On Some Approximation Processes Generated by Integrated Means on Noncompact Real Intervals

The purpose of the present paper is to carry out a detailed study of a sequence of positive linear operators acting on continuous function spaces on an arbitrary real interval and constructed by means of (Borel) integrated means with respect to two families of probability Borel measures on the underlying interval and a positive real parameter. The study is mainly focused on their approximation properties in weighted spaces of continuous functions with respect to wide classes of weights. Pointwise estimates as well as weighted norm estimates are also established. In the final section a weighted asymptotic formula is obtained.


Introduction
In the papers [11,12,14] the authors introduced and studied a wide class of positive linear operators acting on spaces of continuous functions defined on a general convex compact subset of a locally convex space.Their construction is carried out in terms of (Borel) integrated means with respect to two families of probability Borel measures on the underlying domain and a positive real parameter.In the finite dimensional setting and for a special choice of the involved parameters, these operators turn into the classical Kantorovich operators and this was the reason why the authors referred to them as generalized Kantorovich operators.Furthermore, this class of operators extends the one of Bernstein-Schnabl operators, widely studied, e.g., in [1,2] and the references therein, and, together with them, they share a nonmarginal relevance in facing, within such general domains, some approximation problems for continuous functions as well as for the solutions of special classes of initial-boundary value differential problems.
Very recently, in the paper [3] we used similar (Bochner) integrated means in Banach space settings in order to achieve some representation/approximation formulae for strongly continuous operator semigroups acting on such spaces.The method we employed led us to consider the following sequence of positive linear operators defined by where J is an arbitrary interval, x ∈ J, n ≥ 1, r ≥ 0, μ x and μ n are probability Borel measures on J and f is a continuous real-valued function on J with at most quadratic growth.In the case where J is a compact interval, these operators are just the generalized Kantorovich operators we have mentioned previously.Moreover, for r = 0, they turn into the Bernstein-Schnabl operators on noncompact intervals already studied in [9].Actually, in the paper [3] the operators C n , n ≥ 1, have an ancillary role and we limited ourselves to investigate very few approximation properties of them.In the present paper we carry out a more detailed study of such operators because they reveal to have an interest on their own in approximating wide classes of real-valued functions defined on noncompact intervals.Moreover, according to the particular examples we exhibit in Examples 2, the operators C n , n ≥ 1, generalize, by means of an unifying approach, several well known approximation processes on noncompact intervals, such as Szász-Mirakjan operators, Baskakov operators, Post-Widder operators, and Gauss-Weierstrass operators.
We mainly focus in studying their approximation properties in weighted function spaces of continuous functions on J with respect to wide classes of weights.We also establish pointwise estimates for uniformly continuous bounded functions as well as with respect to weighted norms.
Due to the generality of the parameters involved in the definition, the operators C n , n ≥ 1, can be also used for approximating p-fold integrable functions (1 ≤ p < +∞) and we reserve such possible development in a forthcoming paper.
In the final section we establish a weighed asymptotic formula, which could be possibly used in studying some classes of evolution equations on

Notations and Preliminary Results
Throughout the paper we shall denote by J an arbitrary noncompact real interval with endpoints r 1 := inf J ∈ R ∪ {−∞} and r 2 := sup J ∈ R ∪ {+∞}.
Let F(J, R) be the space of all real valued functions defined on J.As usual we shall denote by C(J) (resp., C b (J)) the space of all real valued continuous (resp., continuous and bounded) functions on J.The space C b (J), endowed with the natural (pointwise) order and the sup-norm • ∞ , is a Banach lattice.We shall also consider the (closed) subspaces of C b (J) Observe that a function f ∈ C(J) belongs to C 0 (J) if for every ε > 0 there exists a compact subset K of J such that |f (x)| ≤ ε for every x ∈ J \ K. Moreover we shall consider the space K(J) of all real valued continuous functions f : We also recall that the symbol UC b (J) (resp., UC 2 b (J)) stands for the space of all uniformly continuous and bounded functions on J (resp., the space of all twice differentiable functions on J with uniformly continuous and bounded second-order derivative).
A bounded weight on J is a function w ∈ C b (J) such that w(x) > 0 for every x ∈ J. Then the symbol C w b (J) (resp., C w 0 (J)) will stand for the Banach lattice of all functions f ∈ C(J) such that wf ∈ C b (J) (resp., wf ∈ C 0 (J)) endowed with the natural order and the weighted norm • w defined by J) as well.Now let B(J) be the σ-algebra of all Borel subsets of J and denote by M + (J) (resp., M + b (J), M + 1 (J)) the cone of all Borel (resp., bounded, probability Borel) measures on J.If μ ∈ M + (J), we shall denote by L 1 (J, μ) the space of all Borel measurable functions such that f 1 := J |f | dμ < +∞.We also denote by λ 1 the Borel-Lebesgue measure on J and, for every x ∈ J, by ε x the point-mass measure concentrated at x, i.e., for every B ∈ B(J), The symbol 1 denotes the constant function with constant value 1.Furthermore, for every k ∈ N and x ∈ J, we shall set e k (t) := t k and ψ x (t) := t − x (t ∈ J). ( Finally note that, if μ ∈ M + 1 (J) and e k ∈ L 1 (J, μ) for some k ≥ 2, then

