Abstract
In this paper we introduce a general class of integral operators that fix exponential functions, containing several recent modified operators of Gauss–Weierstrass, or Picard or moment type operators. Pointwise convergence theorems are studied, using a Korovkin-type theorem and a Voronovskaja-type formula is obtained.
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1 Introduction
The classical Bohman–Korovkin theorem (see [10, 22, 23]) is one of the pivotal results of approximation theory and several convergence theorems known in literature employ this basic tool. It states that a sequence of positive linear operators \(T_nf\) acting on the set of the continuous functions over a compact interval of the real line converges to the identity operator only if it converges on a finite number of test functions which form a so-called Chebyshev system. A complete treatment of the Korovkin theorem can be found in the monographies [2, 3].
In this respect, if a sequence of operators \(T_nf\) is such that \(T_n\varphi = \varphi \) for some continuous function \(\varphi ,\) then to obtain the convergence appears very simple, if the functions \(\varphi \) belong to a Chebyshev system. Thus, it is of interest to define sequences of operators that have this property. In literature the so-called King type operators, have this property, especially in case of discrete operators (see e.g. [21]). In this paper we define an entire class of positive linear integral operators which fix exponential functions. This kind of results were obtained by Agratini, Aral and Deniz [1], by Aral [6] and recently by Yilmaz, Uysal and Aral [26], by considering modifications of specific operators (Picard, Gauss–Weiertsrass and moment-type operators). All the operators considered in these papers are special case of our present theory. Our approach includes also all the integral operators having a compactly supported kernel. Thus it applies for example to spline-type operators [11, 24]. Related results can be found in [16, 18, 19] and [17].
For our operators, we apply the Gadjiev version of Bohman–Korovkin theorem (see [14, 15]) in case of unbounded domains, for functions belonging to certain weighted spaces of continuous functions. Setting for every \(n \in {\mathbb {N}}\) and \(a>0,\)
we define
where \(\{K_n\}\) is a family of non-negative functions (kernel) belonging to a suitable function space. Setting for \(a>0,\) \(\exp _a (x) := e^{ax}\) for \(x\in {\mathbb {R}},\) we show that \((T_n \exp _a)(x) = \exp (ax)\) and \((T_n \exp _{2a})(x) = \exp (2ax)\). Moreover we show that \(T_ne_j \rightarrow e_j\) where \(e_j(x) = x^j,\) \(j=0,1,2,\) so obtaining two uniform convergence theorems in weighted spaces of continuous functions. Then, using certain moduli of continuity we obtain certain quantitative estimates of the convergence and finally a Voronovskaja-type asymptotic formula. These kinds of asymptotic formulae are very useful also for applications, especially for discrete operators, like e.g. sampling-type operators (see, e.g., [11,12,13]. For integral operators of Mellin type see [7, 8]). The last section is devoted to several examples.
We recall here, that general approaches to convergence of integral operators was recently given in [9], and more recently in [20] and [25] in the frame of nonlinear operators.
2 Basic Notations
Let us denote by \({\mathbb {R}}\) the set of real numbers, and by \({\mathbb {N}}\) the set of the positive integers. By \(L^1({\mathbb {R}})\) we denote the space comprising all the Lebesgue integrable functions on \({\mathbb {R}}\) with respect to the Lebesgue measure, and by \(L^\infty ({\mathbb {R}})\) the space comprising all the essentially bounded functions on \({\mathbb {R}}.\) By \(C({\mathbb {R}})\) we denote the space of all the continuous functions defined on \({\mathbb {R}}.\) Finally, for \(r \in {\mathbb {N}},\) we will say that a function \(f: {\mathbb {R}} \rightarrow {\mathbb {R}}\) is locally of class \(C^r\) at a point \(x\in {\mathbb {R}}\) if there is a neighbourhood U of x such that f is \((r-1)\)-fold continuously differentiable in U and \(f^{(r)}(x)\) exists.
3 The Class of Integral Operators
For any \(a>0,\) we define the function \(\exp _a(t):= e^{at},\) for \(t \in {\mathbb {R}}.\)
We define now the function space
Let now \(\{K_n\}\) be a non negative kernel, that is, for every \(n \in {\mathbb {N}},\) \(K_n(t) \ge 0,\) for every \(t \in {\mathbb {R}},\) \(K_n \in L^1({\mathbb {R}}),\) and
In what follows we assume that \(K_n \in L_{\exp _a}({\mathbb {R}}),\) for every sufficiently large \(n \in {\mathbb {N}},\) namely for \(n\ge n_0,\) with \(n_0\in {\mathbb {N}}.\) We set, for \(n\ge n_0,\)
Note that from the assumption \(K_n \in L_{\exp _a}({\mathbb {R}}),\) we have \(A_{a,n}< + \infty ,\, A_{-a,n} < +\infty \) and if moreover for every \(n \in {\mathbb {N}},\) \(K_n\) is an even function, that is \(K_n(t) = K_n(-t),\) for every \(t \in {\mathbb {R}},\) one has easily \(A_{a,n} = A_{-a,n}.\) We introduce now the sequence of integral operators defined by
for every function f belonging to the domain \(D:= \bigcap _{n \ge n_0} \mathrm{dom}\,T_n,\) where \(\mathrm{dom}\,T_n\) denotes the set of all the Lebesgue measurable functions f such that \((T_n |f|)(x)\) is convergent for almost all \(x \in {\mathbb {R}}.\)
Note that if \(f \in L^\infty ({\mathbb {R}}),\) then \(f \in D.\) Moreover, the functions \(\exp _a\) and \(\exp _{2a}\) both belong to D. We have the following
Proposition 3.1
Under the established assumptions on the kernel \(\{K_n\},\) we have
Proof
We have
Analogously, we have
that is the assertion \(\square \)
Remark 3.2
In order to establish a connection with classical integral operators of convolution type of the form
for continuous and bounded functions \(f \in D\) we have
whenever
Indeed, under the above assumptions on the kernel \( \{K_n\},\) it is easy to show that \(\lim _{a \rightarrow 0^+} A_{a,n} = 1,\) and \(\lim _{a \rightarrow 0^+} \lambda _n(x) = x.\) Thus under the assumptions on the function f the assertion follows by the Lebesgue theorem of dominated convergence.
