Approximation process of a positive linear operator of hypergeometric type

Abstract

In this article, we construct a new sequence of positive linear operators Hn:B[0,1]→C[0,1]{H}_{n}:B{[}0,1]\to C{[}0,1] using the hypergeometric distribution of probability theory and the rational values of f at the equally spaced control points k∕nk/n (k=0,1,…,n)\left(k=0,1,\ldots ,n) of the unit interval [0,1]. Moreover, we obtain some approximation properties of these operators. It is important to note that hypergeometric distribution has a special interest in probability theory because of its natural behaviour. Namely, unlike all other discrete distributions, the previous steps in the hypergeometric distribution affect the next steps. In other discrete distributions, the process starts from the beginning at each stage, whereas in the hypergeometric distribution, the previous steps determine the structure of the next steps, since the previous steps are not replaced.

Keywords

• hypergeometric distribution
• convergence
• korovkin theorem
• modulus of continuity
• 41a36
• 41a25