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Home 9 Author: George A. Anastassiou
George A. Anastassiou
Open AccessArticle

q-Deformed and L-parametrized hyperbolic tangent function relied complex valued multivariate trigonometric and hyperbolic neural network approximations

Annals of Communications in Mathematics 2023

, 6 (3)

, 141-164

DOI: https://doi.org/10.62072/acm.2023.060301

AbstractHere we study the multivariate quantitative approximation of complex valued continuous functions on a box of RN , N ∈ N, by the multivariate normalized type neural network operators. We investigate also the case of approximation by iterated multilayer neural network operators. These approximations are achieved by establishing multidimen-sional Jackson type inequalities involving the multivariate moduli of continuity of the en- gaged function and its partial derivatives. Our multivariate operators are defined by using a multidimensional density function induced by a q-deformed and λ-parametrized hyper-bolic tangent function, which is a sigmoid function. The approximations are pointwise and uniform. The related feed-forward neural network are with one or multi hidden layers. The basis of our theory are the introduced multivariate Taylor formulae of trigonometric and hyperbolic type.
Open AccessArticle

Trigonometric and hyperbolic Poincare, Sobolev and Hilbert-Pachpatte type inequalities

Annals of Communications in Mathematics 2023

, 6 (3)

, 191-198

DOI: https://doi.org/10.62072/acm.2023.060304

AbstractIn this article based on trigonometric and hyperbolic type Taylor formulae we establish Poincare, Sobolev and Hilbert-Pachpatte type inequalities of different kinds specific and general.
Open AccessArticle

Trigonometric generated Lp degree of approximation

Annals of Communications in Mathematics 2023

, 6 (4)

, 209-219

DOI: https://doi.org/10.62072/acm2023060401

AbstractIn this article we continue the study of smooth Picard singular integral operators that started in [3], see there chapters 10-14. This time the foundation of our research is a trigonometric Taylor’s formula. We establish the Lp convergence of our operators to the unit operator with rates via Jackson type inequalities engaging the first Lp modulus of continuity. Of interest here is a residual appearing term. Note that our operators are not positive.
Open AccessArticle

q-Deformed and β-parametrized half hyperbolic tangent based Banach space valued ordinary and fractional neural network approximation

Annals of Communications in Mathematics 2023

, 6 (1)

, 1-16

DOI: https://doi.org/10.62072/acm2023060101

AbstractHere we research the univariate quantitative approximation, ordinary and fractional, of Banach space valued continuous functions on a compact interval or all the real line by quasi-interpolation Banach space valued neural network operators. These approximations are derived by establishing Jackson type inequalities involving the modulus of continuity of the engaged function or its Banach space valued high order derivative of fractional derivatives. Our operators are defined by using a density function generated by a q-deformed and β-parametrized half hyperbolic tangent function, which is a sigmoid function. The approximations are pointwise and of the uniform norm. The related Banach space valued feed-forward neural networks are with one hidden layer.
Open AccessArticle

Parametrized error function based Banach space valued univariate neural network approximation

Annals of Communications in Mathematics 2023

, 6 (1)

, 31-43

DOI: https://doi.org/10.62072/acm2023060104

AbstractHere we research the univariate quantitative approximation of Banach space valued continuous functions on a compact interval or all the real line by quasi-interpolation Banach space valued neural network operators. We perform also the related Banach space valued ractional approximation. These approximations are derived by establishing Jackson type inequalities involving the modulus of continuity of the engaged function or its Banach space valued high order derivative or fractional derivaties. Our operators are defined by using a density function induced by a parametrized error function. The approximations are pointwise and with respect to the uniform norm. The related Banach space valued feed-forward neural networks are with one hidden layer. We finish with a convergence analysis.
Open AccessArticle

Sequential Generalized Fractional Ostrowski and Grüss type inequalities for several Banach algebra valued functions

Annals of Communications in Mathematics 2021

, 4 (3)

, 207-225

DOI: https://doi.org/10.62072/acm.2021.040301

AbstractEmploying sequential generalized Caputo fractional left and right vectorial Taylor formulae we establish mixed sequential generalized fractional Ostrowski and Gruss ¨ type inequalities for several Banach algebra valued functions. The estimates are with respect to all norms k·kp , 1 ≤ p ≤ ∞. We finish with applications.
Open AccessArticle

Multivariate right side Caputo fractional Taylor formula and Landau inequalities

Annals of Communications in Mathematics 2020

, 3 (3)

, 185-192

DOI: https://doi.org/10.62072/acm.2020.030301

AbstractHere we present a multivariate right side Caputo fractional Taylor’s formula with fractional integral remainder. Based on this we give three multivariate right side Caputo fractional Landau’s type inequalities. Their constants are precisely calculated and we give best upper bounds.
Open AccessArticle

Generalized Canavati type g-fractional Iyengar and Ostrowski type

Annals of Communications in Mathematics 2019

, 2 (2)

, 57-72

DOI: https://doi.org/10.62072/acm.2019.020201

AbstractWe present here generalized Canavati type g-fractional Iyengar and Ostrowski type inequalities. Our inequalities are with respect to all Lp norms: 1 ≤ p ≤ ∞. We finish with applications.