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n-widths of Sobolev spaces in \(L^ p\). (English) Zbl 0582.41018

Let \(W_ p^{(r)}=\{f: f\in C^{r-1}[0,1]\), \(f^{(r-1)}\) abs. cont., \(\| f^{(r)}\|_ p<\infty \}\), and set \(B_ p^{(r)}=\{f: f\in W_ p^{(r)}\), \(\| f^{(r)}\|_ p\leq 1\}\). We find the exact Kolmogorov, Gel’fand, linear, and Bernstein n-widths of \(B_ p^{(r)}\) in \(L^ p\) for all \(p\in (1,\infty)\). For the Kolmogorov n-width we show that for \(n\geq r\) there exists an optimal subspace of splines of degree r-1 with n-r fixed simple knots depending on p.

MSC:

41A45 Approximation by arbitrary linear expressions
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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