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Refined rates of bias convergence for generalized L-Statistics in the i.i.d. case

George AnastassiouTomasz Rychlik — 1999

Applicationes Mathematicae

Using tools of approximation theory, we evaluate rates of bias convergence for sequences of generalized L-statistics based on i.i.d. samples under mild smoothness conditions on the weight function and simple moment conditions on the score function. Apart from standard methods of weighting, we introduce and analyze L-statistics with possibly random coefficients defined by means of positive linear functionals acting on the weight function.

Approximation by perturbed neural network operators

George A. Anastassiou — 2015

Applicationes Mathematicae

This article deals with the determination of the rate of convergence to the unit of each of three newly introduced perturbed normalized neural network operators of one hidden layer. These are given through the modulus of continuity of the function involved or its high order derivative that appears in the right-hand side of the associated Jackson type inequalities. The activation function is very general, in particular it can derive from any sigmoid or bell-shaped function. The right-hand sides of...

Left general fractional monotone approximation theory

George A. Anastassiou — 2016

Applicationes Mathematicae

We introduce left general fractional Caputo style derivatives with respect to an absolutely continuous strictly increasing function g. We give various examples of such fractional derivatives for different g. Let f be a p-times continuously differentiable function on [a,b], and let L be a linear left general fractional differential operator such that L(f) is non-negative over a closed subinterval I of [a,b]. We find a sequence of polynomials Qₙ of degree ≤n such that L(Qₙ) is non-negative over I,...

General uniform approximation theory by multivariate singular integral operators

George A. Anastassiou — 2012

Annales Polonici Mathematici

We study the uniform approximation properties of general multivariate singular integral operators on N , N ≥ 1. We establish their convergence to the unit operator with rates. The estimates are pointwise and uniform. 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 to which this theory can be applied directly.

Ostrowski Type Inequalities over Spherical Shells

Anastassiou, George A. — 2008

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 26D10, 26D15. Here are presented Ostrowski type inequalities over spherical shells. These regard sharp or close to sharp estimates to the difference of the average of a multivariate function from its value at a point.

Most General Fractional Representation Formula for Functions and Implications

Anastassiou, George A. — 2014

Serdica Mathematical Journal

Here we present the most general fractional representation formulae for a function in terms of the most general fractional integral operators due to S. Kalla, [3], [4], [5]. The last include most of the well-known fractional integrals such as of Riemann-Liouville, Erdélyi-Kober and Saigo, etc. Based on these we derive very general fractional Ostrowski type inequalities. 2010 Mathematics Subject Classification: 26A33, 26D10, 26D15.

On some shift invariant integral operators, univariate case

George A. AnastassiouHeinz H. Gonska — 1995

Annales Polonici Mathematici

In recent papers the authors studied global smoothness preservation by certain univariate and multivariate linear operators over compact domains. Here the domain is ℝ. A very general positive linear integral type operator is introduced through a convolution-like iteration of another general positive linear operator with a scaling type function. For it sufficient conditions are given for shift invariance, preservation of global smoothness, convergence to the unit with rates, shape preserving and...

Representation formulae for (C₀) m-parameter operator semigroups

Mi ZhouGeorge A. Anastassiou — 1996

Annales Polonici Mathematici

Some general representation formulae for (C₀) m-parameter operator semigroups with rates of convergence are obtained by the probabilistic approach and multiplier enlargement method. These cover all known representation formulae for (C₀) one- and m-parameter operator semigroups as special cases. When we consider special semigroups we recover well-known convergence theorems for multivariate approximation operators.

On global smoothness preservation in complex approximation

George A. AnastassiouSorin G. Gal — 2002

Annales Polonici Mathematici

By using the properties of convergence and global smoothness preservation of multivariate Weierstrass singular integrals, we establish multivariate complex Carleman type approximation results with rates. Here the approximants fulfill the global smoothness preservation property. Furthermore Mergelyan's theorem for the unit disc is strengthened by proving the global smoothness preservation property.

Fractional Trigonometric Korovkin Theory in Statistical Sense

Anastassiou, George A.Duman, Oktay — 2010

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 41A25, 41A36. In the present paper, we improve the classical trigonometric Korovkin theory by using the concept of statistical convergence from the summability theory and also by considering the fractional derivatives of functions. We also show that our new results are more applicable than the classical ones.

Multidimensional Opial inequalities for functions vanishing at an interior point

George A. AnastassiouGisèle Ruiz GoldsteinJerome A. Goldstein — 2004

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In this paper we generalize Opial inequalities in the multidimensional case over balls. The inequalities carry weights and are proved to be sharp. The functions under consideration vanish at the center of the ball.

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