A Class of Integral Operators that Fix Exponential Functions

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.


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 n f 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 n f is such that T n ϕ = ϕ for some continuous function ϕ, then to obtain the convergence appears very simple, if the functions ϕ 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 ∈ N and a > 0, 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 ∈ R, we show that (T n exp a )(x) = exp(ax) and (T n exp 2a )(x) = exp(2ax). Moreover we show that T n e j → 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.

The Class of Integral Operators
For any a > 0, we define the function exp a (t) := e at , for t ∈ R.
We define now the function space Let now {K n } be a non negative kernel, that is, for every n ∈ N, K n (t) ≥ 0, for every t ∈ R, K n ∈ L 1 (R), and In what follows we assume that K n ∈ L exp a (R), for every sufficiently large n ∈ N, namely for n ≥ n 0 , with n 0 ∈ N. We set, for n ≥ n 0 , Note that from the assumption K n ∈ L exp a (R), we have A a,n < +∞, A −a,n < +∞ and if moreover for every n ∈ N, K n is an even function, that is K n (t) = K n (−t), for every t ∈ 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 := n≥n0 dom T n , where 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 ∈ R.
Note that if f ∈ L ∞ (R), then f ∈ D. Moreover, the functions exp a and exp 2a both belong to D. We have the following Proof. We have Analogously, we have that is the assertion for continuous and bounded functions f ∈ D we have Indeed, under the above assumptions on the kernel {K n }, it is easy to show that lim a→0 + A a,n = 1, and lim a→0 + λ n (x) = x. Thus under the assumptions on the function f the assertion follows by the Lebesgue theorem of dominated convergence.

Pointwise and Uniform Convergence
In this section we will study the pointwise convergence of T n f 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 (R), in which the above moments are well-defined exp a (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 ∈ R. Obviously, e j ∈ D, for j = 0, 1, 2.
We have the following Proof. We have Next, Finally, The proof is completed and Proof. It is an immediate consequence of Proposition 4.1 Using Proposition 4.1 and Corollary 4.2, we now give a uniform convergence result for the operators (T n f ) when f belongs to suitable weighted spaces of continuous functions. The key tool is a Korovkin-type theorem proved in [14] (see also [15]).
We will consider two particular cases: ϕ 1 (x) = x, and ϕ 2 (x) = exp a (x), and we set ρ 1 (x) := 1 + x 2 and ρ 2 (x) := 1 + exp 2a (x). Let us consider the spaces, for j = 1, 2 We define a norm on the space C 0 ρj (R) on setting We are ready to prove the main theorem of this section.
Proof. First, consider the case j = 1. The test functions e j obviously belong to C 0 ρ1 (R) and moreover and the last term tends to 0 as n → +∞. Next, 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

Quantitative Estimates
For K n ∈ L * exp a (R), we define the absolute exponential moment of order 1 of K n as In the spaces C 0 ρj (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 ω defined by As it is well-known, for any uniformly continuous function f one has lim δ→0 ω(f, δ) = 0. We have the following estimate , for sufficiently large values of n ∈ N, and let f ∈ C 0 ρj (R). Then for every δ > 0, we have Proof. For every x ∈ R and δ > 0 we have Thus we have to estimate only I 1 . In order to do that, we use the following well-known property of ω (see e.g. [4]) .
Passing to norm, we get the desired result 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 ∈ L * exp a (R) setting In order to establish rate of convergence, we will use special kind of modulus of continuity ω which is compatible with the space C 0 ρ1 (R). This weighted modulus of continuity was first introduced in [27] for f ∈ C 0 ρ1 (R + 0 ), then considered in [5] for f ∈ C 0 ρ1 (R) as follows: For any function f ∈ C 0 ρ1 (R), m ∈ N and λ, δ ∈ R + , ω has the following properties (see [5,27]): 1. ω (f, δ) is a monotonically increasing function of δ, provided that a −1 log A a,n ≤ C −a,n for sufficiently large n ∈ N.
Proof. In view of Proposition 4.1, we have For any δ > 0, properties of ω enable us to write In view of this observation, there holds For J 1 , using the well-known inequality |a − b| By the Cauchy-Schwarz inequality for J 2 , we have
Collecting all inequalities we get Choosing δ = (C −a,n ) 1/2 with a −1 log A a,n ≤ δ for sufficiently large n, dividing both sides by 1 + x 2 and taking supremum over all x, we obtain that is the assertion

