A new approximation operator generalizing Meyer-König and Zeller's power series

Benny Levikson
1976 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
Introduction. In this paper we introduce a new approximation operator of the Arato-Renyi type and study its properties. Special cases of our operator are the power-series, of W. Meyer-Kônig and K. Zeller (see [9] ) and the generalized Berenstein power series introduced by A. Jakimovski and D. Leviatan in [5] and analyzed by them in [6] . We prove that our approximation operator converges uniformly to the approximated function provided this function is continuous. Using Liapounov's central limit
more » ... nov's central limit theorem we analyze the behavior of the operator near discontinuity points. We use these theorems along with some other probablistic arguments to give new results for the generalized Berenstein power series. The motivation for using probablistic methods came from the interesting paper by M. Arato and A. Renyi [1] . Such methods simplify proofs, give insight to them, and thus enable a better understanding of the approximation mechanism. In the first section we summarize the main results, while their proofs along with some additional lemmas are given in Section 2.
doi:10.4153/cjm-1976-032-6 fatcat:knmne3wq7zhzflqywap7465qfa