4 Pointwise and Uniform Convergence
In this section we will study the pointwise convergence of \(T_nf\) to f, where f belongs to a suitable weighted space of continuous functions, as introduced in [14], using a Korovkin-type theorem, established in [14] (see also [15]). In order to do that, we introduce the following constants
We call these constants the exponential moments of orders 1 and 2 respectively. At the same way, we refer to \(A_{-a,n}\) as the exponential moment of order 0. We introduce the following subspace of \(L_{\exp _a}({\mathbb {R}}),\) in which the above moments are well-defined
with c, d positive constants. If \(K_n \in L^*_{\exp _a}({\mathbb {R}}),\) then \(B_{-a,n}, \,C_{-a,n}\) exist finite.
Let us introduce now the test functions \(e_0(t) = 1,\,e_1(t) = t,\,e_2(t) = t^2,\,\,t \in {\mathbb {R}}.\) Obviously, \(e_j \in D,\) for \(j=0,1,2.\)
We have the following
Proposition 4.1
Let \(K_n \in L^*_{\exp _a}({\mathbb {R}}),\) for sufficiently large \(n \in {\mathbb {N}}.\) Then we have
and
Proof
We have
Next,
Finally,
The proof is completed \(\square \)
Corollary 4.2
Under the assumptions of Proposition 4.1, if moreover
and
then
Proof
It is an immediate consequence of Proposition 4.1\(\square \)
Using Proposition 4.1 and Corollary 4.2, we now give a uniform convergence result for the operators \((T_nf)\) when f belongs to suitable weighted spaces of continuous functions. The key tool is a Korovkin-type theorem proved in [14] (see also [15]).
Given a continuous, strictly increasing function \(\varphi : {\mathbb {R}} \rightarrow {\mathbb {R}},\) we define the function \(\rho (x) := 1 + \varphi ^2(x),\) and assume that \(\lim _{x \rightarrow \pm \infty } \rho (x) = +\infty .\)
We will consider two particular cases: \(\varphi _1(x) = x,\) and \(\varphi _2(x) = \exp _a(x),\) and we set \(\rho _1(x) := 1 + x^2\) and \(\rho _2(x) := 1 + \exp _{2a}(x).\)
Let us consider the spaces, for \(j=1,2\)
Note that if \(f \in C^0_{\rho _j}({\mathbb {R}}),\) then \(|f(x)| \le M_f \rho _j(x),\) for a suitable constant \(M_f >0\) and \(x \in {\mathbb {R}}.\) We define a norm on the space \(C^0_{\rho _j}({\mathbb {R}})\) on setting
We are ready to prove the main theorem of this section.
Theorem 4.3
Let \(j=1,2\) and let \(f \in C^0_{\rho _j}({\mathbb {R}}).\) Then, under the assumptions of Proposition 4.1 and Corollary 4.2, we have
Proof
First, consider the case \(j=1.\) The test functions \(e_j\) obviously belong to \(C^0_{\rho _1}({\mathbb {R}})\) and moreover
and the last term tends to 0 as \(n \rightarrow +\infty .\) Next,
and so
Analogously, one can see that
For \(j=2,\) taking into account of (4.2) and Proposition 3.1, we obtain again the convergence on the test functions \(\exp ^k(ax),\) for \(k=0,1,2.\) Applying Theorem 2 in [14] we obtain the assertion \(\square \)
5 Quantitative Estimates
For \(K_n \in L^*_{\exp _a}({\mathbb {R}}),\) we define the absolute exponential moment of order 1 of \(K_n\) as
In the spaces \(C^0_{\rho _j}({\mathbb {R}}),\) \(j=1,2\) we can define various modulus of continuity. We begin with an estimate of the convergence expressed by Theorem 4.3, in terms of the classical modulus of continuity \(\omega \) defined by
As it is well-known, for any uniformly continuous function f one has \(\lim _{\delta \rightarrow 0}\omega (f,\delta ) = 0.\) We have the following estimate
Theorem 5.1
Let \(K_{n}\in L^*_{\exp _a}({\mathbb {R}}),\) for sufficiently large values of \(n\in {\mathbb {N}},\) and let \(f\in C^0_{\rho _{j}} ( {\mathbb {R}}).\) Then for every \(\delta >0,\) we have
Proof
For every \(x\in {\mathbb {R}}\) and \(\delta >0\) we have
Thus we have to estimate only \(I_{1}.\) In order to do that, we use the following well-known property of \(\omega \) (see e.g. [4])
