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.
Highlights
1 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 (Kn)n∈N of positive linear operators on the space C[0, 1] is an approximation process, i.e., Kn(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
Summary
The Szász operators involving Brenke type polynomials were studied in [17]. The positive linear operators involving pk(y) polynomials are introduced while keeping in consideration the above restrictions by the following manner:. Lemma 3.1 From the generating function of the Brenke type polynomials given by (2.1), we obtain pk(ny) = g(1)B(ny), k=0 ∞. Lemma 3.3 For Tn(f ; y) operators and for y ∈ [0, ∞), the following identities are satisfied: Tn(s – y; y) =. Lemma 4.1 (Gavrea and Raşa [5]) Let h ∈ C2[0, a] and (Km)m≥0 be a sequence of positive linear operators with the property Km(1; y) = 1.
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