1 Introduction
Let
I be a closed real interval with nonempty interior set
I̊. A function
\(\sigma:I\to[0,\infty)\) is called a weight function if
\(\sigma(y) >0\),
\(y\in\mathring{I}\). The usual second order differences of a function
\(f:I\to \mathbb {R}\) are defined as
$$ \Delta_{h}^{2} f(y)=f(y+h)-2f(y)+f(y-h), \quad [y-h,y+h] \subseteq I, h\geq0. $$
Recall (
cf. [
1]) that the Ditzian-Totik modulus of smoothness of
f with weight function
σ is defined by
$$ \omega_{\sigma}^{2} (f;\delta)=\sup\bigl\{ \bigl\vert \Delta_{h\sigma(y)} f(y)\bigr\vert : 0\leq h\leq\delta, \bigl[y-h\sigma(y),y+h \sigma(y)\bigr]\subseteq I\bigr\} , \quad \delta\geq0. $$
If
\(\sigma\equiv1\), we simply denote by
\(\omega^{2} (f;\cdot)=\omega _{\sigma}^{2} (f;\cdot)\) the usual second modulus of smoothness of
f. Also, we denote by
\(\mathcal{C}(I)\) the set of continuous functions
\(f:I\to \mathbb {R}\) such that
\(0<\omega_{\sigma}^{2} (f;\delta)<\infty\),
\(\delta>0\).
It is well known (see, for instance, [
2‐
7], and [
8]) that many sequences
\((L_{n}, n=1,2,\ldots)\) of positive linear operators acting on
\(\mathcal{C}(I)\) satisfy direct and converse inequalities of the form
$$ K_{1} \omega_{\sigma}^{2} \biggl(f; \frac{1}{\sqrt{n}} \biggr)\leq\sup_{x\in I} \bigl\vert L_{n} f(x)-f(x)\bigr\vert \leq K_{2} \omega_{\sigma}^{2} \biggl(f;\frac{1}{\sqrt {n}} \biggr),\quad n=1,2,\ldots, $$
(1)
where
\(f\in\mathcal{C}(I)\),
\(K_{1}\) and
\(K_{2}\) are absolute constants, and
σ is an appropriate weight function depending on the operators under consideration. From a probabilistic perspective, the weight
σ can be understood as follows. Let
\(n=1,2,\ldots\) and
\(x\in I\), and suppose that we have the representation
$$ L_{n}f(x)=Ef\bigl(Y_{n}(x)\bigr), \quad f\in \mathcal{C}(I), $$
where
E stands for mathematical expectation and
\(Y_{n}(x)\) is an
I-valued random variable whose mean and standard deviation are, respectively, given by
$$ E\bigl(Y_{n}(x)\bigr)=x, \qquad \sqrt{E \bigl(Y_{n}(x)-x\bigr)^{2}}=\frac{\sigma(x)}{\sqrt{n}}. $$
(2)
In such a case, we can write
$$ L_{n}f(x)=E f \biggl( x+ \frac{\sigma(x)}{\sqrt{n}} Z_{n}(x) \biggr),\quad Z_{n}(x)=\frac{Y_{n}(x)-x}{\sigma(x)/ \sqrt{n}}. $$
Since the standard deviation of
\(Z_{n}(x)\) equals 1, it seems natural to choose in (
1) the weight function
σ defined in (
2).
Several authors have obtained estimates of the upper constant
\(K_{2}\) in (
1) for the ordinary second modulus of smoothness,
i.e., for
\(\sigma\equiv1\). For instance, with regard to the Bernstein polynomials, Gonska [
9] showed that
\(1\leq K_{2} \leq 3.25\), Păltănea [
10] obtained
\(K_{2}=1.094\), and finally this last author closed the problem in [
11] by showing that
\(K_{2}=1\) is the best possible constant. For the Weierstrass operator, Adell and Sangüesa [
12] gave
\(K_{2}=1.385\). Finally, for a certain class of Bernstein-Durrmeyer operators preserving linear functions, we refer the reader to Gonska and Păltănea [
13].
