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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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