Constructive Approximation [Hardcover]

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 present book deals with some basic problems of Approximation Theory: with properties of polynomials and splines, with approximation by poly- mials, splines, linear operators. It also provides the necessary material ab out different function spaces. In some sense, this is a modern version of the corre? sponding parts of the book of one of us (Lorentz [A-1966]). We have tried to give a complete exposition of the main, basic theorems of the theory, without going into too much detail, treating the most general cases or discussing very special problems. There are essential limitations: this is a book about approximation of functions of one real variable. But Approxima? tion Theory of functions of several real or of complex variables would require new books. Very little is given on interpolation. But even with these restric? tions, proofs of some deep and important results, like Korneichuk's theorems about approximation in Lipschitz spaces, could not be included. Another book, with different authors, Constructive Approximation, Advanced Problems , is in preparation. There is an extensive bibliography, which can be used also as an Author Index: each paper is supplied with references for the page or pages where it has been used.1. Theorems of Weierstrass.- 2. Spaces of Functions.- 3. Best Approximation.- 4. Properties of Polynomials.- 5. Splines.- 6. K-Functionals and Interpolation Spaces.- 7. Central Theorems of Approximation.- 8. Influence of Endpoints in Polynomial Approximation.- 9. Approximation by Operatorló!