Approximating fractional calculus operators with general analytic kernel by Stancu variant of modified Bernstein–Kantorovich operators

The main aim of this paper is to approximate the fractional calculus (FC) operator with general analytic kernel by using auxiliary newly defined linear positive operators. For this purpose, we introduce the Stancu variant of modified Bernstein–Kantorovich operators and investigate their simultaneous approximation properties. Then we construct new operators by means of these auxiliary operators, and based on the obtained results, we prove the main theorems on the approximation of the general FC operators. We also obtain some quantitative estimates for this approximation in terms of modulus of continuity and Lipschitz class functions. Additionally, we exhibit our approximation results for the well‐known FC operators such as Riemann–Liouville integral, Caputo derivative, Prabhakar integral, and Caputo–Prabhakar derivative.


INTRODUCTION
Fractional calculus (FC) has its roots in 1695 and has been an attractive investigation area of mathematics.Obtained results during the last 328 years are important for pure mathematics and applied fields.Especially in the last three decades, several application areas of FC have been appeared.Because of this reason, lots of numerical approximation techniques have been developed for FC operators in various function spaces [1][2][3].Recently, linear approximation operators have been introduced with the help of auxiliary linear positive operators to approximate the bivariate FC operator based on the bivariate Mittag-Leffler function [4,5].
On the other hand, in Fernandez et al. [6], general FC operator has been proposed based on the general analytic kernel.More precisely, letting [ a, b ] an interval, ,  ∈ C with positive real parts, and the radius R satisfies the condition (b − a) Re() < R, considering the analytic function A on the disc D (0, R) which is represented by a locally uniformly convergent power series A(x) = ∑ ∞ n=0 a n x n , where the coefficients a n may depend on  and , the authors introduced the general analytic kernel fractional operators as follows: where  ∶ [ a, b ] → R is chosen from an appropriate function space (e.g., C [a, b]).For our purposes, we express this operator in the following form.For fixed r ∈ N 0 ∶= {0, 1, 2, • • •} and for all  ∈ C r [0, B] , where C r [0, B] denotes the space of continuous functions such that  (r) is continuous on [0, B], we consider A I ,;(r) These operators act as fractional integral in the case r = 0 and as Caputo type fractional derivative for r ∈ N. At this stage, we list some fractional operators contained in (1).
In the case r = 0, A(x) = x∕Γ(), we recover the Riemann-Liouville integral operator For r = 0, choosing we get the generalized proportional fractional integral (GPF) [7] A I ,1 0 +  (x) = GPF I , 0 +  (x) = Letting r = 0 and choosing where E  , (x) being the three-parameter Mittag-Leffler function, the operator in (1) turns out to be the Prabhakar integral [8,9] given by Now letting Re() > 0, r = ⌊Re()⌋ + 1 and choosing A(x) = x∕Γ(r − ), we get the Caputo derivative operator [10] A I 1,r−−1;(r) For Re() > 0, r = ⌊Re()⌋ + 1 and choosing A(x) = E − ,r− (x), we get the Caputo-Prabhakar [11] derivative given by A I r−,;(r) The main purpose of this paper is to approximate the FC operator with general analytic kernel by using auxiliary newly defined linear positive operators, namely, Stancu variant of modified Bernstein-Kantorovich operators.Our first intention is to investigate the simultaneous approximation properties of this interesting newly defined auxiliary operators and based on the obtained results to prove the main theorems on the approximation of these general FC operators.We also obtain some quantitative estimates for this approximation in terms of modulus of continuity and Hölder continuous function class.Additionally, we exhibit our approximation results for well-known fractional calculus operators such as Riemann-Liouville integral, Caputo derivative, Prabhakar integral, and Caputo-Prabhakar derivative.

