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Zbl 0536.46021
Burenkov, V.I.
Mollifying operators with variable step and their application to approximation by infinitely differentiable functions.
(English)
[A] 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]. \par 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 $E\sb n$ of points $x=(x\sb 1,...,x\sb n)$ 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: \par Theorem 3.1. Let $\Omega$ be an open set in $E\sb n$, $1\le p<\infty$ an $f\in W\sb p\sp{\ell}(\Omega)$. Then there is such a sequence of functions $\phi\sb s(x)\in C\sp{\infty}(\Omega) (\phi\sb s(x)$ linearly depends on f and is independent of p) that $\lim\sb{s\to \infty}\Vert f- \phi\sb s\Vert\sb{W\sb p\sp{\ell}(\Omega)}=0$ and $\lim\sb{s\to \infty}\Vert(D\sp{\alpha}f-D\sp{\alpha}\phi\sb s)\Lambda(x)\sp{\vert \alpha \vert -\ell}\Vert\sb{L\sb p(\Omega)}=0$ for $\vert \alpha \vert \le \ell$ while $\Vert D\sp{\alpha}\phi\sb s\Lambda(x)\sp{\vert \alpha \vert -\ell}\Vert\sb{L\sb p(\Omega)}\le C\sb{n,s}\Vert f\Vert\sb{W\sb p\sp{\ell}(\Omega)}$ for $\vert \alpha \vert>\ell$, with $C\sb{\alpha,s}$ independent of f and $\Omega$. \par 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)].
[S.P.Singh]
MSC 2000:
*46E35 Sobolev spaces and generalizations
41A65 Abstract approximation theory
41A30 Approximation by other special function classes

Keywords: Minkowski inequality; approximation theorems; mollifying operators with variable step; partition of unity; regularized distance

Cited in: Zbl 0641.46018 Zbl 0625.46042

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