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1
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Approximation by perturbed neural network operators

100%
Anastassiou G.
Applicationes Mathematicae
|
2015
|
tom 42
|
nr 1
57-81
EN
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 our convergence inequalities do not depend on the activation function. The sample functionals are of Stancu, Kantorovich or quadrature types. We give applications for the first derivative of the function involved.
2
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Left general fractional monotone approximation theory

100%
Anastassiou G.
Applicationes Mathematicae
|
2016
|
tom 43
|
nr 1
117-131
EN
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, and furthermore f is approximated uniformly by Qₙ over [a,b]. The degree of this constrained approximation is given by an inequality using the first modulus of continuity of $f^{(p)}$. We finish with applications of the main fractional monotone approximation theorem for different g. On the way to proving the main theorem we establish useful related general results.
3
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General uniform approximation theory by multivariate singular integral operators

100%
Anastassiou G.
Annales Polonici Mathematici
|
2012
|
tom 103
|
nr 1
15-25
EN
We study the uniform approximation properties of general multivariate singular integral operators on $ℝ^{N}$, 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.
4
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On global smoothness preservation in complex approximation

64%
Anastassiou G., Gal S.
Annales Polonici Mathematici
|
2002
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tom 79
|
nr 3
199-205
EN
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.
5
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Representation formulae for (C₀) m-parameter operator semigroups

64%
Zhou M., Anastassiou G.
Annales Polonici Mathematici
|
1996
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tom 63
|
nr 3
247-272
EN
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.
6
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On some shift invariant integral operators, univariate case

64%
Anastassiou G., Gonska H.
Annales Polonici Mathematici
|
1995
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tom 61
|
nr 3
225-243
EN
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 preservation of continuous probabilistic functions. Finally, four examples of very general specialized operators are presented fulfilling all the above properties; in particular, the inequalities for global smoothness preservation are proven to be sharp.
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