Mathematics of approximation (Q2898735)

This book provides an introduction to the theoretical foundations of numerical approximation methods. It is meant to be used as a textbook for teaching advanced undergraduate and beginning graduate courses or as a self-studying tool for students from these groups. The standard material is covered in a detailed fashion, starting from polynomial interpolation and approximation (with a very thorough discussion of Runge's counterexample in equispaced polynomial interpolation), mainly in the \(L_\infty\) and to some extent also in the \(L_2\) norm, and continues via numerical integration (where the focus is on interpolatory methods) to trigonometric and spline approximation and interpolation. The presentation mostly follows the commonly accepted standards. In particular, abstract concepts from functional analysis are used where appropriate. A particular feature of the book is that these concepts are not simply cited from other books, but they are developed within the text itself, thus making the exposition self-contained which is certainly useful for the target group that the author had in mind. On the other hand, the author's decision to proceed in this way implies that a few interesting elementary and many advanced issues had to be omitted. This includes, e.g., the converse (Bernstein) theorems of approximation theory, the connection between the best approximations in the algebraic and the trigonometric setting, and a more detailed description of the unpleasant properties of the Newton-Cotes integration method. The text also does not contain any explicit pointers to advanced textbooks or monographs.