Certain approximation properties of Brenke polynomials using Jakimovski–Leviatan operators

In this article, we establish the approximation by Durrmeyer type Jakimovski–Leviatan operators involving the Brenke type polynomials. The positive linear operators are constructed for the Brenke polynomials, and thus approximation properties for these polynomials are obtained. The order of convergence and the weighted approximation are also considered. Finally, the Voronovskaya type theorem is demonstrated for some particular case of these polynomials.


Introduction and preliminaries
The Korovkin approximation process plays a crucial role in a wide variety of problems in measure theory, functional analysis, probability theory, and partial differential equations. Korovkin [11] established a well-known simple criterion to decide whether a given sequence (K n ) n∈N of positive linear operators on the space C[0, 1] is an approximation process, i.e., K n (f ) → f uniformly on [0, 1] for every f ∈ C[0, 1]. Taking into account this significant result, mathematicians all across the globe have extended this theorem named after Korovkin to other abstract spaces, such as Banach spaces, lattices, algebras, etc. The work of Korovkin laid a foundation and basis for a new theory, mainly referred to as Korovkintype approximation theory.
Favard and Szasz [16] introduced the following example of positive linear operator: Leviatan and Jakimovski [8] established the following generalization: where an analytic function h(v) = ∞ l=0 h l v l ; |v| < R, R > 1, and 0 = g (1), and derived the approximation properties for these above operators.

Construction of operators
Motivated by the work of Ali Karaisa [9], here we develop the positive linear operators containing the Brenke polynomials [2,6], which possess generating relation of the form where analytic functions g and B are given by ∞ r=0 g r t r = g(t), g 0 = 0, and possess explicit expansions: k r=0 g k-r b r y r = p k (y), k = 0, 1, 2, . . . . (2.4) Restraining ourselves to p k (y), i.e., the Brenke polynomials satisfying: , 0≤ r ≤ k, k = 0, 1, 2, . . . , Further, the positive linear operators involving p k (y) polynomials are introduced while keeping in consideration the above restrictions by the following manner: where y ≥ 0 and n ∈ N.

Approximation properties of T n operators
Korovkin [11,12] derived the results concerning the convergence of sequences (K m (g, y)) ∞ m=1 , where K m (g, y) are positive linear operators. For instance, if K m (g, y) uniformly converges to g for some particular cases 1, t, t 2 ≡ g(t), likewise it performs such activity for each g, being continuous and real. Shisha and Mond in [14,15] described the rate of convergence for K m (g, y) in terms of the moduli of continuity of g.
Our aim is to derive the convergence theorem and the order of convergence of operators T n (f ; y) given by expression (2.6). T n (e 1 ; y) = y B (ny) B(ny) T n (e 2 ; y) = 1 -1

Order of convergence
Now we recall the following definitions.
It is clear that the following inequality holds for all δ > 0. The constant M is independent of g and δ.  . Let g h be the second-order Steklov function attached to the function g. Then the following inequalities are satisfied: Now, we compute the rates of convergence of the operators T n (f ; y) to f by means of a classical approach, the modulus of continuity, and Peetre's K-functional. The following result gives the rates of convergence of the sequence T n (f ; y) to f by means of modulus of continuity.

Theorem 4.1 For f ∈ C[0, a], the following inequality is satisfied:
, which on using inequality (4.7) becomes Taking into account that f h ∈ C 2 [0, a], from Lemma 4.1, it follows that which in view of inequality (4.8) becomes Further, the Landau inequality (4.14) Using inequality (4.14) in inequality (4.13) and taking h = 4 T n ((sy) 2 ); y, we find Making use of inequality (4.15) in inequality (4.11) can lead to assertion (4.9).
which completes the proof. and M ≥ 0 is a constant, which is independent of the functions f and δ. Also, ξ n (y) is the same as in Theorem 4.2.
Proof Suppose that g ∈ C 2 B [0, ∞), from previous Theorem 4.2, we have Since the l.h.s of the above inequality does not depend on the function g ∈ C 2 B [0, ∞), where K is Peetre's functional defined by (4.3). By using relation (4.5) in (4.22), the inequality holds.

Weighted approximation
Here, some properties of approximation for the operator T n of a space of weighted continuous functions are given, for which the succeeding class of functions is defined on [0, ∞). Consider The norm on C * y 2 [0, ∞) is given as follows: |h(y)| 1 + y 2 . which concludes the proof.
By using Lemma 5.1, one can see that the operator T n defined by (2.6) acts from C y 2 [0, ∞) to B y 2 [0, ∞). Theorem 5.1 Lthe5.1 Let T n be the sequence of positive linear operators defined by (2.6) and ρ(y) = 1 + y 2 be a weight function, then for each f ∈ C * y 2 [0, ∞), Proof By using the weighted Korovkin theorem presented by Gadzhiev [4], it is enough to verify the following conditions: Hence the proof is completed.

Special cases of the operators T n and further properties
The Gould-Hopper polynomials p d+1 k (y; h) given by the identity These p d+1 k (y; h) polynomials are p k (y) Brenke polynomials for g(t) = e ht d+1 and B(t) = e t in expression (2.1). Thus, for g(t) = e ht d+1 and B(t) = e t in equation (2.6) gives the following Durrmeyer type Jakimovski-Leviatan operators T * n (f ; y) involving the p d+1 k (y; h) polynomials: beneath the presumption h ≥ 0. Now, to prove the Voronovskaya theorem for operators (6.3), first we prove the succeeding results.