Towards the best constant in front of the Ditzian-Totik modulus of smoothness

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].

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.

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].

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]).

The second one is the following lemma proved in Păltănea [11], Lemma 2.5.7 or in Bustamante [16].

With these notations, we enunciate our first main result.

In the case \(\sigma=1\) and \(I=\mathbb {R}\), Theorem 3.1 takes on a simpler form.

With the notations given in (36) and (37), we state the following lemma.

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).