Setting \(\lambda :=\frac{\left| t-\frac{1}{a}\log A_{a,n}\right| }{ \delta },\) we have
Passing to norm, we get the desired result
\(\square \)
Corollary 5.2
Let the assumptions of Theorem 5.1 be satisfied, and (4.1) holds. If moreover there is \(\alpha >0\) such that \((n^\alpha {\widetilde{B}}_{-a,n})_n,\) and \((n^\alpha |\log A_{a,n}|)_n\) are bounded sequences, then there exists an absolute constant \(M>0\) such that
for sufficiently large values of \(n \in {\mathbb {N}}.\)
Proof
It is a consequence of Theorem 5.1 on setting \(\delta = n^{-\alpha } \,\,\) \(\square \)
Remark 5.3
Note that, under the assumptions of Corollary 5.2, if the function f satisfies a Lipschitz condition of order 1, and \(|A_{a,n}\,A_{-a,n} -1|= {{\mathcal {O}}}(n^{-\alpha })\) as \(n \rightarrow +\infty ,\) then
Now we state another estimate using a suitable weighted modulus of continuity. In order to do that, we introduce the exponential moment of order 4 of \(K_{n} \in L^*_{\exp _{a}}({\mathbb {R}})\) setting
In order to establish rate of convergence, we will use special kind of modulus of continuity \({\widetilde{\omega }}\) which is compatible with the space \(C_{\rho _{1}}^{0} ({\mathbb {R}}) \). This weighted modulus of continuity was first introduced in [27] for \(f\in C_{\rho _{1}}^{0} ({\mathbb {R}} _{0}^{+} ) \), then considered in [5] for \(f\in C_{\rho _{1}}^{0} ({\mathbb {R}}) \) as follows:
For any function \(f\in C_{\rho _{1}}^{0} ( {\mathbb {R}}),\) \(m\in {\mathbb {N}} \) and \(\lambda ,\delta \in {\mathbb {R}} ^{+},\) \({\widetilde{\omega }}\) has the following properties (see [5, 27]):
-
1.
\({\widetilde{\omega }}\left( f,\delta \right) \) is a monotonically increasing function of \(\delta ,\)
-
2.
\(\lim _{\delta \rightarrow 0^{+}}{\widetilde{\omega }}\left( f,\delta \right) =0,\)
-
3.
\({\widetilde{\omega }}\left( f,m\delta \right) \le m \widetilde{\omega }\left( f,\delta \right) ,\) for every \(m\in {\mathbb {N}},\)
-
4.
\({\widetilde{\omega }}\left( f,\lambda \delta \right) \le \left( 1+\lambda \right) {\widetilde{\omega }}\left( f,\delta \right) \) for every \(\lambda >0.\)
Theorem 5.4
Let \(K_n \in L^*_{\exp _a}\) be such that \(E_{-a,n}\) exists finite. For \(f\in C_{\rho _{1}}^{0} ( {\mathbb {R}}) \), there holds
provided that \(\left| a^{-1}\log A_{a,n}\right| \le \sqrt{C_{-a,n}}\) for sufficiently large \(n\in {\mathbb {N}}.\)
Proof
In view of Proposition 4.1, we have
For any \(\delta >0,\) properties of \({\widetilde{\omega }}\) enable us to write
In view of this observation, there holds
For \(J_{1},\) using the well-known inequality \(\left| a-b\right| ^{p}\le 2^{p-1}\left( \left| a\right| ^{p}+\left| b\right| ^{p}\right) ,\) \(a,b \in {\mathbb {R}},\) \(p\ge 1\), we have
By the Cauchy–Schwarz inequality for \(J_2,\) we have
Repeated application of the inequality \(\left| a-b\right| ^{p}\le 2^{p-1}\left( \left| a\right| ^{p}+\left| b\right| ^{p}\right) ,\) \(a,b \in {\mathbb {R}},\) \(p\ge 1\) yields
Using above inequality, we have
Collecting all inequalities we get
Choosing \(\delta =\left( C_{-a,n}\right) ^{1/2}\) with \(\left| a^{-1}\log A_{a,n}\right| \le \delta \) for sufficiently large n, dividing both sides by \(1+x^{2}\) and taking supremum over all x, we obtain
that is the assertion \(\square \)
Remark 5.5
The estimate of Theorem 5.4 gives an evaluation of the convergence in terms of the modulus of continuity with parameter \(\sqrt{C_{-a,n}},\) under the further assumptions that \(C_{-a,n} \rightarrow 0\) and \(|A_{a,n}\,A_{-a,n} - 1| \rightarrow 0\) as \(n \rightarrow +\infty .\) Thus one can obtain a corresponding corollary, as for Theorem 5.1 (see Corollary 5.2 and Remark 5.3).
