Quantitative Approximation Properties for Iterates of Moment Operator

AbstractHere we state a quantitative approximation theorem by means of nets of certain modified Hadamard integrals, using iterates of moment type operators, for functions f defined over the positive real semi-axis ]0, +∞[, having Mellin derivatives. The main tool is a suitable K-functional which is compatible with the structure of the multiplicative group ]0, +∞[. Some numerical examples and graphical representations are illustrated.


Introduction
The theory of the so-called moment (or average) operators in approximation theory was first considered in the papers [4,31]. These operators have many applications to various fields of pure and applied mathematics: from Calculus of Variations (see [2,28]) to fractional calculus [6,20] and stochastic processes [14,18]. In the frame of approximation theory by sequences or nets of moment operators we quote the papers [3,13,27,29,30]. Recently in [6,7,8,9,10,11], the asymptotic behaviour was studied, obtaining Voronovskaya type formulae. In particular in [9], linear combinations of moment type operators are considered, which generate sequences of linear operators with a better order of pointwise and uniform approximation. In [11] special kinds of iterated moment operators were considered and it was remarked that linear combinations of iterated moment operators further improve the order of approximation.
In the present paper we point out the deep connections between of our iterated operators and the generalized Hadamard integrals, introduced firstly in [22], and widely used in Mellin analysis, especially in fractional calculus in Mellin transform setting (see e.g. [5,17,23]). This link enables us to obtain corresponding results concerning the approximation of a function f defined over the positive real semi-axis ]0, +∞[, by means of nets of slightly modified generalized Hadamard operators. The main result is Theorem 2, which gives a quantitative estimate of the pointwise and uniform convergence in certain classes of differentiable functions in the Mellin sense (see [5,16]). We use here a suitable Peetre K-functional, and a technique introduced in [21]. The last section considers some suitable examples which illustrate the results by means of graphical representations and numerical calculations. These calculations point out that without certain global regularity assumptions on the function f we cannot obtain the uniform convergence over ]0, +∞[. This gives the main motivation for the assumptions of Theorems 1 and 2.

Preliminaries
Let R + be the set of all positive real numbers endowed with the measure µ(E) = E dt t , where E is any (Lebesgue) measurable set. We denote R + 0 = R + ∪ {0}, and by N, N 0 we denote the set of positive integers and non negative integers respectively. We will denote by L p (µ), 1 ≤ p < +∞ the Lebesgue spaces of p-integrable functions with respect to the measure µ and by L ∞ (µ) the space of all the essentially bounded functions. We will denote by f p and f ∞ , the corresponding norms.
For m ∈ N 0 we denote by C m = C m (R + ) the space of all functions f : R + → R with continuous m-order derivative in R + . In particular C 0 will denote the space of all the continuous functions over R + . Moreover, we will say that f ∈ C m locally at the point s ∈ R + if there is a neighbourhood U s of the point s such that f is (m − 1)-times continuously differentiable in U s and the derivative of order m exists at the point s.
In the following we will say that a function f is log-uniformly continuous in R + if for every ε > 0 there exists η > 0 such that |f (s 1 ) − f (s 2 )| < ε whenever |log s 1 − log s 2 | < η. This notion was firstly given, in an equivalent form, in [25, page 7], in terms of the dilation operator. Note that there are functions uniformly continuous in the usual sense but not in the log-sense and conversely. For example the function f (u) = sin u is obviously uniformly continuous but not in the log-sense, while the function g(u) = sin(log u) is log-uniformly continuous but not in the usual sense. However, the two notions are equivalent on every bounded interval [a, b] with a > 0.
We denote by BC 0 the subspace of C 0 containing the log-uniformly continuous and bounded functions in R + and by BC m the space of all the functions such that the k-order derivatives, k = 1, . . . m, are in BC 0 . We recall that the Mellin derivatives of f are defined recursively in the following way [5,16,25] Θf (s) = sf (s), In particular, we have also the following Taylor formula of order m for a function f in BC m (see [10,25]), where h s (t) → 0 as t → 1. If f is bounded, we can assume that h s is also bounded.

The Moment Operator and Its Iterates
It is easy to see that K is homogeneous of degree zero, i.e. K(λs, λt) = K(s, t), for every λ, s, t > 0, and For every j ∈ N we define the logarithmic moment of order j ∈ N of K as and, using the change of variable z = t/s, we have easily For the absolute logarithmic moment of order j, namely The Mellin-Fejer kernel generated by K is given by, for w > 0, The corresponding moment operator is given by We obtain that the corresponding logarithmic moments of order j of the functions K w are Given the function K we define for every n ∈ N the iterated kernel of order n of K in the following way: for n = 2, and for n > 2, It turns out that the iterated functions K n exist and for the corresponding Mellin-Fejer kernels we have the following result Proof. For n = 2 we have and the general case follows by an easy induction. The formulae for the moments follow using Proposition 7 in [11].
The iterated moment operator of order n is defined as for every f for which the integral is well defined as a Lebesgue integral. The above integrals are strictly related with the so-called Hadamard-type integrals, introduced for the first time in [22], and then used in the frame of Mellin analysis, (see e.g. [5,17,23]). Here we give a slightly different version of such integrals.
For α > 0 and c > 0, we define the modified generalized Hadamard integral of order α of a function f : R + → R as: As recently proved in [5], if α > 1, the domain of the operator J α 0+,c is contained in the space Moreover, it contains the space As an immediate consequence of the definition of the modified Hadamard integrals, we have

