This monograph is a testimony of the impact over Computational Analysis of some new trigonometric and hyperbolic types of Taylor’s formulae with integral remainders producing a rich collection of approximations of a very wide spectrum.

This volume covers perturbed neural network approximations by themselves and with their connections to Brownian motion and stochastic processes, univariate and multivariate analytical inequalities (both ordinary and fractional), Korovkin theory, and approximations by singular integrals (both univariate and multivariate cases). These results are expected to find applications in the many areas of Pure and Applied Mathematics, Computer Science, Engineering, Artificial Intelligence, Machine Learning, Deep Learning, Analytical Inequalities, Approximation Theory, Statistics, Economics, amongst others. Thus, this treatise is suitable for researchers, graduate students, practitioners and seminars of related disciplines, and serves well as an invaluable resource for all Science and Engineering libraries.

Contents:

• Trigonometric and Hyperbolic Taylor Formulae and Opial and Ostrowski Inequalities


• Perturbed A-Generalized Logistic Function-Activated Complex-Valued Trigonometric and Hyperbolic Neural Network High Order Approximation


• Perturbed A-Generalized Logistic Function-Activated Complex-Valued Multivariate Trigonometric and Hyperbolic Neural Network Approximation


• Perturbed Hyperbolic Tangent Function-Activated Complex-Valued Trigonometric and Hyperbolic Neural Network High Order Approximation


• Perturbed Hyperbolic Tangent Function-Activated Complex-Valued Multivariate Trigonometric and Hyperbolic Neural Network Approximation


• General Sigmoid Function-Activated Complex-Valued Trigonometric and Hyperbolic Neural Network High Order Approximation


• General Multiple Sigmoid Functions Activated Complex Valued Multivariate Trigonometric and Hyperbolic Neural Network Approximation


• Approximation of Multiple Time Separating Random Functions by Neural Networks


• Brownian Motion Approximation by Perturbed Neural Networks


• Approximation of Brownian Motion over Simple Graphs


• Multivariate Fuzzy-Random with Perturbed Neural Network Approximation


• Trigonometric and Hyperbolic Korovkin Properties


• Integral Inequalities Using the New Conformable Derivatives


• About Proportional Fractional Calculus and Inequalities


• New Radial Ostrowski Inequalities on the Ball


• New Ostrowski Inequalities on a Spherical Shell


• Trigonometric and Hyperbolic Polya Inequalities


• New Opial and Polya Inequalities Over a Spherical Shell


• Trigonometric and Hyperbolic Poincaré, Sobolev and Hilbert-Pachpatte Inequalities


• Trigonometric-Induced Degree of Approximation by Smooth Picard Singular Integral Operators


• Trigonometric-Derived Lp Degree of Approximation by Smooth Picard Singular Integral Operators


• Parametrized and Trigonometric-Induced Quantitative Convergence of Smooth Picard Singular Integral Operators


• Parametrized and Trigonometric Lp Approximation with Rates of Smooth Picard Singular Integral Operators


• Update on Uniform Approximation by Smooth Picard Multivariate Singular Integral Operators Revisited


• Trigonometric-Derived Multivariate Smooth Picard Singular Integrals Lp Quantitative Approximation


• Trigonometric Produced Rate of Approximation of Various Smooth Singular Integral Operators


• Trigonometric-Derived Lp Quantitative Approximation by Various Smooth Singular Integral Operators


• Parametrized and Trigonometric-Produced Uniform Quantitative Approximation by Various Smooth Singular Integral Operators


• Parametrized Trigonometric-Implied Lp Degree of Quantitative Approximation by Various Smooth Singular Integral Operators


• Trigonometric-Implied Multivariate Smooth Gauss–Weierstrass Singular Integrals Quantitative Approximation


• Trigonometric-Based Multivariate Smooth Poisson–Cauchy Singular Integrals Quantitative Approximation


• Trigonometric Implied Multivariate Smooth Trigonometric Singular Integrals Quantitative Approximation

Readership: Graduate students and researchers interested in approximation theory, from the fields of Pure Mathematics, Statistics, Engineering and Computer Science.

George A Anastassiou received his BSc degree in Mathematics from Athens University, Greece in 1975. He received his Diploma in Operations Research from Southampton University, UK in 1976. He also received his MA in Mathematics from the University of Rochester, USA in 1981. He was awarded his Ph D in Mathematics from the University of Rochester, USA in 1984. From 1984–86, he served as a visiting assistant professor at the University of Rhode Island, USA. Since 1986 till now, he is a faculty member at the University of Memphis, USA. He is currently a full Professor of Mathematics since 1994. His research area is Computational Analysis, in a very broad sense. He has published over 700 research articles in international mathematics journals and over 52 monographs, proceedings and textbooks in well-known publishing houses. Several awards have been conferred to George Anastassiou. In 2007, he received an Honorary Doctoral Degree from University of Oradea, Romania. He is associate editor in over 80 international journals in mathematics, and editor in-chief in 3 journals: most notably in the well-known Journal of Computational Analysis and Applications.