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Global Cm - approximations in Rk

Jurzak J. -P.

Journal of Applied Analysis

2004

Vol. 10, nr 2

291-302

The convolution of a function f ∈ Cm(Rk ;R) with a Dirac-sequence gives an approximation of f ∈ C°(Rk ;R) by a sequence of functions fn ∈ C∞(Rk ;R) n ≥ 1 converging uniformly on compact subsets of Rk as well as its derivatives. The paper studies global estimations on Rk , i.e. there exists functions hα ∈ C∞(Rk ; R) and a sequence εα(n) → 0 such that [wzór] for all x ∈ Rk, all | α |≤ m and all n ≥ 1. Special subalgebras of C∞(Rk ;R) are introduced for the study of the dominated behavior of the convolution process. Some extensions for Cm(Ω ;R) are obtained with Ω open subset of Rk.