6 An Asymptotic Formula
Here we establish two asymptotic formulae of Voronovskaya type for functions \(f \in C^0_{\rho _j}({\mathbb {R}}),\) \(j=1,2,\) which are locally of class \(C^2\) at a point x. These kinds of formulae give an exact evaluation of the order of pointwise convergence. We examine the case \(j=1.\)
Theorem 6.1
Let \(f \in C^0_{\rho _1}({\mathbb {R}})\) be locally of class \(C^2\) at a point \(x \in {\mathbb {R}}.\) Let \(K_n \in L^*_{\exp _a}({\mathbb {R}}),\) for sufficiently large values of \(n \in {\mathbb {N}},\) be a kernel satisfying the assumptions of Corollary 4.2. Assume that there exist \(\alpha >0\) such that the kernel \(\{K_n\}\) satisfies further the following conditions:
-
(i)
\(\lim _{n \rightarrow +\infty } n^\alpha (A_{a,n}\,A_{-a,n} - 1) = \ell _0,\) \(\lim _{n \rightarrow +\infty } n^\alpha B_{-a,n} = \ell _1,\) \(\lim _{n \rightarrow +\infty } n^\alpha C_{-a,n} = \ell _2\) and \(\lim _{n \rightarrow +\infty } n^\alpha \frac{1}{a}\log A_{a,n} = \ell _3,\) with \(\ell _j \in {\mathbb {R}},\) for \(j=0,1,2,3.\)
-
(ii)
For every \(\eta >0,\)
$$\begin{aligned} \lim _{n \rightarrow +\infty } n^\alpha \int _{|t| \ge \eta } e^{-at}\left( t - \frac{1}{a}\log A_{a,n}\right) ^2 K_n(t)\mathrm{d}t = 0. \end{aligned}$$
Then,
Proof
Since f has a polynomial growth of order 2, and locally of class \(C^2\) at the point x, using a local Taylor formula of the second order, we can write
where \(r_x(y)\) is a bounded function such that \(\lim _{y \rightarrow 0} r_x(y) = 0.\)
We write
As to \(I_2\) we have by assumptions that \(n^\alpha I_2 \rightarrow \ell _0 f(x)\) as \(n \rightarrow +\infty .\) Therefore we evaluate now the term \(I_1.\) Inserting (6.2) in \(I_1\), we can write
where
Now,
Thus we have to estimate the remainder term \(R_x.\)
Since \(\lim _{y \rightarrow 0}r_x(y) = 0,\) given an arbitrary \(\varepsilon >0,\) there is \(\delta \in ]0,1[\) such that \(|r_x(y)| < \varepsilon \) whenever \(|y| \le \delta .\) We take now an index \({\overline{n}}\) such that for \(n \ge {\overline{n}},\) one has \(a^{-1}|\log A_{a,n}| < \delta /2.\) Thus for \(|t| < \delta /2,\) we have also \(|t- a^{-1} \log A_{a,n}| < \delta .\) Thus
Writing
we have, by (6.4),
Therefore using the assumptions (i) we have, for a suitable absolute positive constant M,
As to the term \(R_2\), using the boundedness of \(r_x\) we can write
and by (ii) we obtain
This implies
that is \(|R_x | \rightarrow 0\) as \(n \rightarrow +\infty .\) Thus the theorem is completely proved \(\square \)
Theorem 5.4 works perfectly in several particular examples, as we will see in the next section. But in certain situations, as for example, the moment-type operators, formula (6.1) becomes
for a suitable constant \(\alpha \) for which all the assumptions are satisfied. In case of the moment-type operator, we have \(\alpha = 1.\) If we try to take \(\alpha =2\) some of the assumptions (i) of Theorem 5.4 are not satisfied. This result is however interesting, but it gives no exact information about the pointwise order of approximation at a point x. Therefore, we will formulate a slight generalization of the above theorem, in order to include also the case of moment-type operators, by changing assumptions (i). We have the following
Theorem 6.2
Let \(f \in C^0_{\rho _1}({\mathbb {R}})\) be locally of class \(C^2\) at a point \(x \in {\mathbb {R}}.\) Let \(K_n \in L^*_{\exp _a}({\mathbb {R}}),\) for sufficiently large values of \(n \in {\mathbb {N}},\) be a kernel satisfying the assumptions of Corollary 4.2. Assume that there exist \(\alpha >0\) such that the kernel \(\{K_n\}\) satisfies assumption (ii) of Theorem 5.4 and the following conditions:
-
(j)
\(\lim _{n \rightarrow +\infty } n^\alpha (A_{a,n}\,A_{-a,n} - 1) = \ell _0,\) \(\lim _{n \rightarrow +\infty } n^\alpha \left( B_{-a,n}-A_{-a,n}\frac{1}{a}\log A_{a,n}\right) = \lambda _1,\)
-
(jj)
\(\lim _{n \rightarrow +\infty } n^\alpha \left( C_{-a,n} + \frac{A_{-a,n}}{a^2} \log ^2 A_{a,n} - \frac{2}{a} B_{-a,n} \log A_{a,n}\right) = \lambda _2\)
Then,
Proof
The proof is clearly exactly the same. \(\square \)
7 Examples
In this section we discuss some examples of kernels \(\{K_n\}\) for which the theory developed can be applied.
