Chlodowsky type $\left( \lambda,q\right)$-Bernstein Stancu operators of Pascal rough triple sequences

Abstract

The fundamental concept of statistical convergence first was put forward by Steinhaus and at the same time but also by Fast \cite{Fast} independently both for complex and real sequences. In fact, the convergence in terms of statistical manner can be seen as a generalized form of the common convergence notion that is in the parallel of the theory of usual convergence. Measuring how large a subset of the set of natural number can be possible by means of asymptotic density. It is intuitively known that positive integers are in fact far beyond the fact that they are perfect squares. This is due to the fact that each perfect square is positive and besides at the same time there are many other positive integers. But it is also known that the set consisting of integers which are positive is not larger than that of those which are perfect squares: both of those sets are countable and infinite and therefore can be considered in terms of $1$-to-$1$ correspondence. However, when the natural numbers are scanned for increasing order, then the squares are seen increasingly scarcity. It is at this point that the concept of natural density comes into out help and this intuition becomes more precise. In this study, the above mentioned statistical convergence and asymptotic density concepts are examined in a new space and an attempt is made to fill a gap in the literature as follows. Stancu type extension of the widely known Chlodowsky type \linebreak$\left( \lambda,q\right) $-operators is going to be introduced. Moreover, the description of the novel rough statistical convergence having Pascal Fibonacci binomial matrix is going to be presented and several general characteristics of rough statistical convergence are taken into consideration. In the second place, the approximation theory is investigated as the rate of the rough statistical convergence of Chlodowsky type $\left(\lambda,q\right)$-operators.

Keywords

• chlodowsky type (λ
• q)-bernstein stancu operators
• rough statistical convergence
• natural density
• triple sequences