Positive Approximation Processes Generated by Integrated Means on Noncompact Real Intervals
In this section we introduce the main object of study of the paper.We begin by presenting some preliminaries.
A continuous selection of probability Borel measures on J is a family (μ x ) x∈J of probability Borel measures on J such that, for every f ∈ C b (J), the function U (f ) defined by is continuous on J.If such a selection is assigned, for every n ≥ 1 and x ∈ J, the symbols μ n x and μ x,n will stand, respectively, for the product measure on J n of μ x with itself n-times and for the image measure of μ n x under the mapping π n : J n → J defined by Moreover, extending formula (2), if f ∈ x∈J L 1 (J, μ x ) we continue to denote by U (f ) the function From now on, we shall fix a real number r ≥ 0, a continuous selection (μ x ) x∈J of probability Borel measures on J satisfying, for each x ∈ J, e 1 ∈ L 1 (J, μ x ) and J e 1 dμ x = x, and a sequence (μ n ) n≥1 of probability Borel measures on J.For every n ≥ 1 and x ∈ J, let λ x,n,r be the image measure of μ n x ⊗ μ n under the mapping σ n,r : J n+1 → J defined, for every (x 1 , . . ., x n+1 ) ∈ J n+1 , by Then, the positive linear operators (C n ) n≥1 we are interested in studying are defined by setting, for every f ∈ n≥1,x∈J L 1 (J, λ x,n,r ), n ≥ 1, and x ∈ J, For the sake of simplicity we shall restrict the operators C n , n ≥ 1, to the subspace of those functions f ∈ C(J) such that f ∈ n≥1,x∈J L 1 (J, λ x,n,r ) and C n (f ) ∈ C(J) for every n ≥ 1.Such functions will be referred to as admissible functions with respect to the selection (μ x ) x∈J , the sequence (μ n ) n≥1 and r ≥ 0. The linear subspace of all of them will be denoted by L a (J).From assumption (2) and the continuity property of the product measure (see [15,Proposition 13.12] and [13,Theorem 30.8]) it follows that C b (J) ⊂ L a (J).We note that for r = 0 the above operators turn into the Bernstein-Schnabl operators B n associated with (μ x ) x∈J , introduced and studied in [9].In this particular case the space of all admissible functions will be denoted by C a (J).More specifically, we have that λ x,n,0 = μ x,n and, for any f ∈ The operators C n are, indeed, related to the operators B n , as the next result shows. where x ); (iii) In (5) one can exchange the integration order.
From (i), by choosing x 1 = x 2 = . . .= x n = x, we get that I n (f ) is well defined and, since f is continuous, it is obvious that x ) and, hence, I n (f ) ∈ L 1 (J, μ x,n ).From this, being n and x arbitrarily chosen, it follows that I n (f ) ∈ C a (J).
As explained in the Introduction, the operators C n , n ≥ 1, have been briefly studied in the setting of spaces of real-valued continuous functions with at most quadratic growth.In the current paper we attempt to study them in their full generality and in wider classes of weighted function spaces.A similar attempt has been carried out in [9] in the case r = 0, i.e., for Bernstein-Schnabl operators defined by ( 6).
Below we discuss some examples (see also [3,Section 5.2]).Due to the generality of the parameters involved in the definition, many other examples can be furnished.For additional ones in the compact framework we refer to [11,12,14].
Examples 2. 1.Let (α p ) p≥1 be a (finite or infinite) sequence of positive continuous functions on J such that ∞ p=1 α p = 1 uniformly on compact subsets of J.Moreover, consider (a p ) p≥1 in J and set, for every x ∈ J, In this case, for every f ∈ C a (J), n ≥ 1 and x ∈ J, we have and hence, for every f ∈ L a (J), In particular, set J = [0, +∞[ and, for every x ≥ 0, let μ x be one of the following measures on [0, +∞[: for every f ∈ L a ([0, +∞[), n ≥ 1 and x ≥ 0 and, in the case (ii), for every f ∈ L a ([0, +∞[), n ≥ 1 and x ≥ 0. Specializing the measures μ n , n ≥ 1, we obtain more specific examples.Let (a n ) n≥1 and (b n ) n≥1 be two sequences of real numbers such that 0 ≤ a n < b n ≤ 1 (n ≥ 1) and set where Then, for r > 0, operators (8) turn into In particular, for r = 1, The above operators are strictly related to the generalization of the Szász-Mirakjan-Kantorovich operators we have introduced and studied in [10] and which are defined by where f ranges in a suitable function space on [0, +∞[.Actually these last operators can be recovered by the operators ( 12) by means of the formula . Similarly, by choosing the sequence (μ n ) n≥1 defined by (10), the operators (9) turn into for every f ∈ L a ([0, +∞[), n ≥ 1 and x ≥ 0. 2. For every x ≥ 0 set where the function ϕ(x, •) is defined on [0, +∞[ by Moreover, fix a family of measures (μ n ) n≥1 in M + 1 ([0, +∞[) and r ≥ 0.Then, for every f ∈ L a (J), If, for instance, we choose μ n = ε bn , where (b n ) n≥1 is a sequence of positive real numbers, we have (b) for every compact subset K ⊂ J there exists h ∈ L 1 (J, μ) such that ϕ(x, y) ≤ h(y) for every x ∈ K and y ∈ J. Set For instance, fix a strictly positive function α ∈ C(R) and, for every x ∈ R, let μ x be the normal distribution in R with mean value x and variance 2α(x), i.e., μ x = ϕ(x, •)λ 1 where, for x, y ∈ R, Then (μ x ) x∈R is a continuous selection of probability Borel measures on R.
The operators C n , n ≥ 1, associated with (μ x ) x∈R and (μ n ) n≥1 are the operators defined by for every f ∈ L a (R), x ∈ R and n ≥ 1 (see [8]).

