Weighted polynomial approximation and numerical methods for integral equations / / Peter Junghanns, Giuseppe Mastroianni, Incoronata Notarangelo

• Autore: Junghanns Peter <1953->
• Pubblicazione: Cham, Switzerland : , : Birkhäuser, , [2021]; ©2021
• Edizione: 1st edition 2021.
• Descrizione fisica: 1 online resource (X, 658 p. 5 illus., 3 illus. in color.)
• Disciplina: 511.4
• Soggetto topico: Approximation theory; Integral equations - Numerical solutions; Teoria de l'aproximació
• Soggetto genere / forma: Llibres electrònics
• Persona (resp. second.): MastroianniG (Giuseppe); NotarangeloIncoronata
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- Preface -- Contents -- 1 Introduction -- 2 Basics from Linear and Nonlinear Functional Analysis -- 2.1 Linear Operators, Banach and Hilbert Spaces -- 2.2 Fundamental Principles -- 2.3 Compact Sets and Compact Operators -- 2.4 Function Spaces -- 2.4.1 Lp-Spaces -- 2.4.2 Spaces of Continuous Functions -- 2.4.3 Approximation Spaces and Unbounded Linear Operators -- 2.5 Fredholm Operators -- 2.6 Stability of Operator Sequences -- 2.7 Fixed Point Theorems and Newton's Method -- 3 Weighted Polynomial Approximation and Quadrature Rules on (-1,1) -- 3.1 Moduli of Smoothness, K-Functionals, and Best Approximation -- 3.1.1 Moduli of Smoothness and K-Functionals -- 3.1.2 Moduli of Smoothness and Best Weighted Approximation -- 3.1.3 Besov-Type Spaces -- 3.2 Polynomial Approximation with Doubling Weights on the Interval (-1,1) -- 3.2.1 Definitions -- 3.2.2 Polynomial Inequalities with Doubling Weights -- 3.2.3 Christoffel Functions with Respect to Doubling Weights -- 3.2.4 Convergence of Fourier Sums in Weighted Lp-Spaces -- 3.2.5 Lagrange Interpolation in Weighted Lp-Spaces -- 3.2.6 Hermite Interpolation -- 3.2.7 Hermite-Fejér Interpolation -- 3.2.8 Lagrange-Hermite Interpolation -- 3.3 Polynomial Approximation with Exponential Weights on the Interval (-1,1) -- 3.3.1 Polynomial Inequalities -- 3.3.2 K-Functionals and Moduli of Smoothness -- 3.3.3 Estimates for the Error of Best Weighted Polynomial Approximation -- 3.3.4 Fourier Sums in Weighted Lp-Spaces -- 3.3.5 Lagrange Interpolation in Weighted Lp-Spaces -- 3.3.6 Gaussian Quadrature Rules -- 4 Weighted Polynomial Approximation and Quadrature Rules on Unbounded Intervals -- 4.1 Polynomial Approximation with Generalized Freud Weights on the Real Line -- 4.1.1 The Case of Freud Weights -- 4.1.2 The Case of Generalized Freud Weights -- 4.1.3 Lagrange Interpolation in Weighted Lp-Spaces. ; 4.1.4 Gaussian Quadrature Rules -- 4.1.5 Fourier Sums in Weighted Lp-Spaces -- 4.2 Polynomial Approximation with Generalized Laguerre Weights on the Half Line -- 4.2.1 Polynomial Inequalities -- 4.2.2 Weighted Spaces of Functions -- 4.2.3 Estimates for the Error of Best Weighted Approximation -- 4.2.4 Fourier Sums in Weighted Lp-Spaces -- 4.2.5 Lagrange Interpolation in Weighted Lp-Spaces -- 4.3 Polynomial Approximation with Pollaczek-Laguerre Weights on the Half Line -- 4.3.1 Polynomial Inequalities -- 4.3.2 Weighted Spaces of Functions -- 4.3.3 Estimates for the Error of Best Weighted Polynomial Approximation -- 4.3.4 Gaussian Quadrature Rules -- 4.3.5 Lagrange Interpolation in L2w -- 4.3.6 Remarks on Numerical Realizations -- Computation of the Mhaskar-Rahmanov-Saff Numbers -- Numerical Construction of Quadrature Rules -- Numerical Examples -- Comparison with the Gaussian Rule Based on Laguerre Zeros -- 5 Mapping Properties of Some Classes of Integral Operators -- 5.1 Some Properties of the Jacobi Polynomials -- 5.2 Cauchy Singular Integral Operators -- 5.2.1 Weighted L2-Spaces -- 5.2.2 Weighted Spaces of Continuous Functions -- 5.2.3 On the Case of Variable Coefficients -- 5.2.4 Regularity Properties -- 5.3 Compact Integral Operators -- 5.4 Weakly Singular Integral Operators with Logarithmic Kernels -- 5.5 Singular Integro-Differential or Hypersingular Operators -- 5.6 Operators with Fixed Singularities of Mellin Type -- 5.7 A Note on the Invertibility of Singular Integral Operators