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Function spaces in Lipschitz domains and optimal rates of convergence for sampling. (English) Zbl 1106.41014
Let $$\Omega \subset {\mathbb R}^d$$ be an arbitrary bounded Lipschitz domain. The authors want to recover a function $$f: \Omega \to \mathbb C$$ in the $$L_r$$-quasi-norm $$(0<r\leq \infty)$$ by a linear sampling method $S_n f = \sum_{j=1}^n f(x_j)\, h_j\,,$ where $$h_j \in L_r(\Omega)$$ and $$x_j\in \Omega$$. Assume that $$f$$ is from the unit ball of a Besov space $$B_{pq}^s(\Omega)$$ or of a Triebel-Lizorkin space $$F_{pq}^s(\Omega)$$ with parameters such that the space is compactly embedded into $$C(\overline{\Omega})$$. The authors prove that the optimal rate of convergence of linear sampling methods is $$n^{-s/d + (1/p - 1/r)}_{+}$$, where $$a_{+} =\max \{a,\,0\}$$ for $$a\in \mathbb R$$. Nonlinear methods do not yield a better rate of convergence. The proof uses a result of H. Wendland [IMA J. Numer. Anal. 21, 285–300 (2001; Zbl 0976.65013)] as well as results concerning the function spaces $$B_{pq}^s(\Omega)$$ and $$F_{pq}^s(\Omega)$$.

##### MSC:
 41A25 Rate of convergence, degree of approximation 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 41A45 Approximation by arbitrary linear expressions 41A63 Multidimensional problems
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