Some Functional-Analytic Properties
In this section we investigate the behaviour of the operators C n , n ≥ 1, in some function spaces and, especially, in weighted function spaces.
From now on we fix a bounded weight w on J and we assume that moreover, we also suppose that and From ( 16) and ( 17) it follows that since, for each n ≥ 1, x ∈ J and x 1 , . . ., x n ∈ J, and hence From (19) it also follows that w −1 λ x,n,r ∈ M + b (J) for every x ∈ J and n ≥ 1.
From now on, we shall assume that for every n ≥ 1 the mapping the weak topology, (20) i.e., for every ϕ ∈ C b (J) the function x ∈ J → J ϕw −1 dλ x,n,r is continuous or, equivalently, the function C n (ϕw −1 ) is continuous.
In order to prove Statement (2), we first remark that, thanks to Part (1), for a given n ≥ 1, C n (w −1 ) ∈ C(J).On the other hand, for every x ∈ J, and hence Part (2) follows.As for Statement (3), let n ≥ 1 and Finally, Statement ( 4) is a consequence of Part (3), assuming w = 1.
Proposition 4. Assume that conditions ( 16), ( 17), ( 18), and (20) are fulfilled and consider a weight w ∈ C 0 (J) Then, for every n ≥ 1, , so that wf ∈ C 0 (J); hence, for a fixed ε > 0, we can find a compact subset K of J such that, for every x ∈ J\K, |w(x)f (x)| ≤ ε Cn(w −1 ) w .Therefore, for every x ∈ J, On the other hand, since w ∈ C 0 (J), there exists a compact subset Arguing as in the proof of Part (3) of Proposition 3, we get To obtain the converse inequality, let (ϕ p ) p≥1 be an increasing sequence in K(J) such that 0 ≤ ϕ p ≤ 1 for every p ≥ 1 and sup p≥1 ϕ p = 1.Then, for every x ∈ J, by using Beppo Levi's theorem, we get and this completes the proof.
In the special case r = 0, Propositions 3 and 4 hold true under simpler assumptions.More precisely, consider a bounded weight w on J verifying (16) and assume that Clearly, condition (18) implies (22) for r = 0.Moreover, condition (20) turns into the following one: for every n ≥ 1 the mapping x ∈ J → w −1 μ x,n ∈ M + b (J) is continuous w.r.t. the weak topology, ( i.e., for every ϕ ∈ C b (J) the function B n (ϕw −1 ) is continuous.Therefore we obtain the following result.
Corollary 5. Assume that conditions ( 16), ( 22) and ( 23) are fulfilled.Then Moreover, if in addition w ∈ C 0 (J), we have (5) for every n ≥ 1, B n maps C w 0 (J) into C w 0 (J), it is continuous and Remark 3. As far as we know, Proposition 5 represents an improvement of the results in [9].
Coming back to the study of operators C n , the following proposition holds.Proposition 6.Let n ≥ 1. Assume that where, recalling that r 1 = inf J ∈ R ∪ {−∞} and r 2 = sup J ∈ R ∪ {+∞}, I(b, r j ) denotes the interval whose endpoints are b and r j .Then, for every In particular, if (24) is satisfied, then Without loss of generality, we may assume that l = 0. Otherwise, since C n (1) = 1, it should be enough to replace f with f − l1.Page 13 of 24 250 Then, for a fixed ε > 0, there exists b ∈ Assume now that sup I(b,rj ) |f | = 0. From (24) it follows that there exists c ∈ • J such that, for every x ∈ I(c, r i ) with i = j, λ x,n,r (I(b, r j ))

