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Functional analysis --- Probability theory --- Approximatietheorie --- Approximation theory --- Probabiliteit--Theorie --- Probabiliteitstheorie --- Probabilities --- Probabilité [Théorie de la ] --- Probabilités --- Théorie des approximations --- Waarschijnlijkheid--Theorie --- Waarschijnlijkheidstheorie

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Approximation by Multivariate Singular Integrals is the first monograph to illustrate the approximation of multivariate singular integrals to the identity-unit operator. The basic approximation properties of the general multivariate singular integral operators is presented quantitatively, particularly special cases such as the multivariate Picard, Gauss-Weierstrass, Poisson-Cauchy and trigonometric singular integral operators are examined thoroughly. This book studies the rate of convergence of these operators to the unit operator as well as the related simultaneous approximation. The last chapter, which includes many examples, presents a related Korovkin type approximation theorem for functions of two variables.   Relevant background information and motivation is included in this exposition, and as a result this book can be used as supplementary text for several advanced courses. The results presented apply to many areas of pure and applied mathematics, such a mathematical analysis, probability, statistics and partial differential equations.  This book is appropriate for researchers and selected seminars at the graduate level.

Fiber-reinforced ceramics. --- Functions of complex variables. --- Singular integrals. --- Approximation theory --- Multivariate analysis --- Singular integrals --- Civil & Environmental Engineering --- Mathematics --- Physical Sciences & Mathematics --- Engineering & Applied Sciences --- Operations Research --- Algebra --- Calculus --- Integrals, Singular --- Mathematics. --- Integral transforms. --- Operational calculus. --- Partial differential equations. --- Probabilities. --- Integral Transforms, Operational Calculus. --- Partial Differential Equations. --- Probability Theory and Stochastic Processes. --- Integral operators --- Integral transforms --- Integral Transforms. --- Differential equations, partial. --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Partial differential equations --- Transform calculus --- Integral equations --- Transformations (Mathematics) --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Operational calculus --- Differential equations --- Electric circuits

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Fractional differentiation inequalities are by themselves an important area of research. They have many applications in pure and applied mathematics and many other applied sciences. One of the most important applications is in establishing the uniqueness of a solution in fractional differential equations and systems and in fractional partial differential equations. They also provide upper bounds to the solutions of the above equations. In this book the author presents the Opial, Poincaré, Sobolev, Hilbert, and Ostrowski fractional differentiation inequalities. Results for the above are derived using three different types of fractional derivatives, namely by Canavati, Riemann-Liouville and Caputo. The univariate and multivariate cases are both examined. Each chapter is self-contained. The theory is presented systematically along with the applications. The application to information theory is also examined. This monograph is suitable for researchers and graduate students in pure mathematics. Applied mathematicians, engineers, and other applied scientists will also find this book useful.

Differential inequalities. --- Fractional calculus. --- Fractional calculus --- Differential inequalities --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Derivatives and integrals, Fractional --- Differentiation of arbitrary order, Integration and --- Differintegration, Generalized --- Fractional derivatives and integrals --- Generalized calculus --- Generalized differintegration --- Integrals, Fractional derivatives and --- Integration and differentiation of arbitrary order --- Mathematics. --- Functional analysis. --- Integral transforms. --- Operational calculus. --- Differential equations. --- Partial differential equations. --- Ordinary Differential Equations. --- Functional Analysis. --- Partial Differential Equations. --- Integral Transforms, Operational Calculus. --- Inequalities (Mathematics) --- Differential Equations. --- Differential equations, partial. --- Integral Transforms. --- Transform calculus --- Integral equations --- Transformations (Mathematics) --- Partial differential equations --- Functional calculus --- Calculus of variations --- Functional equations --- 517.91 Differential equations --- Differential equations --- Operational calculus --- Electric circuits

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This monograph presents recent and original work of the author on inequalities in real, functional and fractional analysis. The chapters are self-contained and can be read independently, they include an extensive list of references per chapter. The book’s results are expected to find applications in many areas of applied and pure mathematics, especially in ordinary and partial differential equations and fractional differential equations. As such this monograph is suitable for researchers, graduate students, and seminars of the above subjects, as well as Science and Engineering University libraries.  .

