There are many books on approximation theory, including interpolation methods that appeared in the last fifty years, but a few of them are devoted only to interpolation processes as is this book
Gradimir V. Milovanovic is Professor of the University of Niš and Corresponding member of the Serbian Academy of Sciences and Arts.
Interpolation of functions is one of the basic part of Approximation Theory. There are many books on approximation theory, including interpolation methods that - peared in the last fty years, but a few of them are devoted only to interpolation processes. An example is the book of J. Szabados and P. Vértesi: Interpolation of Functions, published in 1990 by World Scienti c. Also, two books deal with a special interpolation problem, the so-called Birkhoff interpolation, written by G.G. Lorentz, K. Jetter, S.D. Riemenschneider (1983) and Y.G. Shi (2003). The classical books on interpolation address numerous negative results, i.e., - sultsondivergentinterpolationprocesses,usuallyconstructedoversomeequidistant system of nodes. The present book deals mainly with new results on convergent - terpolation processes in uniform norm, for algebraic and trigonometric polynomials, not yet published in other textbooks and monographs on approximation theory and numerical mathematics. Basic tools in this eld (orthogonal polynomials, moduli of smoothness,K-functionals, etc.), as well as some selected applications in numerical integration, integral equations, moment-preserving approximation and summation of slowly convergent series are also given. The rstchapterprovidesanaccountofbasicfactsonapproximationbyalgebraic and trigonometric polynomials introducing the most important concepts on appro- mation of functions. Especially, in Sect. 1.4 we give basic results on interpolation by algebraic polynomials, including representations and computation of interpolation polynomials, Lagrange operators, interpolation errors and uniform convergence in some important classes of functions, as well as an account on the Lebesgue function and some estimates for the Lebesgue constant.
1. Constructive Elements and Approaches in Approximation Theory.- 1.1 Introduction to Approximation Theory.- 1.1.1 Basic notions.- 1.1.2 Algebraic and trigonometric polynomials.- 1.1.3 Best approximation by polynomials.- 1.1.4 Chebyshev polynomials.- 1.1.5 Chebyshev extremal problems.- 1.1.6 Chebyshev alternation theorem.- 1.1.7 Numerical methods.- 1.2 Basic Facts on Trigonometric Approximation.- 1.2.1 Trigonometric kernels.- 1.2.2 Fourier series and sums.- 1.2.3 Moduli of smoothness, best approximation and Besov spaces.- 1.3 Chebyshev Systems and Interpolation.- 1.3.1 Chebyshev systems and spaces.- 1.3.2 Algebraic Lagrange interpolation.- 1.3.3 Trigonometric interpolation.- 1.3.4 Riesz interpolation formula.- 1.3.5 A general interpolation problem.- 1.4 Interpolation by Algebraic Polynomials.- 1.4.1 Representations and computation of interpolation polynomials.- 1.4.2 Interpolation array and Lagrange operators.- 1.4.3 Interpolation error for some classes of functions.- 1.4.4 Uniform convergence in the class of analytic functions.- 1.4.5 Bernstein's example of pointwise divergence.- 1.4.6 Lebesgue function and some estimates for the Lebesgue constant.- 1.4.7 Algorithm for finding optimal nodes.- 2. Orthogonal Polynomials and Weighted Polynomial Approximation.- 2.1 Orthogonal Systems and Polynomials.- 2.1.1 Inner product space and orthogonal systems.- 2.1.2 Fourier expansion and best approximation.- 2.1.3 Examples of orthogonal systems.- 2.1.4 Basic facts on orthogonal polynomials and extremal problems.- 2.1.5 Zeros of orthogonal polynomials.- 2.2 Orthogonal Polynomials on the Real Line.- 2.2.1 Basic properties.- 2.2.2 Asymptotic properties of orthogonal polynomials.- 2.2.3 Associated polynomials and Christoffel numbers.- 2.2.4 Functions of the second kind and Stieltjes polynomials.- 2.3 Classical Orthogonal Polynomials.- 2.3.1 Definition of the classical orthogonal polynomials.- 2.3.2 General properties of the classical orthogonal polynomials.