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Abstract: In this paper we construct some positive linear operators by means of q-Lagrange polynomials and prove some approximation results via A-statistical convergence. We also define and study the rate of A-statistical approximation of these operators by using the notion of modulus of continuity and Lipschitz class. [Copyright &y& Elsevier]

Abstract In the present paper, two different modifications are proposed for (p, q)-Szász-Mirakjan-Kantorovich operators which preserve some test functions. Some approximation results with the help of better-known Korovkin's theorem and weighted Korovkin's theorem for these operators are presented. Furthermore, convergence properties in terms of modulus of continuity and class of Lipschitz functions are studied. It has been shown that for a given absolute error bound, King type modified (p, q)-Szász-Mirakjan-Kantorovich operators require lesser value of m and elapsed time within some subintervals. Further for comparisons, some graphics and error estimation tables are presented using MATLAB(R2018a). [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new analogue of Bernstein operators and we call it as ( p, q )-Bernstein operators which is a generalization of q -Bernstein operators. We also study approximation properties based on Korovkin’s type approximation theorem of ( p, q )-Bernstein operators and establish some direct theorems. Furthermore, we show comparisons and some illustrative graphics for the convergence of operators to a function. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new analogue of Bernstein–Stancu operators based on ( p, q )-integers which we call as ( p, q )-Bernstein–Stancu operators. We study approximation properties for these operators based on Korovkin’s type approximation theorem and also study some direct theorems. Furthermore, we give comparisons and some illustrative graphics for the convergence of operators to some function. [ABSTRACT FROM AUTHOR]

Orientation estimation is considered as an important task in many subsequent pattern recognition and image enhancement systems. In a noisy environment, the gradient-based estimator provides poor results. A pre-smoothing Gaussian function with an appropriate scale is conventionally used to get improved gradients. Later on, a family of pre-smoothing Gaussian functions with a range of scales is employed for estimation, this is referred to as multi-scale orientation estimator. To provide groundwork for comparison, a more formal framework of multi-scale orientation estimation, based on scale-space axioms, in spatial domain is presented. Then for improvement purposes a Fourier domain approach, where directional filter bank (DFB) structure is embedded in multi-scale orientation estimation framework, is proposed. This is referred to as multi-scale DFB approach. The paper presents the comparison work for estimation of local orientations using multi-scale approaches both in spatial and Fourier domain. In the Fourier-domain approach, two linear combinations are deployed, one across the directional image, and the other across the scales. This is opposed to only one linear combination across the scales, used in simple spatial domain techniques. Further more, the DFB-based Fourier domain approach extracts the best local orientation by comparing and contrasting all possible orientations with their respective strength measures. The strength measure used in Fourier method is based on local variance, free from inaccurate gradient calculation. Simulations are conducted over noisy test images as well as real fingerprints. Our objective results indicate that multi-scale Fourier domain approach always yields better estimates at variable level of noise as compared to stand alone multi-scale spatial domain approaches. The improvements made by Fourier domain estimate can largely be attributed to the use of double linear combination both across the directional bands and across the scales. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we prove some approximation results in statistical sense and establish some direct theorems for the positive linear operators constructed by the means of Lagrange type polynomials. We compute error estimation by using modulus of continuity with the help of Matlab and give its algorithm. Furthermore, we show graphically the convergence of our operators to various functions. [Copyright &y& Elsevier]

Abstract: Korovkin type approximation theorems are useful tools to check whether a given sequence of positive linear operators on of all continuous functions on the real interval is an approximation process. That is, these theorems exhibit a variety of test functions which assure that the approximation property holds on the whole space if it holds for them. Such a property was discovered by Korovkin in 1953 for the functions and in the space as well as for the functions 1, cos and sin in the space of all continuous -periodic functions on the real line. In this paper, we use the notion of weighted statistical convergence to prove the Korovkin approximation theorem for the functions 1, cos and sin in the space of all continuous -periodic functions on the real line and show that our result is stronger. We also study the rate of weighted statistical convergence. [Copyright &y& Elsevier]

In the present paper, we have discussed about Theorem 3.1, Remark 3.1 and Theorem 4.1 studied in our article “Some approximation results by ( p, q )-analogue of Bernstein–Stancu operators” [Appl. Math. Comput. 264 (2015) 392–402]. We give here the correct version of the said Theorem 3.1, Remark 3.1 and Theorem 4.1. [ABSTRACT FROM AUTHOR]