Estimates in direct inequalities for the Szász–Mirakyan operator

This paper deals with the approximation of continuous functions by the classical Szász–Mirakyan operator. We give new bounds for the constant in front of the second order Ditzian–Totik modulus of smoothness in direct inequalities. Asymptotic and non asymptotic results are stated. We use both analytical and probabilistic methods, the latter involving the representation of the operators in terms of the standard Poisson process. A smoothing technique based on a modification of the Steklov means is also applied.


Introduction
The classical Szász-Mirakyan operator S t , introduced independently in the 1940s by G. M. Mirakyan, J. Favard and O. Szász, associates with each number t ≥ 1, and each function f ∈ R [0,∞) , provided the series below is absolutely convergent, the function S t f := S t ( f ) defined by This work is partially supported by Research Project PGC2018-097621-B-I00. The second author is also supported by Junta de Andalucía Research Group FQM-0178. This paper deals with the approximation process of S t f (x) towards f (x), as t → ∞. In this regard, let us introduce some notations. The second order central difference of f ∈ R [0,∞) is defined as Ditzian-Totik modulus of smoothness of f with step-weight function ϕ(x) = √ x is given by Note that the operator S t interpolates at 0, and holds fixed every affine function. We denote by F the set of continuous non affine functions f ∈ R [0,∞) such that ω ϕ 2 ( f ; 1) < ∞. We assume that t ≥ 1 and denote by f A = sup{| f (x)| : x ∈ A}, A ⊆ [0, ∞).
In 1994, Totik [10] obtained the following characterization concerning the rates of uniform convergence for the Szász-Mirakyan operator S n : for some absolute positive constants K and K . The lower and the upper inequalities in (1) are called the converse and the direct inequalities, respectively. Once (1) is established, the subsequent natural question is to estimate the constants K and K . To the best of our knowledge, the only result in this direction was proved in [2], where it is shown that the direct inequality holds for K = 4. It seems that no specific value for K has been provided yet.
In this paper, we give the following estimates.

Theorem 1 We have
Then, Theorems 1 and 2 give us, respectively, non-asymptotic and asymptotic estimates of the constant K . As we will see in Sect. 5, the maximum value of |S t f (x) − f (x)| occurs when the product t x takes intermediate values, specifically, when t x ∈ [1,30]. This is the reason why the asymptotic estimate in Theorem 2 is better than that in Theorem 1.
It may be of interest to compare Theorems 1 and 2 with other known results in the literature referring to the classical Bernstein polynomials. Direct and converse inequalities for such polynomials, analogous to those in (1), were obtained by Ditzian and Ivanov [4], and Totik [9]. Different authors have obtained specific values for the corresponding constant K in the direct inequality. In this regard, Adell and Sangüesa [2] gave K = 4, Gavrea et al. [5] and Bustamante [3] provided K = 3, and finally, Pȃltȃnea [7] obtained K = 2.5, this being the best result up to the date and up to our knowledge. Note that Theorem 1 provides K = 2.43 for the Szász-Mirakyan operator. Finally, a similar result to Theorem 2 for the Bernstein polynomials was obtained in [1].
Two main different tools are used to prove Theorems 1 and 2. In the first place, the following probabilistic representation of S t . Let (N λ ) λ≥0 be the standard Poisson process. Since then S t f can be written as where E stands for mathematical expectation. Note that In the second place, we use a certain smooth approximant Q a h f of f ∈ F, built up by antisymmetrizing near the origin the classical Steklov means of f (see Sect. 4 for more details).
This paper is organized as follows. In the following section, we include some auxiliary results involving estimates of tail probabilities for the standard Poisson process, as well as other estimates in terms of the Ditzian-Totik modulus. Sections 3 and 4 contain estimates of |S t f (x) − f (x)| for small and large values of x, respectively. The last section is devoted to prove Theorems 1 and 2.

Auxiliary results
In this section, we gather some technical results. Specifically, Lemma 3 was already used in [2] and comes from Petrov [8, p. 52], so that we state it without proof. Lemma 4 is similar to Lemma 2.1 in [2], a proof of which is included here for the sake of completeness. Finally, Lemma 5 is a reformulation of Lemma 2.5.7 in [7], adapted to our setting.
From now on, · stands for the ceiling function.
Proof For any k ∈ N, it can be checked that where Since ϕ is increasing, we have, for r = 1, . . .
Thus, we have from (6) and (7), Therefore, the result follows after choosing k to be the smallest integer such that This and the fact that the function because the function y → The result follows from (8) and (9).
It turns out that the first and second derivatives of S t f can be written as and the easy to check inequality Proof Let be the affine function that interpolates f at the points 0 and 1/t, and let g := f − . Since S t = , it goes without saying that and, from (10), Let β be a random variable having the beta density ρ(θ) = 2(1 − θ), 0 ≤ θ ≤ 1. By (11)-(14), and the Taylor's formula, we have Apply Lemma 5 to g with a = 0 and b = 1/t to obtain Finally, from (13)-(16), using the trivial equality ω ϕ 2 (g; ·) = ω ϕ 2 ( f ; ·), we have and the proof is over.

Direct estimates far from the origin
We will apply a smoothing technique by using the classical second order Steklov mean associated with the function f , which, in probabilistic terms, can be written as where V 1 and V 2 are independent identically distributed random variables having the uniform distribution on [−1, 1] (see, for instance, [1] and [6]). Following the lines of the proof of Lemma 2 in [1], it can be seen that and Starting from a, t, x ∈ R, and assuming that we take and define the approximant Note that (19) and (20) imply that h ≤ ax, and that Q a h f is well defined. Moreover, Q a h f is twice differentiable except at the point ax, where it only has sided second derivatives. This implies that its first derivative Q a h f is absolutely continuous, thus allowing us to write, by Taylor's formula, with the obvious understanding that (Q a h f ) is not properly the second derivative of Q a h f .
Denote by 1 A the indicator function of the set A.

Proof of Theorem 1
In the first place, we prove the upper inequality in Theorem 1. Let us consider the following partition of the interval [0, ∞): where the λ i 's are given in Table 1.
We will prove the result for x ∈ [0, ∞), separately for each one of the aforesaid subintervals.
For i = 1, 2, . . . , 11, the function k i (λ) takes values less than 2.43 at the end points of the interval [λ i , λ i+1 ), as Table 3shows. In addition to that, k i (λ) is convex for λ = t x ∈ [λ i , λ i+1 ), since in that interval, one has that k i (λ) > 0. Indeed, it suffices to inspect Table  4 and check that the quantities between curly brackets are positive. For instance, if i = 6, By elementary calculus, one sees that k 12 (λ) decreases for λ = t x ∈ [30, ∞), and also that k 12 (30) = 2.40432 (rounded to six significant figures).
To prove the lower inequality, we proceed as in the proof of Theorem 2 in [1]. Let x ∈ (0, 1/t) and consider the function Observe that ω ϕ 2 ( f x ; 1/ √ t) = 1, as well as This implies that Hence, the lower inequality follows by letting x → 0. The proof is complete.
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