Asymptotic Behaviour of Generalized Hadamard Integrals via Moment Operators
We are interested in the asymptotic behaviour of the generalized Hadamard integrals of order n as w → +∞, using the above relation and the properties of the iterated moment operators. At first, as an immediate consequence of (3.1) and Corollary 4 in [11] we obtain In particular, where the O-term depends on the point s. If f ∈ BC 1 then the estimate is uniform on R + .
Proof. The first assertion follows easily from (3.1) and Corollary 4 in [11]. For the second assertion, if f ∈ BC 1 , then |Θf (s)| ≤ C, for an absolute constant C > 0, and so the O-term is independent of s.
In the next section we will show by an example that the assumption f ∈ BC 1 is essential for the uniform convergence over the whole positive real semiaxis.
Our aim is to obtain a quantitative version of Theorem 1 using a Kfunctional suitable for the Mellin setting.
In general, the (modified) Peetre K-functional is defined in the following way. Let (X, · X ) be a Banach space, Y ⊂ X be a semi-normed subspace, with the seminorm |·| Y such that Y is complete under the norm f Y := f X +|f | Y . Then we put Taking X = BC 0 and Y = BC 1 , our K-functional is defined by for f ∈ BC 0 and ε ≥ 0. By Corollary 1 in [12], it follows that this functional is well defined in the sense that lim ε→0 K(f, ε) = 0. For a general treatment of K-functionals, see [1,15,19,24,26]. The intimate connections of a K-functional with approximation theory were developed in [15] and [19].
Associated with our K-functional, we introduce certain Lipschitz classes of functions, as follows. For α ∈ ]0, 1], we set In [12] the following estimate of the remainder R m (f ; s, t) := h s (t) log m t in the Taylor formula of order m is proved, which extends a result proved in [21] for the classical Taylor formula in a slightly different setting.
Lemma 1. For m ∈ N let f ∈ BC m and s, t ∈ R + . Then we have .
In particular, for m = 1 we have, for functions f ∈ BC 1 , We have the following Proof. We start with the estimate of the difference Using the Taylor formula of the first order, and taking into account that Therefore, For J 1 we have, by the notion of moments and Proposition 1 while for J 2 using Lemma 1 and a simple change of variable, Now, let us consider an arbitrary function g ∈ BC 2 . Then, Taking now the infimum over all functions g ∈ BC 2 , we get Thus, the assertion follows.
As a consequence of Theorem 2, for functions f ∈ BC 2 we have the following direct estimate Proof. If f ∈ BC 2 , then we can write by choosing g = Θf. Therefore the assertion easily follows.
More generally, we have the following further consequence of Theorem 2, whose proof is clear.

Examples and Some Numerical Evaluation
We give two examples to show the asymptotic behaviour of the generalized Hadamard integrals (of the second order); the first involving the function e s (see Figure 1) and the second involving the function sin(log s) (see Figure 2). Note that the function f (s) = e s does not belong to the space BC 1 since its Mellin derivative is not bounded over the whole positive real semi-axis. As we will show, we have a uniform convergence over the compact subsets of R + but we have not the uniform convergence over the positive semi-axis. The second function f (x) = sin(log x) belongs to BC 2 since its Mellin derivative Θ 2 f is given by − sin(log x) and so Θ 2 f ∞ = 1. The estimate of Corollary 1, gives now which gives a uniform estimate of the asymptotic formula. The generalized Hadamard integrals (J 2 0 + ,w f )(s) are computed numerically using a modified trapezoidal rule with points, taking into account the singularity at the point zero; that is, the first interval of the integration is further divided into 10 * w points. We use this method in the spirit of a mere example; this is not the best method for integrating such integrals. Indeed, one of the Referees kindly suggested the use of a Gauss-Legendre quadrature rule. We report the results found by the Referee in the last column of each table. For the graphics of Figures 1 and 2 we took a discretization of 0.01. Tables 1 and 2 report the values of the generalized Hadamard integrals (J 2 0 + ,w f )(s) evaluated  at the point s = 0.5 respectively for the function e x and sin(log x). We end the tabulations when the values start becoming worse, because of the numerical approximation problems. Finally we show that the Hadamard integrals for the function e x do not converge uniformly on ]0, ∞). To this end, we evaluate in Table 3 the difference |(J 2 0 + ,w )(w) − e w | as w → ∞ (as far as it is numerically possible to go!): In this case, the trapezoidal rule has been splitted into two parts: one for the interval [0, 1] and the other for the interval [1, w]; the number of points taken for both the numerical integrations is 100 * w.
For the sake of completeness, we report also the values showing the (slow) pointwise convergence in s = 640 (see Table 4).
we wish to thank the Referee who suggested us a Gauss-Legendre quadrature rule for the evaluation of the integrals. The authors have been partially supported by the Gruppo Nazionale Analisi Matematica, Probabilità e Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM), through the INdAM-GNAMPA Project 2014, and by the Department of Mathematics and Computer Science of University of Perugia.