove the uniform boundedness of W 2k (Lemma 1). The technical details of the proof can be found in the Appendix. Section 4 is devoted to the analog of the classical Bernstein-Nikolsky-Stechkin estimate in terms of W 2k (Theorem 1). The next important result in the paper is Theorem 2, which gives a simple and general proof of the Jackson-Stechkin theorem. We improve and simplify the main constructions from [9]. In the fifth section, we introduce Favard-type operators and show that Favard-type operators give Jackson-Stechkin theorems with almost optimal constants. That result is a consequence of the sharp inequality for the trigonometric approximation. We will show that to prove Jackson-Stechkin theorems with almost optimal constants, it is sufficient to obtain a Favard-type inequality (Theorem 3). Theorem 4 is devoted to approximation by periodic splines. Finally, we give in Theorems 5 and 6 the classical almost sharp variants of Theorems 1 and 3.

Notation
Let I denote either a one-dimensional torus T = [−π, π) = R/(2πZ) or the real line R = (−∞, ∞), and let L(I) be the space of integrable functions f : I → R with the norm f L(I) = I |f (t)| dt. The space of continuous 2π-periodic functions with the norm is denoted by C(T). In this paper, we are interested in the approximation of a real continuous function f ∈ C(T) by trigonometric polynomials τ ∈ T n of degree n: By * we denote the convolution operation in L(R) (see [16,Chap. 1,Sect. 1]): and by the periodic convolution operation in L(T) (see [ We will use the well-known periodization method (see [16,Chap. 7, Sect. 2, (2.1)]), which for a given f ∈ L(R) provides the 2π -periodic function f from L(T) by the formula For a nonnegative f ∈ L(R), the inequality (2.2) changes to the equality For example, the 2π-periodization of χ h is given by the formula and consider the convolution squares To prove equalities (2.6), it is sufficient to apply (2.4) and properties of periodic convolution (see [4, Part 3, Sect. 3.1]). Notably, the convolution f g of the functions f, g from L(T) belongs to L(T), and for the Fourier coefficients of the convolution we have the following identity: where we use the standard notation for Fourier coefficients of f ∈ L(T): The last equality in (2.6) follows from the general fact about periodization (see [ for each h > 0.

Special Modulus of Continuity
For fixed h > 0 and k ∈ N, consider the following operator from C(T) to C(T): where is a central difference of order 2k with the step t. The following representation is valid (cf. [9,Sect. 3]): Here For a 2π-periodic function f ∈ C(T), this gives the following representation: Note that for the positive numbers a j = a j (k), j = 1, . . . , k, we have The formulas (2.6), (3.6) and the first equality in (3.7) imply that T Λ k,h (x) dx = 1 for k = 1, 2, 3, . . . .
We will show that Λ k,h L(T) < 2 for all h > 0 and k ∈ N.
For h > 0 and f ∈ C(T), let The definitions (3.8), (3.9), and (3.1) imply that First, consider the case k = 1. In this case, we have (see (2.5), (2.6)) From this and from (2.7), we obtain Hence, Lemma 1 in the case k = 1 is proved. Let k ∈ N, k ≥ 2. It is sufficient to consider the case h = 1. In this case, Λ k := Λ k,1 is an even, piecewise linear function (see Fig. 1) with vertices at the points The inequalities (see Appendix, Lemma A) imply (see (3.8), (3.10), and (2.2)) that Now Lemma 1 is completely proved.
Let e j (x) := exp(ij x), c j (x) := cos(j x). It is easily seen that Lemma 2 For α ∈ (1, 2], we have the following inequality: Set s := πα/2. We have The main result in this section is the following analog of the Bernstein-Nikolsky-Stechkin inequality.
In particular, It is clear that Theorem 1 is sharp. We have the equality in (4.4) for τ = c n . Theorem 1 implies Bernstein's inequality (4.1) for even derivatives. This follows from Lemma 3.
Proof of Theorem 1 Theorem 1 follows from Lemmas 4, 5 below. Specifically, if and Proof We have the following formula:
After the change of variable, we may assume that u, s ∈ (0, h], h ∈ (0, π]. Consider the case k ≥ 2. First, reduce the value of F 2 by omitting the positive quantity t 2 u 2 − 4 sin 2 (tu/2). Then, let To get (4.5) for k ≥ 2, it is sufficient to show that (4.6) It is sufficient to prove (4.6) for the symmetric function F * 2 defined by To prove (4.6), it is sufficient to prove positivity of the following one-dimensional integral: (max(0, a − π), a/2). Consequently, the function

The function ϕ(u)/sin(u) is increasing, and the function ϕ(a − u)/sin(a − u) is decreasing on the interval
is increasing on (max(0, a − π), a/2) from 0 to 1, and the function has exactly one zero on the interval (max(0, a − π), a/2). The functions sin(u) × sin(a − u), φ h (u)φ h (a − u) are increasing and positive on (max(0, a − π), a/2). These facts and the inequality F * 2 (a/2, a/2) > 0 imply that it is sufficient to consider only the case k = 2 and to prove that the following integral is positive: Furthermore, it is sufficient to prove that I (a) := I π (a) > 0 (0 < a < 2π). The proof of this inequality can be found in the Appendix (see Sect. A.2), where a special simple case k = 1 is also considered.
In the proof of Lemma 3, we will use the following Lemma 6.
Theorem 2 Let f ∈ C(T) and α > 1, k, n ∈ N. Then Proof For arbitrary f ∈ C(T), the representation (5.2) is valid. Using a subadditive property for E n−1 (f ), we obtain Therefore, it is sufficient to estimate the best L-approximation of the smooth functions Λ j k,h . This was done in [9]. Specifically, Lemma 4.2 from [9] contains the result, which reads (in our notation) as follows: Since α > 1 and K 2j ≤ 4/π , for f ∈ C(T) we have the following: Here, we use the well-known expansion for the secant function (cf. [5, pp. 561-562, (6), (8)]).

Favard-Type Operators
Consider a family F := {F n,k : n, k ∈ N} of operators with the properties where the constant 0 < C F < ∞ does not depend on g, k, n. We will call F n,k ∈ F a Favard-type operator.
Theorem 3 Let f ∈ C(T). If F n,k is a Favard-type operator and τ * ∈ T n−1 is the best uniform approximation of f , then .
Proof Suppose that and let .

Theorem 1 and Lemma 1 imply that
Thus,
Therefore, we obtain the following Jackson-Stechkin theorem for periodic splines.

Two Results for the Classical Modulus of Continuity ω r
First, we improve the main result from [9] (see [9, Theorem 2.1]).
Theorem 5 For f ∈ C(T), n, r ∈ N, α > 1, In particular, Proof The proof follows from Theorem 1, keeping in mind the inequalities It is clear that We first show that for arbitrary k ∈ N, we have inequalities By (A.4), these inequalities turn into (A.1) if δ = 0. Further, we will show that (A.5) implies (A.2) if δ = 1/2. In order to prove (A.5), let Observe that The following equality: Let the expression in the square brackets be denoted by A i,k (δ). After simplification, it becomes By change of variables ν = j + 1, j = ν − 1 in S 2 (δ), we obtain Therefore, and (A.3) follows. In order to derive (A.2), we note that the first inequality in (A.1) implies From this and (A.5), we get (A.2). Lemma A is proved.

Lemma B
.
One can prove (A.7) by induction on j = k − i − 1 for fixed k: A.2 Proof of Lemma 5. Computations First, we consider the case k = 1 in Lemma 5.