The lower estimate for linear positive operators, I

The goal of the paper is to present some results concerning the approximation of convex functions by linear positive operators. First, one recalls some results concerning the univariate real valued convex functions. Next, one presents the notion of higher order convexity introduced by Popoviciu [Popoviciu, T., Sur quelques propri´et´ees des fonctions d’une ou deux variable r´eelles, PhD Thesis, La Faculte des Sciences de Paris, 1933 (June)] . The Popoviciu’s famous theorem for the representation of linear functionals associated to convex functions of m−th order (with the proof of author) is also presented. Finally, applications of the convexity to study the monotonicity of sequences of some linear positive operators and also mean value theorems for the remainder term of some approximation formulas based on linear positive operators are presented.

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Inverse theorems for some linear positive operators using Beta and Baskakov basis functions

10.1063/5.0062925 ◽

2021 ◽

Author(s):

Lakshmi Narayan Mishra ◽

A. Srivastava ◽

T. Khan ◽

S. A. Khan ◽

Vishnu Narayan Mishra

Keyword(s):

Positive Operators ◽

Basis Functions ◽

Linear Positive Operators ◽

Inverse Theorems

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Iterative combinations for Srivastava–Gupta operators

Asian-European Journal of Mathematics ◽

10.1142/s1793557121501084 ◽

2020 ◽

pp. 2150108

Author(s):

Prerna Maheshwari Sharma

Keyword(s):

Rate Of Convergence ◽

Modulus Of Continuity ◽

Simultaneous Approximation ◽

Higher Order ◽

Positive Operators ◽

Integral Type ◽

Linear Positive Operators ◽

General Family ◽

Special Cases

In the year 2003, Srivastava–Gupta proposed a general family of linear positive operators, having some well-known operators as special cases. They investigated and established the rate of convergence of these operators for bounded variations. In the last decade for modified form of Srivastava–Gupta operators, several other generalizations also have been discussed. In this paper, we discuss the generalized modified Srivastava–Gupta operators considered in [H. M. Srivastava and V. Gupta, A certain family of summation-integral type operators, Math. Comput. Modelling 37(12–13) (2003) 1307–1315], by using iterative combinations in ordinary and simultaneous approximation. We may have better approximation in higher order of modulus of continuity for these operators.

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TheΘ-transformation of certain positive linear operators

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171296000932 ◽

1996 ◽

Vol 19 (4) ◽

pp. 667-678 ◽

Cited By ~ 1

Author(s):

Aleandru Lupaş ◽

Detlef H. Mache

Keyword(s):

Construction Method ◽

Positive Operators ◽

Linear Operators ◽

Positive Linear Operators ◽

Order Of Approximation ◽

Linear Positive Operators

The intention of this paper is to describe a construction method for a new sequence of linear positive operators, which enables us to get a pointwise order of approximation regarding the polynomial summator operators which have best properties of approximation.

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