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