Abstract
In this paper, we study the Kantorovich-Stancu-type generalization of Szász-Mirakyan operators including Brenke-type polynomials and prove a Korovkin-type theorem via the T-statistical convergence and power series summability method. Moreover, we determine the rate of the convergence. Furthermore, we establish the Voronovskaya- and Grüss-Voronovskaya-type theorems for T-statistical convergence.
Highlights
Introduction and PreliminariesLet K ⊆ N and Km = fi ≤ m : i ∈ Kg
We study a Korovkin-type theorem for the KantorovichStancu-type Szász-Mirakyan operators via power series method
We note that the Korovkin-type theorems are very useful tools in approximation which were studied in several function spaces [3,4,5,6,7,8, 10, 22,23,24,25,26,27,28,29]
Summary
A sequence η = ðηjÞ is said to be T -statistically convergent (see [1]) to real number L if for any >0, limn∑j:jηj−Lj≥εtnj = 0, and we write stT − limη = L. Let ðpjÞ be a sequence of real numbers such that p0 > 0, p1, p2, ⋯ ≥ 0, and the corresponding power series pðuÞ = ∑∞j=0pjuj has radius of convergence R with 0 < R ≤ ∞. We study a Korovkin-type theorem for the KantorovichStancu-type Szász-Mirakyan operators via power series method. We give a Voronovskaya-type theorem for T − statistical convergence Such type of operators is widely studied by several authors (see [15,16,17,18,19]). By Theorem 6 and Lemma 3, we obtain the following result.
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