- 2.3.3 Generating function.- 2.3.4 Jacobi polynomials.- 2.3.5 Generalized Laguerre polynomials.- 2.3.6 Hermite polynomials.- 2.4 Nonclassical Orthogonal Polynomials.- 2.4.1 Semi-classical orthogonal polynomials.- 2.4.2 Generalized Gegenbauer polynomials.- 2.4.3 Generalized Jacobi polynomials.- 2.4.4 Sonin-Markov orthogonal polynomials.- 2.4.5 Freud orthogonal polynomials.- 2.4.6 Orthogonal polynomials with respect to Abel, Lindelöf, and logistic weights.- 2.4.7 Strong non-classical orthogonal polynomials.- 2.4.8 Numerical construction of orthogonal polynomials.- 2.5 Weighted Polynomial Approximation.- 2.5.1 Weighted functional spaces, moduli of smoothness and K-functionals.- 2.5.2 Weighted best polynomial approximation on (-1,1).- 2.5.3 Weighted approximation on the semi-axis.- 2.5.4 Weighted approximation on the real line.- 2.5.5 Weighted polynomial approximation of functions having isolated interior singularities.- 3. Trigonometric Approximation.- 3.1 Approximating Properties of Operators.- 3.1.1 Approximation by Fourier sums.- 3.1.2 Approximation by Fejér and de la Vallée Poussin means.- 3.2 Discrete Operators.- 3.2.1 A quadrature formula.- 3.2.2 Discrete versions of Fourier and de la Vallée Poussin sums.- 3.2.3 Marcinkiewicz inequalities.- 3.2.4 Uniform approximation.- 3.2.5 Lagrange interpolation error in Lp.- 3.2.6 Some estimates of the interpolation errors in L1-Sobolev spaces.- 3.2.7 The weighted case.- 4. Algebraic Interpolation in Uniform Norm.- 4.1 Introduction and Preliminaries.- 4.1.1 Interpolation at zeros of orthogonal polynomials.- 4.1.2 Some auxiliary results.- 4.2 Optimal Systems of Nodes.- 4.2.1 Optimal systems of knots on (-1,1).- 4.2.2 Additional nodes method with Jacobi zeros.- 4.2.3 Other "optimal" interpolation processes.- 4.2.4 Some simultaneous interpolation processes.- 4.3 Weighted Interpolation.- 4.3.1 Weighted interpolation at Jacobi zeros.- 4.3.2 Lagrange interpolation in Sobolev spaces.- 4.3.3 Interpolation at Laguerre zeros.- 4.3.4 Interpolation at Hermite zeros.- 4.3.5 Interpolation of functions with internal isolated singularities.- 5. Applications.- 5.1 Quadrature Formulae.- 5.1.1 Introduction.- 5.1.2 Some remarks on Newton-Cotes rules with Jacobi weights.- 5.1.3 Gauss-Christoffel quadrature rules.- 5.1.4 Gauss-Radau and Gauss-Lobatto quadrature rules.- 5.1.5 Error estimates of Gaussian rules for some classes of functions.- 5.1.6 Product integration rules.- 5.1.7 Integration of periodic functions on the real line with rational weight.- 5.2 Integral Equations.- 5.2.1 Some basic facts.- 5.2.2 Fredholm integral equations of the second kind.- 5.2.3 Nyström method.- 5.3 Moment-Preserving Approximation.- 5.3.1 The standard L2-approximation.- 5.3.2 The constrained L2-polynomial approximation.- 5.3.3 Moment-preserving spline approximation.- 5.4 Summation of Slowly Convergent Series.- 5.4.1 Laplace transform method.- 5.4.2 Contour integration over a rectangle.- 5.4.3 Remarks on some slowly convergent power series.- References.- Index.
The classical books on interpolation address numerous negative results, i.e., results on divergent interpolation processes, usually constructed over some equidistant systems of nodes. The authors present, with complete proofs, recent results on convergent interpolation processes, for trigonometric and algebraic polynomials of one real variable, not yet published in other textbooks and monographs on approximation theory and numerical mathematics. In this special, but fundamental and important field of real analysis the authors present the state of art. Some 500 references are cited, including many new results of the authors. Basic tools in this field (orthogonal polynomials, moduli of smoothness, K-functionals, etc.) as well as some selected applications in numerical integration, integral equations, moment-preserving approximation and summation of slowly convergent series are also given. Beside the basic properties of the classical orthogonal polynomials the book provides new results on nonclassical orthogonal polynomials including methods for their numerical construction.