STANCU VARIANT OF MODIFIED BERNSTEIN-KANTOROVICH OPERATORS
Let x ∈ (0, B] be fixed for B > 0. For a real-valued function  defined and bounded on the interval [0, B], let the operators K , n,;x ( () ; t) be defined by where and , ,  are constants such that  ≥  ≥ 0,  > 0. The operators K , n,;x ( ; t) will be called Stancu variant of modified Bernstein-Kantorovich operators.We should note that in the case  =  = 0, the operators in (2) reduce to the one introduced in Özarslan and Duman [12], which provides better error estimate than the usual Bernstein-Kantorovich operators for certain values of .
It is clear that We continue with couple of lemmas which are needed to prove the main theorems. ) On the other hand, Similar manipulation gives (4).□ Lemma 2. For all n ∈ N and fixed  ≥  ≥ 0,  > 0, t ∈ [0, x], we have ) , where and Proof.The proof follows from the linearity of the operators.□ Corollary 1.For each fixed x ∈ (0, B] , we have the estimate where Now we observe the relation x n Also, we use the notation where s ∈ [0, x] and set h = h x ∶= 1 1+n+ x.Using the above identity, we get where  k,n,,, (z) = k++z  n+1+ x.In a similar way for n = 2, 3, ..., we get More generally, where as stated before Therefore, for n = r, r + 1, … , we have From the triangle inequality, we can write that It should be noted that since and Thus, using the above inequalities, we get The modulus of continuity of  ∈ C [0, B] is defined for  > 0 by It is known that for all  ∈ C [0, B], we have lim →0 + (, ) = 0, and for any  > 0, In the following theorem, we compute the error of simultaneous approximation of the operators K , n,;x in terms of the modulus of continuity.
Theorem 1.For all  ∈ C r [0, B] , we have the following inequality and Proof.Using (5), we can write that Using ( 6) and ( 7), we get , where we choose Application of the Cauchy-Schwarz inequality gives and hence, using Corollary 1, we get Therefore, choosing .
Moreover, for each fixed r ∈ N 0 and for all  ∈ C r [0, B] , we have Now we consider the Hölder continuous function space where Λ > 0, r ∈ N 0 , and  ∈ (0, 1].In the following theorem, we compute the rate of convergence of the operators for the Hölder continuous functions. Theorem 2. For all  ∈ C (;r) [0, B] , the following estimate holds true Proof.Starting from (5), we have for all  ∈ C (;r) [0, B] that Considering (6) for the first term on the right and applying the Hölder inequality with p = 2  and q = 2 2− to the second term, we get Finally, using Corollary 1, we get the result.□

APPROXIMATING FC OPERATORS WITH GENERAL ANALYTIC KERNEL
In this section, we introduce approximation operators expressed by means of Stancu variant of modified Bernstein-Kantorovich operators, which converge to the general operators given in (1).Under the same restrictions and conditions on ,  and the function A in Section 1 and on the operators K , n,;x in Section 2, we introduce the operators Note that in the case r = 0, we have Recalling that where we swapped the order of integration and the series, since the series is locally uniformly convergent and the integral is absolutely convergent under the given conditions.Writing this result in ( 9), we get where Thus, taking  = 1,  =  − 1 and choosing A(x) = x∕Γ(), we have and therefore, ( RL I ;,; where we call the operators in (10) as the Stancu variant of modified Bernstein-Kantorovich-Riemann-Liouville approximation operators. Taking ) .
Therefore, we get ( GPF I ,;,; We call the operators in (11) as the Stancu variant of modified Bernstein-Kantorovich-GPF approximation operators. Setting and hence, is the generalized Wright function ( [13,14]).Recall that the generalized Wright function is absolutely convergent for all z ∈ C [15, pp.440] in the case Thus, we get We call the operators in (12) as the Stancu variant of modified Bernstein-Kantorovich-Prabhakar approximation operators.
We continue in proving our main approximation theorems. , and  1 ,  2 , and M 1 are the same as in Theorem 1.

CONCLUDING REMARKS
Recently, linear positive operators have been used for the first time to approximate to bivariate fractional operators having certain bivariate Mittag-Lefler functions in the kernel [4,5].The same approach has not been used for the Riemann-Liouville integral or other fractional calculus integral operators having special functions in the kernel.Moreover, linear positive operators have not been used to approximate any version of fractional "derivative" operators.
In the present paper, we use certain linear positive operator to approximate to fractional integral and derivative operators.To achieve this, as a representative, we first introduce the Stancu variant of modified Bernstein-Kantorovich operators and investigate there simultaneous approximation properties and obtain quantitative error estimates in terms of modulus of continuity and Lipschitz class functions.Then we solve our original problem by using these approximation theorems.Finally, we exhibit our approximation results for the well-known FC operators such as Riemann-Liouville integral, Caputo derivative, Prabhakar integral, and Caputo-Prabhakar derivative.