tailieunhanh - Approximation of analytic functions of several variables by linear k-positive operators

The approximation of analytic functions of complex variables by linear k-positive operators was first tackled in the work of Gadjiev. We investigate the approximation of analytic functions of several variables in polydiscs by the sequences of linear k-positive operators in the Gadjiev sense. | Turk J Math (2017) 41: 426 – 435 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Approximation of analytic functions of several variables by linear k-positive operators T¨ ulin COS ¸ KUN∗ Department of Mathematics, B¨ ulent Ecevit University, Zonguldak, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: We investigate the approximation of analytic functions of several variables in polydiscs by the sequences of linear k-positive operators in the Gadjiev sense. Key words: Analytic functions, linear k-positive operators, Korovkin-type theorems 1. Introduction The approximation of analytic functions of complex variables by linear k-positive operators was first tackled in the work of Gadjiev [5]. He introduced k-positive operators and formulated theorems of Korovkin’s type for these operators in the space of analytic functions on the unit disc. He proposed a method of proving such theorems, applied further on in many other articles (., [1,3,6–13,15,16]). Some of the results from [1,5–7] were included in a monograph [2,14]. In his recent article [12], Gadjiev proved very general results on convergence of the sequences of linear k-positive operators on a simply connected bounded domain within the space of analytic functions. In this article we extend some of the result of Gadjiev to the approximation of analytic functions of several complex variables by sequences of linear k-positive operators. 2. Preliminaries Let N and Z+ be the respective sets of positive and nonnegative numbers and C be the space of complex numbers. For n ∈ N let Sn := {z = (z1 , . . . , zn ) ∈ Cn : |zi | 1 . Therefore, if |m| = |k|, |fm − fk | ≤ 2M g|m| g|k| < 2 2M g|k| n ∑ (mj − kj )2 . j=1 The last two inequalities for |fm − fk | gives us that for all k and m 2 |fm − fk | ≤ 8M g|k| { √ √ n ( g|m| − g|k| )2 ∑ + (mj − kj )2 }. △2g .