From the reviews:
"The entire book deals almost exclusively with the interpolation of univariate functions ... . I believe that this book has the potential to become a standard reference work in this area. It will certainly be very useful to anyone conducting research in this field, both as a beginner and as an advanced scientist. In addition, it provides a large amount of helpful information to people from other fields, who need to use interpolation methods in their daily work." (Kai Diethelm, ACM Computing Reviews, May, 2009)
"A distinctive feature of the present book is the emphasis on convergent interpolation processes in uniform norm, for algebraic and trigonometric polynomials. ... The authors have made substantial contributions to the subject, and the book deals mainly with new results, not yet published in other textbooks and monographs. Intended for researchers and students in mathematics, physics, and applied sciences, the book will be welcomed by all specialists in these areas." (Ioan Rasa, Zentralblatt MATH, Vol. 1154, 2009)
"It contains mostly theoretical results in a wide area of approximation theory, but in many places numerical applications are also given. ... This is a well-written book containing the most up-to-date information on the subjects covered. It can be useful for both experts in the field as well as those who have a basic knowledge in mathematical analysis. A list of more than 500 publications is provided, and an eight-page index makes the search for topics easier." (J. Szabados, Mathematical Reviews, Issue 2009 i)
Interpolation of functions is one of the basic part of Approximation Theory. There are many books on approximation theory, including interpolation methods that - peared in the last fty years, but a few of them are devoted only to interpolation processes. An example is the book of J. Szabados and P. Vértesi: Interpolation of Functions, published in 1990 by World Scienti c. Also, two books deal with a special interpolation problem, the so-called Birkhoff interpolation, written by G.G. Lorentz, K. Jetter, S.D. Riemenschneider (1983) and Y.G. Shi (2003). The classical books on interpolation address numerous negative results, i.e., - sultsondivergentinterpolationprocesses,usuallyconstructedoversomeequidistant system of nodes. The present book deals mainly with new results on convergent - terpolation processes in uniform norm, for algebraic and trigonometric polynomials, not yet published in other textbooks and monographs on approximation theory and numerical mathematics. Basic tools in thiseld (orthogonal polynomials, moduli of smoothness,K-functionals, etc.), as well as some selected applications in numerical integration, integral equations, moment-preserving approximation and summation of slowly convergent series are also given. The rstchapterprovidesanaccountofbasicfactsonapproximationbyalgebraic and trigonometric polynomials introducing the most important concepts on appro- mation of functions. Especially, in Sect. 1.4 we give basic results on interpolation by algebraic polynomials, including representations and computation of interpolation polynomials, Lagrange operators, interpolation errors and uniform convergence in some important classes of functions, as well as an account on the Lebesgue function and some estimates for the Lebesgue constant.
1. Constructive Elements and Approaches in Approximation Theory.- 1.1 Introduction to Approximation Theory.- 1.2 Basic Facts on Trigonometric Approximation.- 1.3 Chebyshev Systems and Interpolation.- 1.4 Interpolation by Algebraic Polynomials.- 2. Orthogonal Polynomials and Weighted Polynomial Approximation.- 2.1 Orthogonal Systems and Polynomials.- 2.2 Orthogonal Polynomials on the Real Line.- 2.3 Classical Orthogonal Polynomials.- 2.4 Nonclassical Orthogonal Polynomials.- 2.5 Weighted Polynomial Approximation.- 3. Trigonometric Approximation.- 3.1 Approximating Properties of Operators.- 3.2 Discrete Operators.- 4. Algebraic Interpolation in Uniform Norm.- 4.1 Introduction and Preliminaries.- 4.2 Optimal Systems of Nodes.- 4.3 Weighted Interpolation.- 5. Applications.- 5.1 Quadrature Formulae.- 5.2 Integral Equations.- 5.3 Moment-Preserving Approximation.- 5.4 Summation of Slowly Convergent Series.- References.- Index.
