Baskakov operators and Jacobi weights: pointwise estimates

In this paper we present direct results (upper estimates) for Baskakov operators acting in spaces related with Jacobi-type weights. Our results include and extend some known facts related with this problem. The approach is based in the use of a new pointwise K-functional.


Introduction
Let C[0, ∞) be the family of all real continuous functions on the semiaxis and B(0, ∞) the family of all bounded functions in (0, ∞). A family of Baskakov operators is obtained as follows. For λ ∈ R, λ ≥ 1, and a function f : [0, ∞) → R, define (whenever the series converges) where v λ,k (x) = λ + k -1 k Some authors have considered these operators acting in spaces defined with the help of a Jacobi weight in the discrete case (λ = n ∈ N) (see [6,17,18], and [21]).
Let us present some notations. Throughout the paper we set ϕ(x) = √ x(1 + x) and, for x > 0, where a ≥ -1 and b ∈ R are fixed real parameters.
We will study approximation properties of the operators V λ in the weighted spaces of continuous functions © The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. When 0 < a < 1 and b > 0, it was proved in [21] that Baskakov operators are unbounded in C [0, ∞) with the usual weighted norm f = f ∞ := sup x>0 | (x)f (x)|. That is the reason why is better to consider the norm To avoid complications, some authors prefer to work with the space For α ∈ [0, 1], we also consider the space In this paper we present upper estimates for the error (x)|f (x) -V λ (f , x)| assuming that a ≥ -1 and b ∈ R.
In the discrete case, pointwise estimates have been obtained by some authors under more restrictive selections of the parameters a and b. For instance, in the unweighted case (a = b = 0), Xie [20,Theorem 1.1] where ω 2 ϕ α (f , t) is the Ditzian-Totik modulus. The case α = 0 was previously studied by Huo and Xue in [14,Theorem 1]. For 0 < a < 1, b > 0, and α ∈ [0, 1], Wang and Xue [18, Theorem 1.1] verified the inequality The same result appeared in [17]. Our results are not restricted to the consideration of a larger family of parameters a and b. There are two other important facts to be taken into account. First, instead of the classical K -functionals, we use a pointwise K -functional of the form In second place, we present the estimates for operators with the continuous parameter λ.
Why are these two changes necessary? As we will show in another paper, the K -functional (3) is convenient for proving strong converse inequalities. Notice that, when α = 1, the K -functional (3) is just equivalent to the usual one because, for all x > 0, Moreover, for 0 ≤ α ≤ 1 and x > 0, we have Indeed, for y ∈ (0, x], α ∈ [0, 1] and g ∈ C 2,α , we have (y)ϕ 2 (y)g (y) = ϕ 2(1-α) (y) (y)ϕ 2α (y)g (y) ≤ ϕ 2(1-α) (x) ϕ 2α g and then On the other hand, the use of a continuous parameter allows us to apply the results to study other family of operators. For a real c > 0, n > c, and a function f whenever the series converges, where with C(n, k, c) = n(n + c) · · · (n + (k -1)c), and w n,0,c (x) = 1/(1 + cx) n/c . Here n is not necessarily an integer.
The operators (7) have been studied by several authors (see [3,4,10,14], and [22]). Later we show that the operators W n,c (f , x) are related with Baskakov operators, but with a family of not integer parameters. This is a good reason for studying the operators V λ .
The main results are given in Sect. 2 (pending of some auxiliary results that will be proved in Sect. 3). In Sect. 4 we explain how our approach can be used to obtain similar results for the operators (7). Converse results will appear in a separated paper.
The paper contains several references written in Chinese. We include them to provide a review in the topic. Since we do not use any result from these work, the reader can follow our arguments.
(i) Assume k = 0 and λx ≤ 1. Then If b ≥ α, then from (9) we obtain If b < α, then from (9) and the condition b + λ ≥ 0, we get Hence, from Proposition 3.5, (iii) Now we assume that λx ≥ 1. From Corollary 3.4 and Propositions 3.2 (i) and 3.6, one On the other hand where in the last inequality we have used Proposition 3.8. Hence, Proof If x > 0 and g ∈ C 2,α [0, ∞), we use the representation and, if t > x, then Hence, from Proposition 2.1, we have Now, using standard arguments and Proposition 2.2, we have, for x > 0, Thus, the result follows from (11), (12) and the definition of the K -functional given in (3).
Note that the condition a < 1 in Theorem 2.3 is necessary in order to apply Proposition 2.1 with α = 1 to the right side integral in (10). However, writing we only need the condition 0 ≤ a + α < 2 to apply Proposition 2.1 and conclude that Remark 1 The inequality (13)  Taking into account (12), (13), Proposition 2.2 and the definition of the K -functional given in (4) we have immediately the following result.
where K α (f , t) is the K -functional given in (4).
As in [5, page 56], if a = 0, define where , and It is well known (see [5, Th. 6.1.1] that there exist positive constants C 1 and t 0 such that, for f ∈ C 0 [0, ∞) and 0 < t ≤ t 0 , where K α (f , t) is given by (4), and the modulus is defined by (15) when a = 0 and by (16) when a > 0. For a < 0, we do not know a characterization of the K-functional in terms of an appropriate modulus of smoothness.
From Corollary 2.4 and (17), we obtain the following result.
In the discrete and unweighted case (n ∈ N and a = b = 0), the inequality (18) was obtained by Huo and Xue [14,Theorem 1] in terms of the usual modulus of continuity (α = 0): In [20, Theorem 1.1], Xie proved an inequality of the form

