Quantitative Estimates for Positive Linear Operators in terms of the Usual Second Modulus

and Applied Analysis 3 Proof. Assume that y ∈ (c, (c + d)/2], the case y ∈ [(c + d)/2, d) being similar. Set y = y + (y − c) = 2y − c ∈ [c, d]. Then, 󵄨󵄨󵄨󵄨f (y) 󵄨󵄨󵄨󵄨 = 1 2 󵄨󵄨󵄨󵄨f (c) − 2f (y) + f (y) − f (y) 󵄨󵄨󵄨󵄨 ≤ 1 2 ω 2 (f; d − c 2 ) + M 2 . (12) The proof is complete. For any 0 < ε ≤ (b − a)/3, denote by L ε (I) the set of functions inL(I) whose set of nodesN ε = {x i : i = −(m + 1), . . . , k + 1} satisfies ε ≤ min {x i − x i−1 : i = −m, . . . , k + 1} . (13) Lemma 3. Let g ∈ L ε (I), for some 0 < ε ≤ (b − a)/3. Then, ω 2 (g; τ) = τ max −m≤i≤k 󵄨󵄨󵄨󵄨δci 󵄨󵄨󵄨󵄨 , 0 ≤ τ ≤ ε. (14) Proof. Let 0 ≤ h ≤ τ ≤ ε and i = −m, . . . , k. Denote by s i (y) = (y − x i ) + . We claim that Δ 2 h s i (y) = (h − 󵄨󵄨󵄨󵄨y − xi 󵄨󵄨󵄨󵄨)+ =: qi (y) , y ∈ [a + h, b − h] . (15) Formula (15) is obvious if |y−x i | ≥ h; suppose that |y−x i | < h. If y ∈ (x i − h, x i ] ∩ [a + h, b − h], then Δ 2 h s i (y) = (y + h − x i ) + = (h − 󵄨󵄨󵄨󵄨y − xi 󵄨󵄨󵄨󵄨)+ , (16) whereas if y ∈ [x i , x i+h ) ∩ [a + h, b − h], then Δ 2 h s i (y) = −2 (y − x i ) + y + h − x i = (h − 󵄨󵄨󵄨󵄨y − xi 󵄨󵄨󵄨󵄨)+ , (17) thus showing claim (15). By virtue of (10), formula (15) is also true if we replace s i by any one of the functions s i (y) = (y − x i ) − or s i (y) = |y − x i |/2, y ∈ I. We therefore have from (8) and (15)


Introduction
Let  be a closed real interval with nonempty interior set ∘ .The usual second modulus of smoothness of a function  :  → R is defined as where Denote by M() the set of measurable functions  :  → R such that  2 (; ) < ∞,  ≥ 0. Many sequences (  ,  = 1, 2, . ..) of positive linear operators acting on M() allow for a probabilistic representation of the form (cf. [1]) () =  (  ()) ,  ∈ M () ,  ∈ ,  = 1, 2, . . ., where  stands for mathematical expectation and   () is an -valued random variable whose mean and standard deviation are given, respectively, by for some nonnegative function  :  → R. The condition   () =  is equivalent to say that   reproduces linear functions.
The aim of this paper is to give a general method to provide accurate estimates of the constants   (A(), ) satisfying the inequalities        () −  ()     ≤   (A () , )  2 (;  () √ ) ,  ∈ A () ,  ∈ ,  = 1, 2, . . ., (7) where A() is a certain subset of M().Such a problem is meaningful, because in specific examples the estimates of the constants in ( 6) and ( 7) may be quite different, mainly depending on two facts: the degree of smoothness of the functions in the set A() and the distance from the point  to the boundary of .In this way, we complete the general results shown by Pȃltȃnea [5].
The method is based on the approximation of any function  ∈ M() by a quasi interpolating piecewise linear function having an appropriate set of nodes.In doing this, special attention must be paid to the nodes near the endpoints of , if any.The main results are Theorems 6 and 7 stated in Section 3. In particular, Theorem 6 provides inequalities of form (7), where the upper bound consists of various terms involving  2 (; ⋅) evaluated at different lengths.Theorem 7 gives a closed form expression for the best constant in (7) when A() is a certain set of continuous piecewise linear functions.
As illustrative examples, we consider the Szàsz-Mirakyan operator (Section 4) and the Bernstein polynomials (Section 5).Although the kind of estimates is similar in both examples, the results take on a simpler form in the first case, because the interval of definition  = [0, ∞) has only one endpoint.In any case, both examples show that the size of the constants in front of  2 (; ⋅) heavily depends on the set of functions A() under consideration and on the distance from point  to boundary of .
We believe that the methods proposed in this paper could be applied to a wide class of positive linear operators, such as Baskakov operators, Stancu operators, and their -analogues, among others (see [9,10] and the references therein).To obtain accurate estimates of the constants involved in each case, we essentially need to compute second moments (see Theorem 8 in Section 3) and tail probabilities of the underlying random variables defining the operators under consideration (see Lemmas 9 and 11 in Sections 4 and 5, resp.).

