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Density in approximation theory. (English) Zbl 1067.41003
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.

MSC:
41A10 Approximation by polynomials
41-03 History of approximations and expansions
41A15 Spline approximation
41A45 Approximation by arbitrary linear expressions
41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)
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