For the Ditzian-Totik modulus in strict sense,
i.e., for nonconstant
σ, some estimates are also available. In this respect, Adell and Sangüesa [
14] showed that
\(K_{2}=4\) for the Szàsz operators and for the Bernstein polynomials. For such polynomials, the aforementioned estimate was improved by Gavrea
et al. [
15] and by Bustamante [
16], who obtained
\(K_{2}=3\), and finally by Păltănea [
11], who showed that
\(K_{2}=2.5\). Referring to noncentered operators, that is, operators for which the first equality in (
2) is not fulfilled, we mention the estimates for both
\(K_{1}\) and
\(K_{2}\) with regard to gamma operators proved in [
17].
Once it is known that a sequence
\((L_{n}, n=1,2,\ldots)\) satisfies (
1), a natural question is to ask for the uniform constants
$$ \sup \biggl\{ \frac{|L_{n} f(x)-f(x)|}{\omega_{\sigma}^{2} ( f; \frac {1}{\sqrt{n}} )}: f\in\mathcal{A}(I), n=1,2,\ldots,\, x\in I \biggr\} =K\bigl(\mathcal{A}(I)\bigr), $$
(3)
as well as for the local constants
$$ \sup \biggl\{ \frac{|L_{n} f(x)-f(x)|}{\omega_{\sigma}^{2} ( f; \frac {1}{\sqrt{n}} )}: f\in\mathcal{A}(I) \biggr\} =K \bigl(\mathcal {A}(I), n, x\bigr), \quad n=1,2,\ldots,\, x\in I, $$
(4)
where
\(\mathcal{A}(I)\) is a certain subset of
\(\mathcal{C}(I)\). Such questions are meaningful, because in specific examples, the estimates for the constants in (
3) and (
4) may be quite different, mainly depending on the degree of smoothness of the functions in the set
\(\mathcal{A}(I)\) and on the distance from
x to the boundary of
I (see Section
5).
The aim of this paper is to give a general method to provide accurate estimates of the constants in (
3) and (
4) when
\(I=\mathbb {R}\) or
\(I=[0,\infty)\). In this last case, the main assumption is that the weight function
σ is concave and satisfies a simple boundary condition at the origin (see (
9) in Section
2), whereas for
\(I=\mathbb {R}\),
σ is assumed to be constant. In view of the probabilistic meaning of
σ described in (
2), such assumptions do not seem to be very restrictive and are fulfilled in the usual examples. The method relies upon the approximation of any function
\(f\in\mathcal{C}(I)\) by an interpolating continuous piecewise linear function having an appropriate set of nodes, depending on the weight
σ.
The main results are Theorems
3.1 and
3.2, stated in Section
3. To keep the paper in a moderate size, we only consider two illustrative examples. The first one is the classical Weierstrass operator, involving the usual second modulus of smoothness (see Corollary
4.1). In this case, we are able to obtain the exact constants in (
3) and (
4) when the set
\(\mathcal{A}(\mathbb {R})\) is either the set of convex functions or a certain set of continuous piecewise linear functions. The second example refers to the Szàsz-Mirakyan operators (Theorem
5.3). In this case, we give different upper estimates of the aforementioned constants, heavily depending on the set of functions under consideration and on the kind of convergence we are interested in, namely, pointwise convergence or uniform convergence. Both examples are connected in the sense that, roughly speaking, the upper estimates for Szàsz-Mirakyan operators are, asymptotically, the same as those for the Weierstrass operator. This is due to the central limit theorem satisfied by the standard Poisson process, which can be used to represent Szàsz-Mirakyan operators.
2 Continuous piecewise linear functions
If \(I=\mathbb {R}\), we fix \(x\in \mathbb {R}\) and denote by \(\mathcal{N}\) an ordered set of nodes \(\{x_{i}, i\in \mathbb {Z}\}\) with \(x_{0}=x\). If \(I=[0,\infty)\), we fix \(x>0\) and denote by \(\mathcal{N}\) an ordered set of nodes \(\{x_{i}, i\geq-(m+1)\}\) such that \(0=x_{-(m+1)}<\cdots <x_{-1}<x_{0}=x\), for some \(m=0,1,2,\ldots\) . Also, we denote by \(\mathcal {L}(I)\) the set of continuous piecewise linear functions \(g:I\to \mathbb {R}\) whose set of nodes is \(\mathcal{N}\).
