Certain approximation properties of Brenke polynomials using Jakimovski–Leviatan operators

Abstract

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.

Stancu variant of Jakimovski-Leviatan-Durrmeyer operators involving Brenke type polynomials

“…In view of Lemma 3.2 of [43], we have the following lemma for the operators defined by ( 6): Lemma 2.2. For f (t) = e s (t) = t s , s = 0, 1, 2, 3, 4 the operators (6) satisfy the following equalities:…”

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Karaisa [29] presented Jakimovski-Leviatan-Durrmeyer type operators by means of Appell polynomials. In a similar manner, Wani et al [43] proposed a sequence of Jakimovski-Leviatan-Durrmeyer type operators involving Brenke type polynomials which include Appell polynomials and Hermite polynomials. We note that the definitions of the operators given in both these papers are not correct.

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“…We note that the definitions of the operators given in both these papers are not correct. In the present article, we introduce a Stancu variant of the operators considered in [43] after correcting their definition. The definition of the operator proposed in [29] may be similarly corrected.…”

“…In 2020, Braha et al [8] introduced a Stancu type generalization of Szász operators including Brenke-type polynomials and established some approximation properties via power series summability method. Recently, Wani et al [43] developed a sequence of Durrmeyer type Jakimovski-Leviatan operators by means of Brenke-type polynomials in the following way:…”

Statistical Convergence via q-Calculus and a Korovkin’s Type Approximation Theorem

“…The q-analogs of Bernstein operators and other operators significantly lead to more general results on approximations and show a better rate of convergence than the respective classical operators [13]. Recently, approximation properties for Bernstein operators and their different generalizations have been studied in [14][15][16][17][18][19][20][21].…”