An Asymptotic Formula
Here we establish two asymptotic formulae of Voronovskaya type for functions f ∈ C 0 ρj (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. (i) lim n→+∞ n α (A a,n A −a,n − 1) = 0 , lim n→+∞ n α B −a,n = 1 , lim n→+∞ n α C −a,n = 2 and lim n→+∞ n α 1 a log A a,n = 3 , with j ∈ R, for j = 0, 1, 2, 3. 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→0 r x (y) = 0. We write As to I 2 we have by assumptions that n α I 2 → 0 f (x) as n → +∞. Therefore we evaluate now the term I 1 . Inserting (6.2) in I 1 , we can write where Thus we have to estimate the remainder term R x . Since lim y→0 r x (y) = 0, given an arbitrary ε > 0, there is δ ∈]0, 1[ such that |r x (y)| < ε whenever |y| ≤ δ. We take now an index n such that for n ≥ n, one has a −1 | log A a,n | < δ/2. Thus for |t| < δ/2, we have also |t − a −1 log A a,n | < δ. Thus As to the term R 2 , using the boundedness of r x we can write for a suitable constant α for which all the assumptions are satisfied. In case of the moment-type operator, we have α = 1. If we try to take α = 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 Proof. The proof is clearly exactly the same.

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 ∈ N, let us consider the kernel {K n } with 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 Hence A a,n A −a,n = e a 2 /(2n) . Taking α = 1, we obtain Moreover, using partial integrations, one has As to J 1 we can write 1 we have by a suitable substitution, and so by the absolute continuity of the Lebesgue integral, we obtain lim n→+∞ J 1 1 = 0. Analogously, and again, since the integrand in the right-hand side is Lebesgue integrable, we obtain J 2 1 → 0 as n → +∞. Next, we evaluate J 2 . We have immediately J 2 ≤ n a 2 log 2 A a,n A −a,n → 0, (n → +∞). Finally, we evaluate J 3 . Using the same reasonings as for the estimate of J 1 , we have e at te −nt 2 dt =: J 1 3 + J 2 3 . As for J 1 we obtain easily that J 1 3 → 0, J 2 3 → 0 as n → +∞. Concluding, we obtain assumption (ii) of Theorem 5.4. Therefore all the assumptions introduced are satisfied with α = 1. The asymptotic formula of Theorem 5.4 reads which is a result of [26].

(2) The Picard Kernel
For n ∈ N, let us consider the kernel {K n } with Also this kernel is defined by even functions. Let us evaluate the exponential moments of order 0, 1, 2. First we have Now, we evaluate the moment B −a,n . Using elementary calculation, based on partial integration, we can see that Therefore, Finally we check assumption (ii) of Theorem 5.4. We proceed as in the previous example. Let η > 0 be fixed and set We have Using now elementary calculations, based on suitable substitutions, it is easy to see that for every n such that 1 − a/ √ n > 1/2. Thus, J 1 → 0 as n → +∞. Next, let us consider J 2 . We have easily J 2 ≤ n a 2 log 2 n n − a 2 A −a,n , and so J 2 → 0 as n → +∞. Finally, as to J 3 we have for every n such that 1 − a/ √ n > 1/2. Therefore, J 3 → 0 as n → +∞, and assumption (ii) is satisfied. Therefore all the assumptions introduced are satisfied with α = 1. The asymptotic formula of Theorem 5.4 reads which is a result of [6]. 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 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 α = 2. In order to do that, we have to check the assumptions (j) and (jj). As to (j), we first calculate the limit lim n→+∞ n 2 B −a,n − 1 a log A a,n , which can be written as lim n→+∞ n 2 a 2 −ae −a/n + a n a (1 − e −a/n ) − a log n a (e a/n − 1) .