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Bernstein polynomials. --- Approximation theory. --- Bernstein, Polynômes de. --- Approximation, Théorie de l'.

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This book provides comprehensive information on the main aspects of Bernstein operators, based on the literature to date. Bernstein operators have a long-standing history and many papers have been written on them. Among all types of positive linear operators, they occupy a unique position because of their elegance and notable approximation properties. This book presents carefully selected material from the vast body of literature on this topic. In addition, it highlights new material, including several results (with proofs) appearing in a book for the first time. To facilitate comprehension, exercises are included at the end of each chapter. The book is largely self-contained and the methods in the proofs are kept as straightforward as possible. Further, it requires only a basic grasp of analysis, making it a valuable and appealing resource for advanced graduate students and researchers alike.

Mathematics. --- Approximation theory. --- Approximations and Expansions. --- Operator theory. --- Bernstein polynomials. --- Convolutions (Mathematics) --- Convolution transforms --- Transformations, Convolution --- Distribution (Probability theory) --- Functions --- Integrals --- Transformations (Mathematics) --- Polynomials, Bernstein --- Convergence --- Probabilities --- Series --- Functional analysis --- Math --- Science --- Theory of approximation --- Polynomials --- Chebyshev systems

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This monograph presents the first comprehensive treatment in book form of shape-preserving approximation by real or complex polynomials in one or several variables. Such approximation methods are useful in many problems that arise in science and engineering and require an optimal mathematical representation of physical reality. The main topics are structured in four chapters, followed by an appendix: shape-preserving approximation and interpolation of real functions of one real variable by real polynomials; shape-preserving approximation of real functions of several real variables by multivariate real polynomials; shape-preserving approximation of analytic functions of one complex variable by complex polynomials in the unit disk; and shape-preserving approximation of analytic functions of several complex variables on the unit ball or the unit polydisk by polynomials of several complex variables. The appendix treats related results of non-polynomial and non-spline approximations preserving shape including those by complexified operators with applications to complex partial differential equations. Shape-Preserving Approximation by Real and Complex Polynomials contains many open problems at the end of each chapter to stimulate future research along with a rich and updated bibliography surveying the vast literature. The text will be useful to graduate students and researchers interested in approximation theory, mathematical analysis, numerical analysis, computer aided geometric design, robotics, data fitting, chemistry, fluid mechanics, and engineering.

Mathematics. --- Approximations and Expansions. --- Real Functions. --- Functions of a Complex Variable. --- Computational Mathematics and Numerical Analysis. --- Appl.Mathematics/Computational Methods of Engineering. --- Math Applications in Computer Science. --- Computer science. --- Functions of complex variables. --- Computer science --- Engineering mathematics. --- Mathématiques --- Informatique --- Fonctions d'une variable complexe --- Mathématiques de l'ingénieur --- Approximation theory. --- Bernstein polynomials. --- Mathematical optimization. --- Multivariate analysis. --- Approximation theory --- Bernstein polynomials --- Multivariate analysis --- Mathematics --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Operations Research --- Algebra --- Multivariate distributions --- Multivariate statistical analysis --- Statistical analysis, Multivariate --- Polynomials, Bernstein --- Theory of approximation --- Algebra. --- Functions of real variables. --- Computer mathematics. --- Applied mathematics. --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Analysis of variance --- Mathematical statistics --- Matrices --- Convergence --- Integrals --- Probabilities --- Series --- Mathematical and Computational Engineering. --- Engineering --- Engineering analysis --- Mathematical analysis --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Complex variables --- Elliptic functions --- Functions of real variables --- Math --- Science --- Real variables --- Functions of complex variables

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Polynomial and its applications are well known for their proven properties and excellent applicability in interdisciplinary fields of science. Until now, research on polynomial and its applications has been done in mathematics, applied mathematics, and sciences. This book is based on recent results in all areas related to polynomial and its applications. This book provides an overview of the current research in the field of polynomials and its applications. The following papers have been published in this volume: ‘A Parametric Kind of the Degenerate Fubini Numbers and Polynomials’; ‘On 2-Variables Konhauser Matrix Polynomials and Their Fractional Integrals’; ‘Fractional Supersymmetric Hermite Polynomials’; ‘Rational Approximation for Solving an Implicitly Given Colebrook Flow Friction Equation’; ‘Iterating the Sum of Möbius Divisor Function and Euler Totient Function’; ‘Differential Equations Arising from the Generating Function of the (r, β)-Bell Polynomials and Distribution of Zeros of Equations’; ‘Truncated Fubini Polynomials’; ‘On Positive Quadratic Hyponormality of a Unilateral Weighted Shift with Recursively Generated by Five Weights’; ‘Ground State Solutions for Fractional Choquard Equations with Potential Vanishing at Infinity’; ‘Some Identities on Degenerate Bernstein and Degenerate Euler Polynomials’; ‘Some Identities Involving Hermite Kampé de Fériet Polynomials Arising from Differential Equations and Location of Their Zeros.’

