Approximation of functions by a new class of linear operators

Abstract

Various extensions and generalizations of Bernstein polynomials have been considered among others by Szasz [13], Meyer-Konig and Zeller [8], Cheney and Sharma [1], Jakimovski and Leviatan [4], Stancu [12], Pethe and Jain [11]. Bernstein polynomials are based on binomial and negative binomial distributions. Szasz and Mirakyan [9] have defined another operator with the help of the Poisson distribution. The operator has approximation properties similar to those of Bernstein operators. Meir and Sharma [7] and Jam and Pethe [3] deal with generalizations of Szasz-Mirakyan operator. As another generalization, we define in this paper a new operator with the help of a Poisson type distribution, consider its convergence properties and give its degree of approximation. The results for the Szasz-Mirakyan operator can easily be obtained from our operator as a particular case.

Asymptotic behaviour of Jain operators

“…Moreover, the rate of convergence can be estimated by as n → ∞ , provided that β n → 0. Like in Jain's paper [13] therein a suited condition on the growth of f is missing. The main formula [10, Theorem 2.5]…”

“…In 1972 Jain [13] introduced the sequence of linear operators defined, for functions f ∈ C [ 0, +∞ ), by…”

“…, whenever the sum in (1) is convergent. By using Lagrange inversion, Jain [13,Lemma 1] showed that, for 0 < α < +∞ and |β| < 1, ∞ ν=0 ω β (ν, α) = 1.…”

Quantitative estimates for Jain-Kantorovich operators

“…e ( +k ) ; k = 0; 1; 2; :: (1.1) for 0 < < 1; j j < 1; in 1970, G. C. Jain [14] introduced a positive linear operator de…ned for f 2 C (R + ) as As a particular case = 0; we obtain the well-known Szasz-Mirakyan operators studied in [6], [11] and [15];…”

An Asymptotic Formula of Modified Family of Positive Linear Operators

“…In particular = , the operators (1) reduce to Jain operators [2]. Also, if = and = 0 then, the operators [ , ] equal to the classical Szasz-Mirakyan operators [3].…”