From the reviews:
"The entire book deals almost exclusively with the interpolation of univariate functions ... . I believe that this book has the potential to become a standard reference work in this area. It will certainly be very useful to anyone conducting research in this field, both as a beginner and as an advanced scientist. In addition, it provides a large amount of helpful information to people from other fields, who need to use interpolation methods in their daily work." (Kai Diethelm, ACM Computing Reviews, May, 2009)
"A distinctive feature of the present book is the emphasis on convergent interpolation processes in uniform norm, for algebraic and trigonometric polynomials. ... The authors have made substantial contributions to the subject, and the book deals mainly with new results, not yet published in other textbooks and monographs. Intended for researchers and students in mathematics, physics, and applied sciences, the book will be welcomed by allspecialists in these areas." (Ioan Rasa, Zentralblatt MATH, Vol. 1154, 2009)
"It contains mostly theoretical results in a wide area of approximation theory, but in many places numerical applications are also given. ... This is a well-written book containing the most up-to-date information on the subjects covered. It can be useful for both experts in the field as well as those who have a basic knowledge in mathematical analysis. A list of more than 500 publications is provided, and an eight-page index makes the search for topics easier." (J. Szabados, Mathematical Reviews, Issue 2009 i)
Gradimir V. Milovanovic is Professor of the University of Nis and Corresponding member of the Serbian Academy of Sciences and Arts.
Über den Autor
Gradimir V. Milovanovic is Professor of the University of Niš and Corresponding member of the Serbian Academy of Sciences and Arts.
Inhaltsverzeichnis
1. Constructive Elements and Approaches in Approximation Theory.- 1.1 Introduction to Approximation Theory.- 1.1.1 Basic notions.- 1.1.2 Algebraic and trigonometric polynomials.- 1.1.3 Best approximation by polynomials.- 1.1.4 Chebyshev polynomials.- 1.1.5 Chebyshev extremal problems.- 1.1.6 Chebyshev alternation theorem.- 1.1.7 Numerical methods.- 1.2 Basic Facts on Trigonometric Approximation.- 1.2.1 Trigonometric kernels.- 1.2.2 Fourier series and sums.- 1.2.3 Moduli of smoothness, best approximation and Besov spaces.- 1.3 Chebyshev Systems and Interpolation.- 1.3.1 Chebyshev systems and spaces.- 1.3.2 Algebraic Lagrange interpolation.- 1.3.3 Trigonometric interpolation.- 1.3.4 Riesz interpolation formula.- 1.3.5 A general interpolation problem.- 1.4 Interpolation by Algebraic Polynomials.- 1.4.1 Representations and computation of interpolation polynomials.- 1.4.2 Interpolation array and Lagrange operators.- 1.4.3 Interpolation error for some classes of functions.- 1.4.4 Uniform convergence in the class of analytic functions.- 1.4.5 Bernstein's example of pointwise divergence.- 1.4.6 Lebesgue function and some estimates for the Lebesgue constant.- 1.4.7 Algorithm for finding optimal nodes.- 2. Orthogonal Polynomials and Weighted Polynomial Approximation.- 2.1 Orthogonal Systems and Polynomials.- 2.1.1 Inner product space and orthogonal systems.- 2.1.2 Fourier expansion and best approximation.- 2.1.3 Examples of orthogonal systems.- 2.1.4 Basic facts on orthogonal polynomials and extremal problems.- 2.1.5 Zeros of orthogonal polynomials.- 2.2 Orthogonal Polynomials on the Real Line.- 2.2.1 Basic properties.- 2.2.2 Asymptotic properties of orthogonal polynomials.- 2.2.3 Associated polynomials and Christoffel numbers.- 2.2.4 Functions of the second kind and Stieltjes polynomials.- 2.3 Classical Orthogonal Polynomials.- 2.3.1 Definition of the classical orthogonal polynomials.- 2.3.2 General properties of the classical orthogonal polynomials.- 2.3.3 Generating function.- 2.3.4 Jacobi polynomials.- 2.3.5 Generalized Laguerre polynomials.- 2.3.6 Hermite polynomials.- 2.4 Nonclassical Orthogonal Polynomials.- 2.4.1 Semi-classical orthogonal polynomials.- 2.4.2 Generalized Gegenbauer polynomials.- 2.4.3 Generalized Jacobi polynomials.- 2.4.4 Sonin-Markov orthogonal polynomials.- 2.4.5 Freud orthogonal polynomials.- 2.4.6 Orthogonal polynomials with respect to Abel, Lindelöf, and logistic weights.- 2.4.7 Strong non-classical orthogonal polynomials.- 2.4.8 Numerical construction of orthogonal polynomials.- 2.5 Weighted Polynomial Approximation.