-
(1)
The Gauss–Weierstrass kernel For \(n \in {\mathbb {N}},\) let us consider the kernel \(\{K_n\}\) with
$$\begin{aligned} K_n(t) = \sqrt{\frac{n}{\pi }} e^{-n t^2} \qquad (t \in {\mathbb {R}}). \end{aligned}$$This kernel is defined by even functions. First we evaluate the exponential moments of orders 0, 1, 2. In order to do that, we first calculate the coefficients \(A_{a,n} \) using a differentiation under the integral. By solving a simple first order linear differential equation, we obtain
$$\begin{aligned} A_{a,n} = A_{-a,n} = \sqrt{\frac{n}{\pi }} \int _{-\infty }^\infty e^{at} e^{-nt^2}\mathrm{d}t = e^{a^2/(4n)}. \end{aligned}$$Hence \(A_{a,n}\,A_{-a,n} = e^{a^2/(2n)}.\) Taking \(\alpha = 1,\) we obtain
$$\begin{aligned} \lim _{n \rightarrow + \infty } n (A_{a,n}\,A_{-a,n} - 1) = \lim _{n \rightarrow +\infty } n(e^{a^2/(2n)} - 1) = \frac{a^2}{2}. \end{aligned}$$Moreover, using partial integrations, one has
$$\begin{aligned} B_{-a,n} = -\frac{a}{2n} e^{a^2/4n},\,\,C_{-a,n} = \frac{1}{2n} A_{a,n} + \frac{a^2}{4n^2}A_{a,n}, \end{aligned}$$and so
$$\begin{aligned} \lim _{n \rightarrow +\infty } n B_{-a,n} = \frac{1}{2},\,\,\lim _{n \rightarrow +\infty }nC_{-a,n} = \frac{1}{2}. \end{aligned}$$Next, we have
$$\begin{aligned} \lim _{n \rightarrow +\infty } n \log A_{a,n} = \frac{a^2}{4}, \end{aligned}$$therefore we have also
$$\begin{aligned} \lim _{n \rightarrow +\infty } n \log ^2 A_{a,n} = 0. \end{aligned}$$Therefore, \(\ell _1 -\ell _3 = -3a/4,\,\, \ell _2 = 1/2.\) Now we prove that the Gauss–Weierstrass kernel satisfies also assumption (ii) of Theorem 5.4. Let \(\eta >0\) be fixed. Let us set
$$\begin{aligned} J:= n \int _{|t| \ge \eta } e^{-at}\left( t - \frac{1}{a}\log A_{a,n}\right) ^2 K_n(t) \mathrm{d}t. \end{aligned}$$Then
$$\begin{aligned} J&\le n \int _{|t| \ge \eta } e^{-at} t^2 K_n(t) \mathrm{d}t + n \left( \frac{1}{a^2}\log ^2 A_{a,n}\right) \int _{|t| \ge \eta } e^{-at} K_n(t) \mathrm{d}t \\&\quad + 2n \frac{1}{a}|\log A_{a,n}|\int _{|t| \ge \eta } e^{-at} |t| K_n(t) \mathrm{d}t =: J_1+J_2+J_3. \end{aligned}$$As to \(J_1\) we can write
$$\begin{aligned} J_1&= \frac{n \sqrt{n}}{\sqrt{\pi }}\int _{|t| \ge \eta } e^{-at} t^2 e^{-nt^2}\mathrm{d}t\\&= \frac{n \sqrt{n}}{\sqrt{\pi }}\left\{ \int _\eta ^\infty e^{-at} t^2 e^{-nt^2}\mathrm{d}t + \int _\eta ^\infty e^{at} t^2 e^{-nt^2}\mathrm{d}t \right\} \\&=: J_1^1 + J_1^2. \end{aligned}$$For \(J_1^1\) we have by a suitable substitution,
$$\begin{aligned} J^1_1 \le \frac{e^{-a\eta }}{\sqrt{\pi }}\int _{\sqrt{n} \eta }^\infty v^2 e^{-v^2}dv, \end{aligned}$$and so by the absolute continuity of the Lebesgue integral, we obtain
$$\begin{aligned} \lim _{n \rightarrow +\infty } J^1_1 = 0. \end{aligned}$$Analogously,
$$\begin{aligned} J_1^2 \le \frac{1}{\sqrt{\pi }}\int _{\sqrt{n}\eta }^\infty e^{av} v^2 e^{-v^2}dv, \end{aligned}$$and again, since the integrand in the right-hand side is Lebesgue integrable, we obtain \(J_1^2 \rightarrow 0\) as \(n \rightarrow +\infty .\) Next, we evaluate \(J_2.\) We have immediately
$$\begin{aligned} J_2 \le \frac{n}{a^2}\log ^2 A_{a,n} A_{-a,n} \rightarrow 0, \quad (n \rightarrow +\infty ). \end{aligned}$$Finally, we evaluate \(J_3.\) Using the same reasonings as for the estimate of \(J_1,\) we have
$$\begin{aligned} J_3&\le \frac{a \sqrt{n}}{2 \sqrt{\pi }}\left\{ e^{-a\eta }\int _\eta ^\infty t e^{-nt^2}\mathrm{d}t +\int _\eta ^\infty e^{at}t e^{-nt^2}\mathrm{d}t \right\} \\&=: J_3^1 + J_3^2. \end{aligned}$$As for \(J_1\) we obtain easily that \(J_3^1 \rightarrow 0,\,J_3^2 \rightarrow 0\) as \(n\rightarrow +\infty .\) Concluding, we obtain assumption (ii) of Theorem 5.4. Therefore all the assumptions introduced are satisfied with \(\alpha =1.\) The asymptotic formula of Theorem 5.4 reads
$$\begin{aligned} \lim _{n \rightarrow +\infty } n ((T_nf)(x) - f(x) ) = \frac{a^2}{2}f(x) - \frac{3a}{4} f'(x) + \frac{1}{4} f''(x), \end{aligned}$$which is a result of [26].
