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Deferred weighted A-statistical convergence based upon the (p,q)-Lagrange polynomials and its applications to approximation theorems

Srivastava H. M., Jena B. B., Paikray S. K., Misra U. K.

Journal of Applied Analysis

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2018

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Vol. 24, nr 1

1--16

EN

Recently, the notion of positive linear operators by means of basic (or q-) Lagrange polynomials and A-statistical convergence was introduced and studied in [M. Mursaleen, A. Khan, H. M. Srivastava and K. S. Nisar, Operators constructed by means of q-Lagrange polynomials and A-statistical approximation, Appl. Math. Comput. 219 2013, 12, 6911-6918]. In our present investigation, we introduce a certain deferred weighted A-statistical convergence in order to establish some Korovkin-type approximation theorems associated with the functions 1, t and t2 defined on a Banach space C[0,1] for a sequence of (presumably new) positive linear operators based upon (p,q)-Lagrange polynomials. Furthermore, we investigate the deferred weighted A-statistical rates for the same set of functions with the help of the modulus of continuity and the elements of the Lipschitz class. We also consider a number of interesting special cases and illustrative examples in support of our definitions and of the results which are presented in this paper.

2

Order of approximation for nonlinear sampling Kantorovich operators in Orlicz spaces

Costarelli D., Vinti G.

Commentationes Mathematicae

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2013

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Vol. 53, No. 2

171--192

EN

In this paper, we study the rate of approximation for the nonlinear sampling Kantorovich operators. We consider the case of uniformly continuous and bounded functions belonging to Lipschitz classes of the Zygmund-type, as well as the case of functions in Orlicz spaces. We estimate the aliasing errors with respect to the uniform norm and to the modular functional of the Orlicz spaces, respectively. The general setting of Orlicz spaces allows to deduce directly the results concerning the rate of convergence in Lp-spaces, 1 ≤ p < ∞, very useful in the applications to Signal Processing. Others examples of Orlicz spaces as interpolation spaces and exponential spaces are discussed and the particular cases of the nonlinear sampling Kantorovich series constructed using Fejér and B-spline kernels are also considered.

3

Approximation of functions from IP {uj)p by linear operators of their Fourier series

Łenski W., Szal B.

Commentationes Mathematicae

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2011

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Vol. 51, [Z] 2

189-202

EN

We show the results corresponding to theorems of S. Lai [Appl. Math. Comput., 209 (2009) 346-350] on the rate of approximation of functions from the generalized integral Lipschitz classes by matrix summability means of their Fourier series as well as to the authors theorems [Acta Comment. Univ. Tartu. Math., 13 (2009), 11-24] also on such approximations.

4

On the order of BVφ - approximation of convolution integrals over the line group

Bardaro C., Vinti G.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2004

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Vol. 44, nr spec.

47-63

EN

Here we study the rate of BVφ - approximation of higher order for a class of linear integral operators of convolution type over the line group.