- 2.5.1 Weighted functional spaces, moduli of smoothness and K-functionals.- 2.5.2 Weighted best polynomial approximation on (-1,1).- 2.5.3 Weighted approximation on the semi-axis.- 2.5.4 Weighted approximation on the real line.- 2.5.5 Weighted polynomial approximation of functions having isolated interior singularities.- 3. Trigonometric Approximation.- 3.1 Approximating Properties of Operators.- 3.1.1 Approximation by Fourier sums.- 3.1.2 Approximation by Fejér and de la Vallée Poussin means.- 3.2 Discrete Operators.- 3.2.1 A quadrature formula.- 3.2.2 Discrete versions of Fourier and de la Vallée Poussin sums.- 3.2.3 Marcinkiewicz inequalities.- 3.2.4 Uniform approximation.- 3.2.5 Lagrange interpolation error in Lp.- 3.2.6 Some estimates of the interpolation errors in L1-Sobolev spaces.- 3.2.7 The weighted case.- 4. Algebraic Interpolation in Uniform Norm.- 4.1 Introduction and Preliminaries.- 4.1.1 Interpolation at zeros of orthogonal polynomials.- 4.1.2 Some auxiliary results.- 4.2 Optimal Systems of Nodes.- 4.2.1 Optimal systems of knots on (-1,1).- 4.2.2 Additional nodes method with Jacobi zeros.- 4.2.3 Other "optimal" interpolation processes.- 4.2.4 Some simultaneous interpolation processes.- 4.3 Weighted Interpolation.- 4.3.1 Weighted interpolation at Jacobi zeros.- 4.3.2 Lagrange interpolation in Sobolev spaces.- 4.3.3 Interpolation at Laguerre zeros.- 4.3.4 Interpolation at Hermite zeros.- 4.3.5 Interpolation of functions with internal isolated singularities.- 5. Applications.- 5.1 Quadrature Formulae.- 5.1.1 Introduction.- 5.1.2 Some remarks on Newton-Cotes rules with Jacobi weights.- 5.1.3 Gauss-Christoffel quadrature rules.- 5.1.4 Gauss-Radau and Gauss-Lobatto quadrature rules.- 5.1.5 Error estimates of Gaussian rules for some classes of functions.- 5.1.6 Product integration rules.- 5.1.7 Integration of periodic functions on the real line with rational weight.- 5.2 Integral Equations.- 5.2.1 Some basic facts.- 5.2.2 Fredholm integral equations of the second kind.- 5.2.3 Nyström method.- 5.3 Moment-Preserving Approximation.- 5.3.1 The standard L2-approximation.- 5.3.2 The constrained L2-polynomial approximation.- 5.3.3 Moment-preserving spline approximation.- 5.4 Summation of Slowly Convergent Series.- 5.4.1 Laplace transform method.- 5.4.2 Contour integration over a rectangle.- 5.4.3 Remarks on some slowly convergent power series.- References.- Index.
Klappentext
Interpolation of functions is one of the basic part of Approximation Theory. There are many books on approximation theory, including interpolation methods that - peared in the last fty years, but a few of them are devoted only to interpolation processes. An example is the book of J. Szabados and P. Vértesi: Interpolation of Functions, published in 1990 by World Scienti c. Also, two books deal with a special interpolation problem, the so-called Birkhoff interpolation, written by G.G. Lorentz, K. Jetter, S.D. Riemenschneider (1983) and Y.G. Shi (2003). The classical books on interpolation address numerous negative results, i.e., - sultsondivergentinterpolationprocesses,usuallyconstructedoversomeequidistant system of nodes. The present book deals mainly with new results on convergent - terpolation processes in uniform norm, for algebraic and trigonometric polynomials, not yet published in other textbooks and monographs on approximation theory and numerical mathematics. Basic tools in thiseld (orthogonal polynomials, moduli of smoothness,K-functionals, etc.), as well as some selected applications in numerical integration, integral equations, moment-preserving approximation and summation of slowly convergent series are also given. The rstchapterprovidesanaccountofbasicfactsonapproximationbyalgebraic and trigonometric polynomials introducing the most important concepts on appro- mation of functions. Especially, in Sect. 1.4 we give basic results on interpolation by algebraic polynomials, including representations and computation of interpolation polynomials, Lagrange operators, interpolation errors and uniform convergence in some important classes of functions, as well as an account on the Lebesgue function and some estimates for the Lebesgue constant.
There are many books on approximation theory, including interpolation methods that appeared in the last fifty years, but a few of them are devoted only to interpolation processes as is this book
Includes supplementary material: sn.pub/extras