Computer Science --- Engineering & Applied Sciences --- Engineering. --- Artificial intelligence. --- Mathematical analysis. --- Analysis (Mathematics). --- Computational intelligence. --- Computational Intelligence. --- Artificial Intelligence (incl. Robotics). --- Analysis. --- Intelligence, Computational --- Artificial intelligence --- Soft computing --- 517.1 Mathematical analysis --- Mathematical analysis --- AI (Artificial intelligence) --- Artificial thinking --- Electronic brains --- Intellectronics --- Intelligence, Artificial --- Intelligent machines --- Machine intelligence --- Thinking, Artificial --- Bionics --- Cognitive science --- Digital computer simulation --- Electronic data processing --- Logic machines --- Machine theory --- Self-organizing systems --- Simulation methods --- Fifth generation computers --- Neural computers --- Construction --- Industrial arts --- Technology --- Global analysis (Mathematics). --- Artificial Intelligence. --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Inequalities (Mathematics) --- Processes, Infinite

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PLEASE USE THE FILE BACK COVER! Knowledge can be modelled and computed using computational mathematical methods, then lead to real world conclusions. The strongly related to that Computational Analysis is a very large area with lots of applications. This monograph includes a great variety of topics of Computational Analysis. We present: probabilistic wavelet approximations, constrained abstract approximation theory, shape preserving weighted approximation, non positive approximations to definite integrals, discrete best approximation, approximation theory of general Picard singular operators including global smoothness preservation property, fractional singular operators. We also deal with non-isotropic general Picard singular multivariate operators and q-Gauss-Weierstrass singular q-integral operators.We talk about quantitative approximations by shift-invariant univariate and multivariate integral operators, nonlinear neural networks approximation, convergence with rates of positive linear operators, quantitative approximation by bounded linear operators, univariate and multivariate quantitative approximation by stochastic positive linear operators on univariate and multivariate stochastic processes. We further present right fractional calculus and give quantitative fractional Korovkin theory of positive linear operators. We also give analytical inequalities, fractional Opial inequalities, fractional identities and inequalities regarding fractional integrals.We further deal with semigroup operator approximation, simultaneous Feller probabilistic approximation. We also present Fuzzy singular operator approximations.We give transfers from real to fuzzy approximation and talk about fuzzy wavelet and fuzzy neural networks approximations, fuzzy fractional calculus and fuzzy Ostrowski inequality. We talk about discrete fractional calculus, nabla discrete fractional calculus and inequalities.We study the q-inequalities, and q-fractional inequalities. We further study time scales: delta and nabla approaches, duality principle and inequalities. We introduce delta and nabla time scales fractional calculus and inequalities.We finally study convergence with rates of approximate solutions to exact solution of multivariate Dirichlet problem and multivariate heat equation, and discuss the uniqueness of solution of general evolution partial differential equation in multivariate time. The exposed results are expected to find applications to: applied and computational mathematics, stochastics, engineering, artificial intelligence, vision, complexity and machine learning. This monograph is suitable for graduate students and researchers.

Approximation theory. --- Mathematical analysis. --- MATHEMATICS -- General. --- Engineering & Applied Sciences --- Computer Science --- Calculus. --- Analysis (Mathematics) --- Fluxions (Mathematics) --- Infinitesimal calculus --- Limits (Mathematics) --- Theory of approximation --- 517.1 Mathematical analysis --- Mathematical analysis --- Engineering. --- Artificial intelligence. --- Computational intelligence. --- Computational Intelligence. --- Artificial Intelligence (incl. Robotics). --- Intelligence, Computational --- Artificial intelligence --- Soft computing --- AI (Artificial intelligence) --- Artificial thinking --- Electronic brains --- Intellectronics --- Intelligence, Artificial --- Intelligent machines --- Machine intelligence --- Thinking, Artificial --- Bionics --- Cognitive science --- Digital computer simulation --- Electronic data processing --- Logic machines --- Machine theory --- Self-organizing systems --- Simulation methods --- Fifth generation computers --- Neural computers --- Construction --- Industrial arts --- Technology --- Functions --- Geometry, Infinitesimal --- Functional analysis --- Polynomials --- Chebyshev systems --- Artificial Intelligence.