-
(2)
The Picard Kernel For \(n \in {\mathbb {N}},\) let us consider the kernel \(\{K_n\}\) with
$$\begin{aligned} K_n(t) = \frac{\sqrt{n}}{2}e^{-\sqrt{n}|t|} \quad (t \in {\mathbb {R}}). \end{aligned}$$Also this kernel is defined by even functions. Let us evaluate the exponential moments of order 0, 1, 2. First we have
$$\begin{aligned} A_{a,n} = A_{-a,n} = \frac{n}{n-a^2},\,\,\text{ and }\,\,A^2_{a,n} = \left( \frac{n}{n-a^2}\right) ^2 \end{aligned}$$and from sufficiently large values of n, \(A_{a,n}\) is positive. Moreover
$$\begin{aligned} \lim _{n \rightarrow +\infty } n(A_{a,n} A_{-a,n} - 1) = 2a^2,\,\,\lim _{n \rightarrow +\infty } n \log A_{a,n} = a^2. \end{aligned}$$Now, we evaluate the moment \(B_{-a,n}.\) Using elementary calculation, based on partial integration, we can see that
$$\begin{aligned} B_{-a,n} = -\frac{2an}{(n-a^2)^2},\,\, C_{-a,n} = \sqrt{n} \left( \frac{1}{(a + \sqrt{n})^3} + \frac{1}{(\sqrt{n}-a)^3}\right) . \end{aligned}$$Therefore,
$$\begin{aligned} \lim _{n \rightarrow +\infty } nB_{-a,n} = -2a,\,\,\lim _{n \rightarrow +\infty } n C_{-a,n} = 2. \end{aligned}$$Finally we check assumption (ii) of Theorem 5.4. We proceed as in the previous example. Let \(\eta >0\) be fixed and set
$$\begin{aligned} J:= n\int _{|t| \ge \eta }e^{-at} \left( t-\frac{1}{a}\log A_{a,n}\right) ^2 K_n(t) \mathrm{d}t. \end{aligned}$$We have
$$\begin{aligned} J&\le \frac{n\sqrt{n}}{2}\left\{ \int _{|t| \ge \eta } e^{-at} t^2 e^{-\sqrt{n}|t|}\mathrm{d}t + \frac{1}{a^2}\log ^2 \frac{n}{n-a^2}\int _{|t| \ge \eta } e^{-at} e^{-\sqrt{n}|t|}\mathrm{d}t\right. \\&\quad \left. + \frac{2}{a}\log \frac{n}{n-a^2}\int _{|t| \ge \eta } e^{-at} |t| e^{-\sqrt{n}|t|} \mathrm{d}t\right\} =: J_1 + J_2 + J_3. \end{aligned}$$Using now elementary calculations, based on suitable substitutions, it is easy to see that
$$\begin{aligned} J_1 \le \frac{e^{-a\eta }}{2}\int _{\sqrt{n} \eta }^\infty u^2 e^{-u/2} du + \frac{1}{2}\int _{\sqrt{n}\eta }^\infty u^2 e^{-u/2}du, \end{aligned}$$for every n such that \(1- a/\sqrt{n} > 1/2.\) Thus, \(J_1 \rightarrow 0\) as \(n \rightarrow +\infty .\) Next, let us consider \(J_2.\) We have easily
$$\begin{aligned} J_2 \le \frac{n}{a^2} \log ^2 \frac{n}{n-a^2} \,A_{-a,n}, \end{aligned}$$and so \(J_2 \rightarrow 0\) as \(n \rightarrow +\infty .\) Finally, as to \(J_3\) we have
$$\begin{aligned} J_3 \le \frac{e^{-a\eta }\sqrt{n}}{a}\log \frac{n}{n-a^2}\int _{\sqrt{n}\eta }^\infty v e^{-v}dv + \frac{\sqrt{n}}{a} \log \frac{n}{n-a^2}\int _{\sqrt{n}\eta }^\infty u e^{-u/2} du, \end{aligned}$$for every n such that \(1- a/\sqrt{n} > 1/2.\) Therefore, \(J_3 \rightarrow 0\) as \(n \rightarrow +\infty ,\) and assumption (ii) is satisfied. Therefore all the assumptions introduced are satisfied with \(\alpha =1.\) The asymptotic formula of Theorem 5.4 reads
$$\begin{aligned} \lim _{n \rightarrow +\infty } n ((T_nf)(x) - f(x) ) = 2a^2 f(x) - 3a f'(x) + f''(x) \end{aligned}$$which is a result of [6].
