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1

Quantitative approximation by shift invariant multivariate sublinear-Choquet operators

Anastassiou G. A.

Journal of Applied Analysis

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2018

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Vol. 24, nr 2

115--126

EN

A very general multivariate positive sublinear Choquet integral type operator is given through a convolution-like iteration of another multivariate general positive sublinear operator with a multivariate scaling type function. For it, sufficient conditions are given for shift invariance, preservation of global smoothness, convergence to the unit with rates. Furthermore, two examples of very general multivariate specialized operators are presented fulfilling all the above properties; the higher order of multivariate approximation of these operators is also studied.

2

Approximation by Max-Product operators

Anastassiou G. A.

Fasciculi Mathematici

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2018

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Nr 60

5--28

EN

Here we study the approximation of functions by a great variety of Max-Product operators under differentiability. These are positive sublinear operators. Our study is based on our general results about positive sublinear operators. We produce Jackson type inequalities under initial conditions. So our approach is quantitative by producing inequalities with their right hand sides involving the modulus of continuity of a high order derivative of the function under approximation. We improve known related results which do not use smoothness of functions.

3

Approximations by multivariate sublinear and Max-product operators under convexity

Anastassiou G. A.

Demonstratio Mathematica

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2018

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Vol. 51, nr 1

85--105

EN

Here we search quantitatively under convexity the approximation of multivariate function by general multivariate positive sublinear operators with applications to multivariate Max-product operators. These are of Bernstein type, of Favard-Szász-Mirakjan type, of Baskakov type, of sampling type, of Lagrange interpolation type and of Hermite-Fejér interpolation type. Our results are both: under the presence of smoothness and without any smoothness assumption on the function to be approximated which fulfills a convexity assumption.

4

Monotone Convergence of Extended Iterative Methods and Fractional Calculus with Applications

Anastassiou G. A., Argyros I. K., Kumar S.

Fundamenta Informaticae

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2017

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Vol. 151, nr 1/4

241--253

EN

We present monotone convergence results for general iterative methods in order to approximate a solution of a nonlinear equation defined on a partially ordered linear topological space. The main novelty of the paper is that the operators appearing in the iterative method are not necessarily linear. This way we expand of the applicability of iterative methods. Some applications are also provided from fractional calculus using Caputo and Canavati type fractional derivatives and other areas.

5

Multivariate and abstract approximation theory for Banach space valued functions

Anastassiou G. A.

Demonstratio Mathematica

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2017

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Vol. 50, nr 1

208--222

EN

Here we study quantitatively the high degree of approximation of sequences of linear operators acting on Banach space valued Fréchet differentiable functions to the unit operator, as well as other basic approximations including those under convexity. These operators are bounded by real positive linear companion operators. The Banach spaces considered here are general and no positivity assumption is made on the initial linear operators for which we study their approximation properties. We derive pointwise and uniform estimates, which imply the approximation of these operators to the unit assuming Fréchet differentiability of functions, and then we continue with basic approximations. At the end we study the special case where the approximated function fulfills a convexity condition resulting into sharp estimates. We give applications to Bernstein operators.

6

Fractional self adjoint operator Poincaré and Sobolev type inequalities

Anastassiou G. A.

Fasciculi Mathematici

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2016

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Nr 57

5--24

EN

We present here many fractional self adjoint operator Poincaré and Sobolev type inequalities to various directions. Initially we give several fractional representation formulae in the self adjoint operator sense. Inequalities are based in the self adjoint operator order over a Hilbert space.

7

Univariate right fractional polynomial high order monotone approxi mation

Anastassiou G. A.

Demonstratio Mathematica

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2016

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Vol. 49, nr 1

1--10

EN

Let (…) and let L* be a linear right fractional differential operator such that L*(f) ≥ 0 throughout [-1, 0]. We can find a sequence of polynomials Qn of degree ≤ n such that L*(Qn)≥ 0 over [-1, 0], furthermore f is approximated right fractionally and simultaneously by Qn on [-1, 1]. The degree of these restricted approximations is given via inequalities using a higher order modulus of smoothness for f(r).

8

Harmonic multivariate Ostrowski and Grüss type inequalities for several functions

Anastassiou G. A.

Demonstratio Mathematica

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2015

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Vol. 48, nr 1

31--52

EN

Here we derive very general multivariate Ostrowski and Grüss type inequalities for several functions by involving harmonic polynomials. Estimates are with respect to all basic norms. We give applications.

9

Fractional representation formulae under initial conditions and fractional Ostrowski type inequalities

Anastassiou G. A.

Demonstratio Mathematica

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2015

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Vol. 48, nr 3

357--378

EN

Here we present very general fractional representation formulae for a function in terms of the fractional Riemann–Liouville integrals of different orders of the function and its ordinary derivatives under initial conditions. Based on these, we derive general fractional Ostrowski type inequalities with respect to all basic norms.

10

Lp-general approximations by multivariate singular integral operators

Anastassiou G. A.

Demonstratio Mathematica

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2013

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Vol. 46, nr 3

491--502

EN

In this article, we study the Lp, 1 ≤ p < ∞ approximation properties of general multivariate singular integral operators over RN, N ≥ 1. We establish their convergence to the unit operator with rates. The established inequalities involve the multivariate higher order modulus of smoothness. We list the multivariate Picard, Gauss–Weierstrass, Poisson Cauchy and trigonometric singular integral operators where this theory can be applied directly.

11

On Gauss-Weierstrass type integral operators

Anastassiou G. A., Aral A.

Demonstratio Mathematica

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2010

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Vol. 43, nr 4

841-849

EN

In this paper, we introduce a generalization of Gauss-Weierstrass operators based on q-integers using the q-integral and we call them q-Gauss-Weierstrass integral operators. For these operators, we obtain a convergence property in a weighted function space using Korovkin theory. Then we estimate the rate of convergence of these operators in terms of a weighted modulus of continuity. We also prove optimal global smoothness preservation property of these operators.