Auxiliary results
Lemma 3.1 For m ∈ N, λ > m, x > 0 and k ∈ N 0 , one has Proof Note that for k = 0 the inequality holds trivially. Next we will consider two cases: Then the result follows taking into account that, if k < m and λ > m, then 1 + k/λ ≤ 1 + m/λ ≤ 2 and then where in the last inequality we have applied that (1 + βp) q ≤ (1 + βq) p for p, q ∈ N 0 , p ≥ q and β ≥ 0. Case II: Now suppose k > m. In this case, we have and then the result follows taking into account that, for j = 0, 1, . . . , m -1, For m = 0, 1, 2, . . . , the central moment of order m of the operator V λ is defined by where e 1 is the function defined by e 1 (t) = t.

Proposition 3.3
Assume r ≥ 0, m, q ∈ R and mr + 1 > 0. Then, for x > 0 and t ≥ 0, one has Proof First note that Now we estimate the last integral in the expression above. If t < x, then, putting An inequality similar to (22) was proved in [8,Lemma 3] with the conditions 0 ≤ r ≤ 2 and mr ≥ 0.
the result follows from Proposition 3.3 with q = bα, m = 1 and r = a + α.
In particular, from (23) we have for t > x and b ≤ α or t < x and b ≥ α. It is well known (see [3]) that the operator V λ satisfies the relation Writing 1 + t = (1 + x) + (tx) and applying (25), we have Using (25) twice, it is easy to show that An inequality similar to (28) was proved by Becker [2, Lemma 6] and Zhang [22, Lemma 1.5] with a unspecific constant. We include here a simpler proof and give the optimal constant.
Since P 1 (x) = 1 + x, we deduce by induction that P m (x) is a polynomial of degree m which can be written as with c m,k (λ) ≥ 0, k = 1, . . . , m.
In what follows we denote by (P) the leader coefficient of a polynomial P. From (29) we also obtain the recurrence relation (P 1 ) = 1, (P m+1 ) = 1 + m λ (P m ), and this implies that Taking into account and (30), we deduce that Q m is also a polynomial of degree m which can be written as and then, for all x ≥ 0, Thus R m is increasing on [0, +∞) and then, for all x ≥ 0, Finally, for all x ≥ 0, To finish the proof, it is sufficient to note that Putting λ = 1 in the above expression, we obtain a upper bound which is valid for all λ ≥ 1 and x ≥ 0. Hence, we conclude that where C(m) = (1 + m + m 2 )m!.
Using (30) and repeating the same arguments as above, it can be proved that, for all λ ≥ 1, x ≥ 0, In what follows we use a different approach to get an inequality similar to (32) which is valid for all m ∈ R. We will consider the ceiling and floor functions of a number z ∈ R defined by z = min{r ∈ Z : r ≥ z} and z = max{r ∈ Z : r ≤ z}. Proposition 3.6 For d ∈ R, x > 0 and any λ satisfying λ ≥ 2(1 + |d|), one has Proof Note that the above inequality is trivial when d = 0.
The ideas in this work can be used to study other families of operators like Lupas, Szász-Mirakyan and some modifications of Baskakov operators, like the ones considered in [1,7,12,13]. For instance, in [13] only some special polynomial weights were presented.
In our setting, for the case of Szász-Mirakyan operators, the authors will present similar results in a forthcoming work.