Continuous Piecewise Linear Functions
Throughout this paper,  is a closed real interval of positive length and ∘  is the interior set of .If  = [, ], we denote by N a finite ordered set of nodes  =  −(+1) <  − < ⋅ ⋅ ⋅ <  −1 <  0 <  1 < ⋅ ⋅ ⋅ <   <  +1 = , for some ,  = 0, 1, . ... If  is an infinite interval, N could also be infinite.In such a case, the finite endpoint of , if any, is always in N. We denote by L() the set of continuous piecewise linear functions  :  → R whose set of nodes is N. Unless otherwise specified, we assume from now on that  = [, ].Given a sequence (  ,  ∈ Z), we denote by   =  +1 −   ,  ∈ Z.We set  + = max(0, ),  − = max(0, −) and denote by 1  the indicator function of the set .
Lemma 1.For any  ∈ L(), one has the representations where Proof.The first equality in (8) follows from the fact that the two functions involved have the same Radon-Nikodym derivative in ( −1 ,   ),  = −, . . ., +1, given by the constant   defined in (9).The second equality in (8) follows from the first one and the equalities The proof is complete.
The following auxiliary result is taken from [5, Lemma 2.5.7](see also [11]).We give a simple proof of it for the sake of completeness.Lemma 2. Let  :  → R be a function such that () = () = 0, for some ,  ∈  with  ≤ . ( The proof is complete. For any 0 <  ≤ ( − )/ thus showing that By assumption (13), We thus have from ( 18) This shows the converse inequality to (22) and completes the proof.
We close this section with the following auxiliary result concerning the symmetric functions For any  ∈ R, let ⌊⌋ and ⌈⌉ be the floor and the ceiling of , respectively; that is, Lemma 5. Let  and  be as in ( 27).Then, Proof.Let  ≥ 0.Then, Thanks to (30), the second inequality in Lemma 5 is equivalent to It is easily checked that These equalities imply (31), since ] is convex and  is linear in each interval [,  + 1],  = 0, 1, . ... The proof is complete.
Proof.Let  ∈ L , () with representation (8) and set of nodes as follows from Lemma 5.This completes the proof.
Theorem 8 gives an upper bound for  , () in terms of the variance of the random variable , which is easy to compute in many usual examples.Such an upper bound also suggests the choice (60)

Example 1: The Szàsz Operator
Let (  ,  ≥ 0) the standard Poisson process, that is, a stochastic process starting at the origin, having independent stationary increments such that Let  = 1, 2, . . .and  ≥ 0. Thanks to (61), the classical Szàsz-Mirakyan operator   can be written as where  ∈ M([0, ∞)).It is well known that Concerning the tail probabilities of the standard Poisson process, we give the following lemma.

Example 2: Bernstein Polynomials
Let  = 1, 2, . . .and let (  ) ≥1 be a sequence of independent identically distributed random variables having the uniform distribution on [0, 1].We consider the (uniform) empirical process (  ()/, 0 ≤  ≤ 1) defined as Observe that the random variable   () has the binomial law with parameters  and ; that is, Also observe that the paths of the empirical process are nondecreasing, since we have from (83) It is well known that For any function  : [0, 1] → R, the Bernstein polynomials of  can be written as In view of (86), we define The following auxiliary result will be needed.
Denote by and also This, together with (99) and (100), shows part (a).
We distinguish the following subcases.
To close the paper, let us mention some known results concerning the Bernstein polynomials.Gonska and Zhou [12] showed that there exists a constant 0 <  < 1 such that, for any 1/2 ≤  < 1, there exists () such that sup