Given a sequence
\((c_{i}, i\in \mathbb {Z})\), we denote by
\(\delta c_{i}= c_{i+1}-c_{i}\),
\(i\in \mathbb {Z}\). We set
\(y_{+}=\max(0,y)\),
\(y_{-}=\max (0,-y)\), and denote by
\(1_{A}\) the indicator function of the set
A. For the sake of concreteness, we enunciate the following two lemmas for
\(I=[0,\infty)\), although both of them are also true for
\(I=\mathbb {R}\). We start with the following auxiliary result taken from [
18].
From now on, we make the following assumptions with respect to the weight function
σ. If
\(I=\mathbb {R}\), we assume that
\(\sigma \equiv1\), whereas if
\(I=[0,\infty)\), we assume that
σ is concave (and therefore nondecreasing) and satisfies the boundary condition
$$ \lim_{y\to0} \frac{y}{\sigma(y)}=0. $$
(9)
Assumption (
9) seems to be essential to guarantee a direct inequality (the upper bound in (
1)). Actually, for a weight function
σ not satisfying (
9), it has been constructed in Section 4 of [
14] a sequence
\((L_{n}, n=1,2,\ldots)\) of positive linear operators not satisfying the upper inequality in (
1). On the other hand, the concavity of
σ readily implies that the function
\(r(y)=y/\sigma(y)\),
\(y>0\), is continuous and strictly increasing. Thus, for any
\(\varepsilon>0\), there is a unique number
\(a_{\varepsilon}\) such that
$$ a_{\varepsilon}= \varepsilon\sigma(a_{\varepsilon})>0. $$
(10)
Let
\(\varepsilon>0\). We construct the set of nodes
\(\mathcal {N}_{\varepsilon}\) as follows. If
\(I=\mathbb {R}\), we fix
\(x\in \mathbb {R}\) and define
\(\mathcal{N}_{\varepsilon}=\{x_{i}, i\in \mathbb {Z}\}\) as
$$ x_{i}=x+i\varepsilon, \quad i\in \mathbb {Z}. $$
(11)
If
\(I=[0,\infty)\), we define the new concave weight function
$$ \sigma_{\varepsilon}(y)= \min \biggl( \frac{y}{\varepsilon},\sigma (y) \biggr), \quad y \geq0. $$
We fix
\(x>0\) and define
\(\mathcal{N}_{\varepsilon}=\{x_{i}, i\geq-(m+1)\} \), for some
\(m=0,1,\ldots\) , as follows. We start from the point
\(x_{0}=x\), move to the right by choosing
\(x_{i+1}-x_{i}=\varepsilon\sigma _{\varepsilon}(x_{i+1})\),
\(i=0,1,2,\ldots\) , then move to the left by
\(x_{i+1}-x_{i}=\varepsilon\sigma_{\varepsilon}(x_{i+1})\),
\(i=0,\ldots ,-m\), and lastly setting
\(x_{-(m+1)}=0\). In other words,
$$ x_{-(m+1)}=0,\qquad x_{0}=x, \qquad x_{i+1}-x_{i}=\varepsilon\sigma _{\varepsilon}(x_{i+1}), \quad i\geq-(m+1). $$
(12)
It is easy to check that
\(x_{-m}\) is the unique node in the interval
\((0,a_{\varepsilon}]\). On the other hand, the weight
\(\sigma_{\varepsilon}\) is very appropriate to simplify notations near the origin. For instance, we always have
\(y-h\sigma_{\varepsilon}(y)\geq0\),
\(y\geq0\),
\(0\leq h\leq\varepsilon\). Finally, we mention that the procedure to build up the set
\(\mathcal{N}_{\varepsilon}\) defined in (
12) is close in spirit to the so-called ‘canonical sequence’ in Păltănea [
11], Section 2.5.1 (see also Gonska and Tachev [
19,
20] and Bustamante [
16]).
