Density in approximation theory. (English) Zbl 1067.41003

Pinkus, Allan

Surv. Approx. Theory 1, 1-45 (2005).

Approximation theory is concerned with the ability to approximate functions by simpler and more easy calculated functions. It is important that the set of functions from which one plans to approximate is dense in the set of continuous functions. In this interesting work, the author surveys some of the main density results and density methods. Starting with the Weierstrass approximation theorems, he discusses numerous generalizations (as Hahn-Banach theorem, Stone-Weierstrass theorem, Bohman-Korovkin theorem, Müntz theorem, Mergelyan theorem). Many historical hints and nice proofs of main results are presented. The author mainly considers univariate functions. Finally, some multivariate density results are given.

Reviewer: Manfred Tasche (Rostock)

MSC:

41A10	Approximation by polynomials
41-03	History of approximations and expansions
41A15	Spline approximation
41A45	Approximation by arbitrary linear expressions
41A63	Multidimensional problems

Keywords:

approximation; polynomial approximation; spline approximation; density; Weierstrass theorems; Stone-Weierstrass theorem; Hahn-Banach theorem; Bohman-Korovkin theorem; Müntz theorem