Mollifying operators with variable step and their application to approximation by infinitely differentiable functions. (English) Zbl 0536.46021

Nonlinear analysis, function spaces and applications, Vol. 2, Proc. Spring Sch., Pisek/Czech. 1982, Teubner-Texte Math. 49, 5-37 (1982).
[For the entire collection see Zbl 0488.00011].
The main object of the paper is to study some approximation theorems in the Sobolev space of functions defined on an open set \(\Omega\) in the n- dimensional Euclidean space \(E_ n\) of points \(x=(x_ 1,...,x_ n)\) by mollifying operators with variable step. In order to prove the main theorems, the author has used some results on the partition of unity, nonlinear mollifier with variable step, linear mollifier with variable step and regularized distance. An interesting result is cited as follows:
Theorem 3.1. Let \(\Omega\) be an open set in \(E_ n\), \(1\leq p<\infty\) an \(f\in W_ p^{\ell}(\Omega)\). Then there is such a sequence of functions \(\phi_ s(x)\in C^{\infty}(\Omega) (\phi_ s(x)\) linearly depends on f and is independent of p) that \(\lim_{s\to \infty}\| f- \phi_ s\|_{W_ p^{\ell}(\Omega)}=0\) and \(\lim_{s\to \infty}\|(D^{\alpha}f-D^{\alpha}\phi_ s)\Lambda(x)^{| \alpha | -\ell}\|_{L_ p(\Omega)}=0\) for \(| \alpha | \leq \ell\) while \(\| D^{\alpha}\phi_ s\Lambda(x)^{| \alpha | -\ell}\|_{L_ p(\Omega)}\leq C_{n,s}\| f\|_{W_ p^{\ell}(\Omega)}\) for \(| \alpha |>\ell\), with \(C_{\alpha,s}\) independent of f and \(\Omega\).
Further references can be seen from the author’s earlier works [Trudy Mat. Inst. Steklov 131, 39-50, 244-245 (1974; Zbl 0313.46033); Trudy Mat. Inst. Steklov 150, 24-66 (1979; Zbl 0417.46036/37/38); Math. Physics Inst. Mat. Akad. Nauk SSSR, Sibirsk. Otdel., Novosibirsk (1975)].
Reviewer: S.P.Singh


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
41A30 Approximation by other special function classes
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