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The approximation of a continuous function by either an algebraic polynomial, a trigonometric polynomial, or a spline, is an important issue in application areas like computer-aided geometric design and signal analysis. This book is an introduction to the mathematical analysis of such approximation, and, with the prerequisites of only calculus and linear algebra, the material is targeted at senior undergraduate level, with a treatment that is both rigorous and self-contained.The topics include polynomial interpolation; Bernstein polynomials and the Weierstrass theorem; best approximations in the general setting of normed linear spaces and inner product spaces; best uniform polynomial approximation; orthogonal polynomials; Newton-Cotes , Gauss and Clenshaw-Curtis quadrature; the Euler-Maclaurin formula ; approximation of periodic functions; the uniform convergence of Fourier series; spline approximation,with an extensive treatment of local spline interpolation,and its application in quadrature. Exercises are provided at the end of each chapter
List of contents
Polynomial Interpolation Formulas.- Error Analysis For Polynomial Interpolation.- Polynomial Uniform Convergence.- Best Approximation.- Approximation Operators.- Best Uniform Polynomial Approximation.- Orthogonality.- Interpolatory Quadrature.- Approximation of Periodic Functions.- Spline Approximation.
Summary
The approximation of a continuous function by either an algebraic polynomial, a trigonometric polynomial, or a spline, is an important issue in application areas like computer-aided geometric design and signal analysis. This book is an introduction to the mathematical analysis of such approximation, and, with the prerequisites of only calculus and linear algebra, the material is targeted at senior undergraduate level, with a treatment that is both rigorous and self-contained.
The topics include polynomial interpolation; Bernstein polynomials and the Weierstrass theorem; best approximations in the general setting of normed linear spaces and inner product spaces; best uniform polynomial approximation; orthogonal polynomials; Newton-Cotes , Gauss and Clenshaw-Curtis quadrature; the Euler-Maclaurin formula ; approximation of periodic functions; the uniform convergence of Fourier series; spline approximation,with an extensive treatment of local spline interpolation,and its application in quadrature. Exercises are provided at the end of each chapter
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From the reviews:
“The current book presents a compilation of lecture notes delivered by Professor Johan de Villiers at Stellen Bosch University, South Africa. This hardbound treatise titled Mathematics of approximation is a reader-friendly volume spanning over 425 pages. … The content is self-contained and can be used for both classroom teaching and self-study. … The theory is illustrated by a number of solved examples. This is one of the many graces of the book.” (Sarjoo Prasad Yadav, Mathematical Reviews, February, 2013)
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From the reviews:
"The current book presents a compilation of lecture notes delivered by Professor Johan de Villiers at Stellen Bosch University, South Africa. This hardbound treatise titled Mathematics of approximation is a reader-friendly volume spanning over 425 pages. ... The content is self-contained and can be used for both classroom teaching and self-study. ... The theory is illustrated by a number of solved examples. This is one of the many graces of the book." (Sarjoo Prasad Yadav, Mathematical Reviews, February, 2013)