with Cauchy and Mellin Kernels -- 5.8 Solvability of Nonlinear Cauchy Singular Integral Equations -- 5.8.1 Equations of the First Type -- 5.8.2 Equations of the Second Type -- 5.8.3 Equations of the Third Type -- 6 Numerical Methods for Fredholm Integral Equations -- 6.1 Collectively Compact Sequences of Integral Operators -- 6.2 The Classical Nyström Method. ; 6.2.1 The Case of Jacobi Weights -- 6.2.2 The Case of an Exponential Weight on (0,∞) -- 6.2.3 The Application of Truncated Quadrature Rules -- 6.3 The Nyström Method Based on Product Integration Formulas -- 6.3.1 The Case of Jacobi Weights -- 6.3.2 The Case of an Exponential Weight on (0,∞) -- 6.3.3 Application to Weakly Singular Integral Equations -- 6.4 Integral Equations with Logarithmic Kernels -- 6.4.1 The Well-posed Case -- 6.4.2 The Ill-posed Case -- 6.4.3 A Collocation-Quadrature Method -- 6.4.4 A Fast Algorithm -- 7 Collocation and Collocation-Quadrature Methods for Strongly Singular Integral Equations -- 7.1 Cauchy Singular Integral Equations on an Interval -- 7.1.1 Collocation and Collocation-Quadrature Methods -- 7.1.2 Weighted Uniform Convergence -- Collocation Methods -- Collocation-Quadrature Methods -- 7.1.3 Fast Algorithms -- Weighted L2-Convergence -- Computational Complexity of the Algorithm -- Weighted Uniform Convergence -- 7.2 Hypersingular Integral Equations -- 7.2.1 Collocation and Collocation-Quadrature Methods -- 7.2.2 A Fast Algorithm -- First Step of the Algorithm -- Second Step of the Algorithm -- A More General Situation -- 7.3 Integral Equations with Mellin Type Kernels -- 7.4 Nonlinear Cauchy Singular Integral Equations -- 7.4.1 Asymptotic of the Solution -- 7.4.2 A Collocation-Quadrature Method -- 7.4.3 Convergence Analysis -- 7.4.4 A Further Class of Nonlinear Cauchy Singular Integral Equations -- A Collocation-Quadrature Method -- Convergence Analysis -- 8 Applications -- 8.1 A Cruciform Crack Problem -- 8.1.1 The Integral Equations Under Consideration -- 8.1.2 Solvability Properties of the Operator Equations -- Equation (I+MH0)u0=f0 -- Equation (I+MH1)u1=f1 -- Equation (I+H2)u2=f2 -- 8.1.3 A Quadrature Method -- 8.2 The Drag Minimization Problem for a Wing -- 8.2.1 Formulation of the Problem. ; 8.2.2 Derivation of the Operator Equation -- 8.2.3 A Collocation-Quadrature Method -- 8.2.4 Numerical Examples -- 8.3 Two-Dimensional Free Boundary Value Problems -- 8.3.1 Seepage Flow from a Dam -- The Linear Case -- The Nonlinear Case -- 8.3.2 Seepage Flow from a Channel -- Generating Gaussian Rules -- The Application of Product Integration Rules -- Numerical Results -- 9 Hints and Answers to the Exercises -- 10 Equalities and Inequalities -- 10.1 Equalities and Equivalences -- 10.2 General Inequalities -- 10.3 Marcinkiewicz Inequalities -- Bibliography -- Index.
• Sommario/riassunto: The book presents a combination of two topics: one coming from the theory of approximation of functions and integrals by interpolation and quadrature, respectively, and the other from the numerical analysis of operator equations, in particular, of integral and related equations. The text focusses on interpolation and quadrature processes for functions defined on bounded and unbounded intervals and having certain singularities at the endpoints of the interval, as well as on numerical methods for Fredholm integral equations of first and second kind with smooth and weakly singular kernel functions, linear and nonlinear Cauchy singular integral equations, and hypersingular integral equations. The book includes both classic and very recent results and will appeal to graduate students and researchers who want to learn about the approximation of functions and the numerical solution of operator equations, in particular integral equations.
• Titolo autorizzato: Weighted Polynomial Approximation and Numerical Methods for Integral Equations
• ISBN: 3-030-77497-X
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 996466386903316
• Lo trovi qui: Univ. di Salerno
• Serie: Pathways in Mathematics, . 2367-346X