On the stability of some positive linear operators from approximation theory

Abstract

Recently, Popa and Raşa have shown the stability/ instability of some classical operators defined on $$[0,1]$$ and obtained the best constant when the positive linear operators are stable in the sense of Hyers–Ulam. In this paper we show that the Kantorovich–Stancu type operators, King’s operator, Bernstein–Stancu type operators, and Kantorovich–Bernstein–Stancu type operators with shifted knots are Hyers–Ulam stable. Further we find the best Hyers–Ulam stability constants for some of these operators. We also prove that Szász–Mirakjan and Kantorovich–Szász–Mirakjan type operators are unstable in the sense of Hyers and Ulam.