Mathematics of Approximation

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

Prerequisites of only calculus and linear algebra Self-contained and rigorous treatment of the mathematics of approximation Includes new results,in particular those on local spline interpolation, and its connection to quadrature A new method of proof of the Euler-Maclaurin formula is presented The topic of quadrature formulas and their error analysis is given an extensive treatment The Weierstrass theorem is rigorously proved for both algebraic and trigonometric polynomials Both Fourier series and the Gram-Schmidt procedure are developed from best approximation Includes supplementary material: sn.pub/extras

Inhalt

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.