On Sequences of J. P. King-Type Operators

Abstract

This survey is devoted to a series of investigations developed in the last fifteen years, starting from the introduction of a sequence of positive linear operators which modify the classical Bernstein operators in order to reproduce constant functions and x2 on [0,1]. Nowadays, these operators are known as King operators, in honor of J. P. King who defined them, and they have been a source of inspiration for many scholars. In this paper we try to take stock of the situation and highlight the state of the art, hoping that this will be a useful tool for all people who intend to extend King’s approach to some new contents within Approximation Theory. In particular, we recall the main results concerning certain King-type modifications of two well known sequences of positive linear operators, the Bernstein operators and the Szász-Mirakyan operators.

Gamma operators reproducing exponential functions

“…Recently, the concept of the degenerate gamma function and degenerate Laplace transformation was introduced by Kim-Kim [16]. For each λ ∈ (0, ∞), the degenerate gamma function for the complex variable s with 0 < Re(s) < 1/λ as follows: Approximation of target function g(x) on an equally spaced evaluation grid of [5,6] where β(x, y) is the Beta function. They studied some properties of the degenerate gamma and degenerate Laplace transformation and obtained their properties.…”

“…Example 1 We shall now illustrate the convergence of the new Gamma operator based on its classical counterparts. The new construction of Gamma operator and its standardversion algorithm is applied to the test function f (x) : [5,6] → R, with f (x) = x log(1+x) . Here RMS Error for G n (f ; x) and RMS Error for Γ ω n (f ; x) are root-mean-square (L-norm) errors for G n (f ; x) and Γ ω n (f ; x), respectively.…”

“…Further modifications and their approximation results showed the effectiveness of operators preserving some exponential functions in certain senses, hence many researchers have focused on the introduction and investigation of exponential-type operators. Among others, we refer the readers to [4][5][6][7][8][9][10]20] and the references therein.…”

A Sequence of Kantorovich-Type Operators on Mobile Intervals

“…Subsequently, other modifications of the classical Bernstein operators, as well as of many other well-known operators, that fix suitable functions were introduced (see [2] and the references quoted therein). Here we limit ourselves to mention that, for example, in [9], the authors considered a family of sequences of operators (B n,α ) n≥1 , α ≥ 0, that preserve the constants and the function x 2 + αx.…”

Approximation of functions by a new construction ofBernstein‐Chlodowskyoperators: Theory and applications

“…and studied in [6,7] in detailed. Besides, Bernstein type linear and positive operators introduced in Reference [8], which preserve the set of {1, , 2 Similar generalization of the mentioned linear and positive operators were also introduced in literature. For instance, Durrmeyer type modification introduced in Reference [9] and Szász-Mirakyan type generalization studied in Reference [10,11].…”