Approximation Properties
In this section we shall discuss the approximation properties of the sequence (C n ) n≥1 in the setting of weighted function spaces.
In [3,Remark 3.2] it has been already proven that, for J = [0, +∞[, if e 1 ∈ L a (J) and if there exists C ≥ 0 such that J e 1 dμ n ≤ C for every n ≥ 1 and J e 1 dμ x = x for every x ∈ J, then, for every uniformly continuous and bounded function f , we have that for every x ∈ J.Moreover, under the additional assumptions that e 2 ∈ L a (J), sup x∈K J e 2 dμ x < +∞ for every compact subinterval K of J and sup n≥1 J e 2 dμ n < +∞, then, for every uniformly on compact subintervals (see [3,Proposition 3.1]).
In order to deepen such approximation properties in weighted function spaces, from now on we assume that From ( 27), by simple calculations, it follows that e 1 ∈ L a (J).
We preliminarily remark that, if for some m ≥ 1, The following lemma concerning the operators B n will be useful.For a detailed proof see [9,Lemma 4.1].

Lemma 7. For a given
where

kn! and the operator U is defined by (4). Therefore, if U (e h ) is continuous for every
where the function ψ x is defined in (1).
After these preliminaries we are ready to prove the following lemma.
Concerning Part (2), in order to show that e m ∈ L a (J), it sufficies to prove that C n (e m ) ∈ C(J); this happens (see (32)) if B n (e i ) ∈ C(J), i.e., e i ∈ C a (J), for every i = 1, . . ., m.By applying Lemma 7 this last condition is verified under (35).The converse follows directly from (32).
In order to achieve the desired approximation properties, we shall appeal to the following Korovkin-type theorem which has been obtained in [7] (see [7, Example 4.9, 1] and [6,Example 2.3,3] or, more directly, [4, Corollaries 6.13 and 6.14]).Theorem Let w ∈ C 0 (J) be a weight such that e 2 ∈ C w 0 (J).
From now on, we assume that so that sup n≥1 J e 1 dμ n < +∞.
Theorem 10.The following statements hold true: (1) Under assumption (39), let w ∈ C 0 (J) be a weight on J such that ( 16), ( 17), ( 18), and ( 20) are fulfilled and assume that e 2 ∈ C w 0 (J).Then, for every and the convergence is uniform on compact subsets of J.In particular, for every f ∈ C b (J), lim n→∞ C n (f ) = f uniformly on compact subsets of J. (2) Assume that J = [0, +∞[ and that (21) holds true.Furthermore suppose that Then Proof.
(1) First of all we note that e 2 ∈ L a (J) ∩ C a (J) since U (e 2 ) ∈ C(J) (see (35)).Furthermore, (C n ) n≥1 is equibounded by virtue of Propositions 3 and 4. By Theorem 9, in order to get (40), it is sufficient to prove that lim n→∞ C n (h) = h with respect to • w , for every h ∈ {1, e 1 , e 2 }.
We end this section by stating some estimates of the convergence in ( 25) and (26) as well as the one in Theorem 10.The estimates will be given in terms of the ordinary moduli of smoothness of the first and second order ω 1 and ω 2 (see, e.g., [1,Section 5.1]).Furthermore, they will be mainly stated in the special case where J = [0, +∞[.Perhaps, for other kinds of noncompact intervals, other different techniques would be implemented.
