Constructive Approximation (Grundlehren der mathematischen Wissenschaften (303))

The purpose of this book and its sequel is to give a connected, unified exposition of Approximation Theory for functions of one real variable. Great care has been taken to provide reliable and good proofs, and to establish a logical selection of material, with emphasis on principal results. The history of the subject is not neglected. Methods of functional analysis are used when necessary, as are complex variable methods for real problems. Practical algorithms of approximation are included. Important old results are not missing, but at least half of the material has never yet appeared in books. The first book describes spaces of functions: Sobolev, Lipschitz, Besov rearrangement-invariant function spaces, interpolation of operators. Then we have Weierstrass and best approximation theorems, properties of polynomials and of splines: inequalities, interpolation (also Birkhoff), zero properties. Approximation by operators treats positive operators, projections, also those of minimal norm, saturation phenomena, a modern theory of Bernstein polynomials. Chapters on splines deal with the selection of knots, splines with equidistant, dyadic, fixed and free knots, with identification of approximation spaces, further with orthogonal projections onto splines, cardinal splines, shape preserving algorithms. With no equally comprehensive and up-to-date competitor in the available literature, these two volumes will be a welcome reference for a wide audience of mathematicians and engineers. They can be used as the basis for courses, or as a means to enter research in the subject. Coupled with its sequel, this book gives a connected, unified exposition of Approximation Theory for functions of one real variable. It describes spaces of functions such as Sobolev, Lipschitz, Besov rearrangement-invariant function spaces and interpolation of operators. Other topics include Weierstrauss and best approximation theorems, properties of polynomials and splines. It contains history and proofs with an emphasis on principal results. The book describes function spaces used in approximation, the needed functional analytic tools. It develops basic properties of polynomials, splines, linear operators that are used for approximation (and interpolation). The book takes the reader up to and into the recent great advances of Approximation Theory and emphasizes good proofs, logical selection of material, avoidance of superfluous details