To close this section, we give the following two auxiliary results. The first one is concerned with the symmetric function
$$ \psi(y)=\frac{1}{2}\vert y\vert +\sum _{i=1}^{\infty}\bigl(\vert y\vert -i\bigr)_{+}, \quad y \in \mathbb {R}. $$
(13)
Also, denote by
\(\lfloor x \rfloor\) and
\(\lceil x \rceil\) the floor and the ceiling of
\(x\in \mathbb {R}\), respectively, that is,
$$ \lfloor x \rfloor=\sup\{k\in \mathbb {Z}: k\leq x\},\qquad \lceil x \rceil=\inf\{k \in \mathbb {Z}: k\geq x\}. $$
The second one is the following lemma proved in Păltănea [
11], Lemma 2.5.7 or in Bustamante [
16].
5 The Szàsz-Mirakyan operator
In this section, we will apply Theorem
3.1 to the classical Szàsz-Mirakyan operators
\((L_{n}, n=1,2,\ldots)\). From a probabilistic viewpoint, such operators can be represented as follows. Let
\((N_{\lambda}, \lambda\geq0)\) be the standard Poisson process,
i.e., a stochastic process starting at the origin, having independent stationary increments and nondecreasing paths such that
$$ P(N_{\lambda}=k)=e^{-\lambda}\frac{\lambda^{k}}{k!},\quad k=0,1,\ldots ,\, \lambda\geq0. $$
(33)
Let
\(n=1,2,\ldots\) and
\(x\geq0\). Thanks to (
33), the Szàsz-Mirakyan operator
\(L_{n}\) can be written as
$$ L_{n}f(x)=\sum_{k=0}^{\infty}f \biggl( \frac{k}{n} \biggr) e^{-nx} \frac {(nx)^{k}}{k!}=Ef \biggl( \frac{N_{nx}}{n} \biggr), $$
(34)
where
\(f\in\mathcal{C}([0,\infty))\). It is well known that
$$ E \biggl( \frac{N_{nx}}{n} \biggr)=x, \qquad E \biggl( \frac {N_{nx}}{n}-x \biggr)^{2}=\frac{x}{n}. $$
(35)
Accordingly, we choose in this case (recall (
9), (
10), and (
12), as well as the subsequent comments)
$$ \sigma_{\varepsilon}(y)=\min \biggl( \frac{y}{\varepsilon},\sigma (y) \biggr),\qquad \sigma(y)=\sqrt{y},\quad y\geq0; \qquad \varepsilon = \frac{1}{\sqrt{n}}, \qquad a_{\varepsilon}=\frac{1}{n}. $$
(36)
As follows from (
12), the set of nodes
\(\mathcal{N}_{\varepsilon}=\{x_{i}, i\geq-(m+1)\}\), for
\(\varepsilon=1/\sqrt{n}\), is given by
$$ x_{-(m+1)}=0, \qquad x_{0}=x, \qquad x_{i+1}-x_{i}=\sqrt{\frac {x_{i+1}}{n}},\quad i\geq-m, $$
(37)
\(x_{-m}\) being the unique node in the interval
\((0,1/n]\). In order to apply Theorem
3.1 to Szàsz-Mirakyan operators, we need to estimate the quantities
\(\delta_{\varepsilon}\) and
\(Eg_{\varepsilon}(N_{nx}/n)\), for
\(\varepsilon=1/\sqrt{n}\). In this regard, the following two auxiliary results will be useful.
With the notations given in (
36) and (
37), we state the following lemma.
Denote
$$ K_{n}(x)=Eg_{1/\sqrt{n}} \biggl( \frac{N_{nx}}{n} \biggr),\quad n=1,2,\ldots,\, x>0, $$
(47)
where
\(g_{\varepsilon}\) is defined in (
17). For the Szàsz-Mirakyan operator defined in (
34), we enunciate the following result.
As mentioned in the Introduction, Theorem
5.3 illustrates that the estimates of the general constants in (
3) and (
4) may be quite different. Such estimates mainly depend on two facts: the set of functions under consideration (parts (a)-(c) in Theorem
5.3), and the kind of estimate we are interested in, namely, pointwise estimate or uniform estimate (see equations (
48) and (
49), respectively).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors read and approved the final manuscript.