-
(3)
The moment kernel For \(n \in {\mathbb {N}}\) let us consider the kernel \(\{K_n\}\) with
$$\begin{aligned} K_n(t) = n \chi _{[0, 1/n]}(t) \qquad (t \in {\mathbb {R}}). \end{aligned}$$This kernel is not even, and the functions \(K_n\) have compact support. We calculate now the coefficients \(A_{a,n},\,A_{-a,n},\, B_{-a,n}\) and \(C_{-a,n}\) and some their properties. We have
$$\begin{aligned} A_{a,n} = n \int _0^{1/n} e^{at} \mathrm{d}t = \frac{n}{a}(e^{a/n} -1), \, A_{-a,n} = \frac{n}{a}(1 - e^{-a/n}), \end{aligned}$$and so
$$\begin{aligned} \lim _{n \rightarrow +\infty } A_{a,n} = \lim _{n \rightarrow +\infty } A_{-a,n} = 1, \quad \lim _{n \rightarrow +\infty }\frac{1}{a} n \log A_{a,n} = \frac{1}{2}. \end{aligned}$$Moreover,
$$\begin{aligned} \lim _{n \rightarrow +\infty } n (A_{a,n} A_{-a,n} - 1) = 0. \end{aligned}$$Next,
$$\begin{aligned} B_{-a,n} = -\frac{e^{-a/n}}{a} + \frac{n}{a^2} (1 - e^{-a/n}),\,\, C_{-a,n} = -\frac{e^{-a/n}}{an} + \frac{2}{a}B_{-a,n}, \end{aligned}$$and both tend to zero as \(n \rightarrow +\infty .\) Now,
$$\begin{aligned} \lim _{n \rightarrow +\infty } nB_{-a,n} = 1/2,\,\,\lim _{n \rightarrow +\infty } n C_{a,n} = 0. \end{aligned}$$Concerning assumption (ii) of Theorem 5.4, since the functions \(K_n\) have supports [0, 1/n] for any given \(\eta >0,\) one can find an integer \(n_0\) such that for any \(n \ge n_0\) the set \(\{t: |t| \ge \eta \}\) does not intersect the interval [0, 1/n], and so denoting \(H_{n,\eta } := \{t : |t| \ge \eta \} \cap [0, 1/n],\) we have
$$\begin{aligned} \int _{H_n} e^{-at} \left( t - \frac{1}{a} \log A_{a,n}\right) ^2 \mathrm{d}t = 0 \end{aligned}$$This implies that assumption (ii) is trivially satisfied. Therefore all the assumptions introduced are satisfied with \(\alpha =1.\) The asymptotic formula of Theorem 5.4 reads
$$\begin{aligned} \lim _{n \rightarrow +\infty } n ((T_nf)(x) - f(x) ) = 0. \end{aligned}$$This result is not fully satisfactory, due to the fact that we have no a precise order of pointwise approximation. Therefore we now employ Theorem 6.2 with \(\alpha = 2.\) In order to do that, we have to check the assumptions (j) and (jj). As to (j), we first calculate the limit
$$\begin{aligned} \lim _{n \rightarrow +\infty } n^2\left( B_{-a,n} - \frac{1}{a} \log A_{a,n}\right) , \end{aligned}$$which can be written as
$$\begin{aligned} \lim _{n \rightarrow +\infty }\frac{n^2}{a^2}\left( -a e^{-a/n} + a \frac{n}{a} (1 - e^{-a/n}) - a \log \left( \frac{n}{a}(e^{a/n} -1)\right) \right) . \end{aligned}$$This can be interpreted as a restriction of the limit
$$\begin{aligned} \lim _{x \rightarrow 0} \frac{a}{x^2}\left( -e^{-x} + \frac{1-e^{-x}}{x} - \log \left( \frac{e^x -1}{x}\right) \right) . \end{aligned}$$The above limit can be solved using elementary techniques, based on the L’Hospital rule and its value is given by \(-3a/8.\) Now, in order to calculate \(\lambda _1\) we write
$$\begin{aligned}&n^2\left( B_{-a,n} - A_{-a,n}\frac{1}{a}\log A_{a,n}\right) \\&\quad = n^2\left( B_{-a,n} - \frac{1}{a}\log A_{a,n}\right) + n^2 \frac{1}{a}\log A_{a,n}\,(1 - A_{-a,n}), \end{aligned}$$thus we consider only the limit
$$\begin{aligned} \lim _{n \rightarrow +\infty }n^2 \frac{1}{a}\log A_{a,n}\,(1 - A_{-a,n}). \end{aligned}$$Since as we have seen before \(\lim _{n \rightarrow +\infty } (n/a) \log A_{a,n} = 1/2,\) we have to calculate only \(\lim _{n \rightarrow +\infty } n( 1- A_{-a,n})\) and it is not difficult to see that its value is given by a/2. Thus, finally we obtain \(\lambda _1 = -a/8.\) Now we proceed to the calculation of \(\lambda _2.\) At first we calculate the limit
$$\begin{aligned} \lim _{n \rightarrow +\infty } n^2 C_{-a,n}. \end{aligned}$$Again this limit can be considered as a restriction of the limit
$$\begin{aligned} \lim _{x \rightarrow 0} \frac{1}{x^2}\left( -xe^{-x} - 2e^{-x} + 2 \frac{1-e^{-x}}{x}\right) , \end{aligned}$$and using an analogous elementary approach, its value is given by 1/3. Next, using the previous results, we have immediately
$$\begin{aligned} \lim _{n \rightarrow +\infty } n^2 \frac{A_{-a,n}}{a^2} \log ^2A_{a,n} = \frac{1}{4}, \end{aligned}$$and
$$\begin{aligned} \lim _{n \rightarrow +\infty } \frac{2}{a} n^2 B_{-a,n}\log A_{a,n} = 1/2. \end{aligned}$$Thus, we have \(\lambda _2 = 1/12.\) The last limit \(\ell _0 = \lim _{n \rightarrow +\infty } n^2 (A_{a,n}\,A_{-a,n} -1)\) can be obtained with the same reasonings as a restriction of the limit