Research & information: general --- Mathematics & science --- differential equations, heat equation --- Hermite Kampé de Fériet polynomials --- Hermite polynomials --- generating functions --- degenerate Bernstein polynomials --- degenerate Bernstein operators --- degenerate Euler polynomials --- variational methods --- fractional Choquard equation --- ground state solution --- vanishing potential --- positively quadratically hyponormal --- quadratically hyponormal --- unilateral weighted shift --- recursively generated --- Fubini polynomials --- Euler polynomials --- Bernoulli polynomials --- truncated exponential polynomials --- Stirling numbers of the second kind --- differential equations --- Bell polynomials --- r-Bell polynomials --- (r, β)-Bell polynomials --- zeros --- Möbius function --- divisor functions --- Euler totient function --- hydraulic resistance --- pipe flow friction --- Colebrook equation --- Colebrook–White experiment --- floating-point computations --- approximations --- Padé polynomials --- symbolic regression --- orthogonal polynomials --- difference-differential operator --- supersymmetry --- Konhauser matrix polynomial --- generating matrix function --- integral representation --- fractional integral --- degenerate Fubini polynomials --- Stirling numbers

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Polynomial and its applications are well known for their proven properties and excellent applicability in interdisciplinary fields of science. Until now, research on polynomial and its applications has been done in mathematics, applied mathematics, and sciences. This book is based on recent results in all areas related to polynomial and its applications. This book provides an overview of the current research in the field of polynomials and its applications. The following papers have been published in this volume: ‘A Parametric Kind of the Degenerate Fubini Numbers and Polynomials’; ‘On 2-Variables Konhauser Matrix Polynomials and Their Fractional Integrals’; ‘Fractional Supersymmetric Hermite Polynomials’; ‘Rational Approximation for Solving an Implicitly Given Colebrook Flow Friction Equation’; ‘Iterating the Sum of Möbius Divisor Function and Euler Totient Function’; ‘Differential Equations Arising from the Generating Function of the (r, β)-Bell Polynomials and Distribution of Zeros of Equations’; ‘Truncated Fubini Polynomials’; ‘On Positive Quadratic Hyponormality of a Unilateral Weighted Shift with Recursively Generated by Five Weights’; ‘Ground State Solutions for Fractional Choquard Equations with Potential Vanishing at Infinity’; ‘Some Identities on Degenerate Bernstein and Degenerate Euler Polynomials’; ‘Some Identities Involving Hermite Kampé de Fériet Polynomials Arising from Differential Equations and Location of Their Zeros.’

differential equations, heat equation --- Hermite Kampé de Fériet polynomials --- Hermite polynomials --- generating functions --- degenerate Bernstein polynomials --- degenerate Bernstein operators --- degenerate Euler polynomials --- variational methods --- fractional Choquard equation --- ground state solution --- vanishing potential --- positively quadratically hyponormal --- quadratically hyponormal --- unilateral weighted shift --- recursively generated --- Fubini polynomials --- Euler polynomials --- Bernoulli polynomials --- truncated exponential polynomials --- Stirling numbers of the second kind --- differential equations --- Bell polynomials --- r-Bell polynomials --- (r, β)-Bell polynomials --- zeros --- Möbius function --- divisor functions --- Euler totient function --- hydraulic resistance --- pipe flow friction --- Colebrook equation --- Colebrook–White experiment --- floating-point computations --- approximations --- Padé polynomials --- symbolic regression --- orthogonal polynomials --- difference-differential operator --- supersymmetry --- Konhauser matrix polynomial --- generating matrix function --- integral representation --- fractional integral --- degenerate Fubini polynomials --- Stirling numbers