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On the order of approximation of the Sobolev class \(W^ r_ q\) by linear forms in \(L_ p\) for 1\(\leq q\leq p\leq 2\). (Russian) Zbl 0734.41033
The author proves the following theorem. Let \(f\in W^ r_ q([0,1]^ 1)\) with \(\| f\|_ q\leq 1\) and \(\| f\|^ q_ r:=\sum_{| \alpha | =r}\| f^{(\alpha)}\|^ q_ q\leq 1.\) For \(d\geq 1-d\), let \(G^ M:=\{g=\sum^{M}_{k=1}\phi_ k(x_ 1,...,x_ d)\psi_ k(x_{d+1},...,x_ 1)\},\) where \(\phi_ k\in L_ p([0,1]^ d)\), \(\psi_ k\in L_ p([0,1]^{1-d})\). If \(1\leq q\leq p\leq 2\) and \(r>1/q-1/p\), then \[ \inf \{\| f-g\|_ p:g\in G^ M\}\asymp \exp \{[1/(d-1)](r-d/q+d/p)\log M\}. \]

MSC:
41A63 Multidimensional problems
41A45 Approximation by arbitrary linear expressions
Keywords:
Sobolev classes
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