$$\begin{aligned} \lim _{x \rightarrow 0} a^2 \frac{e^x + e^{-x} -2 -x^2}{x^4} \end{aligned}$$which is given by \(a^2/12.\) Concluding, the Voronovskaya formula (6.5) with \(\alpha =2\) for the moment type operator is given by (see also [26])
$$\begin{aligned} \lim _{n \rightarrow +\infty } n^2 [(T_nf)(x) - f(x)] = \frac{a^2}{12} f(x) -\frac{a}{8} f'(x) + \frac{1}{24} f''(x). \end{aligned}$$ -
(4)
A spline kernel For \(n \in {\mathbb {N}}\) let us consider the kernel \(\{K_n\}\) with
$$\begin{aligned} K_n(t) = n (1-|nt|)\chi _{[-1/n, 1/n]}(t)\qquad (t \in {\mathbb {R}}). \end{aligned}$$For this kernel we have, by elementary calculations
$$\begin{aligned} A_{a,n} = A_{-a,n} = n\int _{-1/n}^{1/n} e^{at} (1 - n|t|)\mathrm{d}t = \frac{2n^2}{a^2}(\cosh (a/n) -1), \end{aligned}$$and so
$$\begin{aligned} \lim _{n \rightarrow +\infty } A_{\pm a,n} = 1,\quad \lim _{n \rightarrow +\infty } n^2( A_{a,n} A_{-a,n} -1) = \frac{a^2}{6}, \end{aligned}$$and \(\lim _{n \rightarrow +\infty } n^2\log A_{a,n} = a^2/12.\) Next,
$$\begin{aligned} B_{-a,n} = n\int _{-1/n}^{1/n} t e^{-at}(1-n|t|) \mathrm{d}t = -\frac{2n}{a^2}\sinh (a/n) + \frac{4n^2}{a^3}(\cosh (a/n) -1), \end{aligned}$$and
$$\begin{aligned} \lim _{n \rightarrow +\infty } n^2 B_{-a,n} = -\frac{a}{6}. \end{aligned}$$As to \(C_{-a,n}\) we have
$$\begin{aligned} C_{-a,n}&= n\int _{-1/n}^{1/n} t^2 (1-n|t|) e^{-at}\mathrm{d}t \\&= \frac{2}{a^2}\cosh (a/n) - \frac{8n}{a^3} \sinh (a/n) + \frac{12 n^2}{a^4}(\cosh (a/n) -1), \end{aligned}$$and so
$$\begin{aligned} \lim _{n \rightarrow +\infty } n^2 C_{-a,n} = \frac{1}{6}. \end{aligned}$$Finally, since the functions \(K_n\) have compact supports \([-1/n, 1/n],\) we easily see, as in the previous example, that (ii) of Theorem 5.4 is satisfied. Thus, all the assumptions used in the previous theory are satisfied with \(\alpha = 2.\) The corresponding asymptotic formula is given by
$$\begin{aligned} \lim _{n \rightarrow +\infty } n^2 ((T_nf)(x) - f(x) ) = \frac{a^2}{6} f(x) - \frac{a}{4} f'(x) + \frac{1}{12} f''(x). \end{aligned}$$
Remark 7.1
The kernel of Example (4) is generated by the spline function of second order, defined by
The functions \(K_n\) are given by \(K_n(x) = n \beta _2 (nx),\) for \(x \in {\mathbb {R}}.\) The spline functions of order k are given by the formula (see e.g. [11, 24])
where for any real number r, \(r_+\) denotes its positive part. The theory may be applied also to any kernel generated by the spline \(\beta _k.\)
Remark 7.2
Note that the kernels of any of the previous examples satisfy the assumptions of Theorem 5.4, in particular the inequality
for every \(a>0\) and sufficiently large \(n \in {\mathbb {N}}.\) First, note that since in any of the above examples \(A_{a,n} \ge 1,\) we have \(\log A_{a,n} \ge 0.\) For the modified Gauss–Weierstrass kernel, one has easily
For the modified Picard kernel, one can employ the calculations of the limits in Example (2): since \(n C_{-a,n} \rightarrow 2\) as \(n \rightarrow +\infty ,\) and
for sufficiently large values of n we obtain the assertion. Similar arguments can be used for the remaining examples.
Remark 7.3
Further examples can be obtained using Phillips type operators and Post-Widder type operators which act on functions defined over the positive real axis, (see [16, 18, 19] in which some modified version are introduced).
Remark 7.4
Our approach may be applied also in case of general exponential functions of the form \(a^{-x}\) for \(a>1\) as studied in [17] for Baskakov-type operators that act on functions defined over the positive real axis.
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Carlo Bardaro and Ilaria Mantellini have been partially supported by the “Gruppo Nazionale per l’Analisi Matematica e Applicazioni (GNAMPA)” of the “Instituto di Alta Matematica (INDAM)” as well as by the project “Ricerca di Base 2019 of University of Perugia (title: Integrazione, Approssimazione Analisi non Lineare e loro Applicazioni)”
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Bardaro, C., Mantellini, I., Uysal, G. et al. A Class of Integral Operators that Fix Exponential Functions. Mediterr. J. Math. 18, 179 (2021). https://doi.org/10.1007/s00009-021-01819-0
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DOI: https://doi.org/10.1007/s00009-021-01819-0