where M 1 is a constant independent on f , n ≥ 1 and x ∈ J.
In order to prove Part (2), we shall appeal to [3,Theorem 4.3], where we have established a similar estimate for the approximation of operators semigroups on Banach spaces.More precisely, consider the Banach space X := UC b ([0, +∞[) endowed with the sup-norm and denote by (T (t)) t≥0 the translation semigroup defined on it, i.e., for every x ≥ 0, f ∈ X and ξ ≥ 0, T (x)(f )(ξ) := f (x + ξ).Clearly T (x) ≤ 1 for every x ≥ 0.Moreover, for every n ≥ 1 and x ≥ 0, consider the bounded linear operator K n (x) : X → X defined by setting, for every f ∈ X and ξ ≥ 0, Thus, for every f ∈ X and x ≥ 0, T (x)(f = f (x) and K n (f )(0) = C n (f )(x).Therefore, from formula (4.4) of [3,Theorem 4.3], we get In order to present some estimates of the rate of convergence with respect to the weighted norm (see ( 42)), we shall use a similarity technique.
Generally speaking, given an approximation process (L n ) n≥1 on some Banach space X, if R : X → Y is an isometric isomorphism between X and another Banach space Y with inverse S : Y → X, then it is possible to construct an approximation process (L * n ) n≥1 on Y by setting, for any n ≥ 1, In such a case, (L n ) n≥1 and (L * n ) n≥1 are said to be similar or isomorphic.Clearly, for every u ∈ X, Clearly, its inverse S : For every n ≥ 1, we consider the similar positive linear operator . By virtue of (43), we have that, for every Moreover (C * n ) n≥1 is an approximation process in C([0, 1]).We are now ready to state some estimates of the rates of convergence with respect to the weighted norm • w .

An Asymptotic Formula
In this last section, under suitable conditions, we shall establish an asymptotic formula for the operators C n .To this end, from now on, we assume that e 4 ∈ x∈J L 1 (J, μ x ) ∩ n≥1 L 1 (J, μ n ) and Before stating the main result of this section, we note that, if e 2 ∈ x∈J L 1 (J, μ x ) ∩ n≥1 L 1 (J, μ n ), then the subspace , then E 2 (J) ⊂ C w b (J).We are now in a position to state the following result which, in the case where J is a compact subset of R p , was shown in [12].Theorem 14.Let w be a bounded weight on J such that ( 16), (17) Proof.We first point out that, given our assumptions, U (e i ) ∈ C w b (J) for every i = 1, . . ., 4 (see (32)).According to [5,Theorem 1], to show (54), we have to prove that (see ( 52) and ( 53 Fix μ ∈ M + (J) and consider a continuous positive function ϕ : J × J → R satisfying (a) J ϕ(x, y) dμ(y) = 1 for every x ∈ • J;

Remark 2 .
Assume that J = [0, +∞[ and denote by C w * ([0, +∞[) the space of all f ∈ C w b (J) such that lim x→+∞ w(x)f (x) ∈ R.Under the same assumptions of Proposition 4, assume that