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Scattered-data interpolation on \(\mathbb R^n\): Error estimates for radial basis and band-limited functions. (English) Zbl 1081.41014

The approximation and interpolation of band-limited functions of the Paley-Wiener class \(B_{\sigma } \) are investigated. Here \(B_{\sigma } =\{f \in L^2({\mathbb R}^n): \text{supp} (\widehat {f}) \subseteq B(0,\sigma)\} \), where \(\widehat {f}\) is the Fourier transform of the function \(f\) and \(B(0,\sigma )\) is the ball with the radius \(\sigma \) and the centre at the origin. The functions \(f \in B_{\sigma } \) are continuous and \(\lim_{x \to \infty }f(x) = 0\), i.e. \(f\in C_0({\mathbb R}^n)\). Let us introduce the norm in the space \(C_0 \cap L^2 \equiv C_0 ({\mathbb R}^n) \cap L^2({\mathbb R}^n)\) by the equality \(\|f\|_{C_0 \cap L^2} = \max (\|f\|_{\infty } ,\|f\|_2 )\), where \(\|\cdot\|_{\infty } \) (or \(\|\cdot\|_{\infty } \)) is the \(sup\)-norm (or \(L^2\)-norm respectively). The separation radius \(q\) of the set \(X = \{x^{j}\}_{j =1}^{N} \subset {\mathbb R}^n\) is introduced by the equation \(q = {1\over2}\min _{j \neq k}\|x^j-x^k\|_2\), where \(\|\cdot\|\) is Euclidean norm in \({\mathbb R}^n\). The diameter \(\text{diam}(X)\) of the set \(X\) is defined by the equality \(\text{diam}(X)= \max_{j \neq k}\|x^{j} - x^k\|_2 \). The main results of the paper are the following two theorems.
Theorem 3.5. Let \(X = \{ x^{j} \}_{j = 1}^{N} \subset {\mathbb R}^n\) is a set of distinct points with the separation radius \(q\) and \(\text{diam}(X)\leq1\). If \(f \in C_0 \cap L^2\) and \(\sigma \geq \sigma _0 = {24\over q}\{ {\sqrt \pi \over3}\Gamma ( {n+2\over 2} )\}^{2\over n+1}\), then there exists a function \(f_{\sigma }\in B_{\sigma } \) such that \(f|_X = f_{\sigma }|_X \) and \(\|f - f_{\sigma }\|_{C_0 \cap L^2} \leq (5 + 2^{n + 3})\text{dist}_{C_0 \cap L^2} (f,B_{\sigma })\), where \(\text{ dist}_{C_0\cap L^2} (f,B_{\sigma }) = \inf _{g\in B_{\sigma }}\|f - g\|_{C_0 \cap L^2} \). Let \(W_2^k = W_2^k ({\mathbb R}^n)\) is the Sobolev space consisting of all functions \(f \in L^2({\mathbb R}^n)\) with distributional derivatives \(D^{\alpha }f\in C_0({\mathbb R}^n)\), \(|\alpha|=\alpha _1+\dots+\alpha _n\leq k\). Further let \(C_0^k = C_0^k ({\mathbb R}^n)\) be the space of all functions having continuous derivatives \(D^{\alpha }f\in C_0({\mathbb R}^n)\), \(|\alpha|\leq k\).
Theorem 3.8. Let \(k > 0\) be an integer and \(\sigma > 0\). If \(f \in C_0^k \cap W_2^k \), then there exists a constant \(C=C(k,n)>0\) such that \(\text{dist}_{C_0 \cap L^2} (f,B_{\sigma }) \leq C\sigma ^{ -k}\|f\|_{C_0^k \cap W_2^k } \).
Corollary 3.9. Let \(f \in C_0^k \cap W_2^k \) and \(f_\sigma \) be as in Theorem 3.5. Then \(\|f - f_{\sigma }\|_{C_0 \cap L^2} \leq C\sigma ^{- k}\|f\|_{C_0^k \cap W_2^k } \), where \(C = C(k,n)>0\).

MSC:

41A30 Approximation by other special function classes
41A25 Rate of convergence, degree of approximation
41A05 Interpolation in approximation theory
41A63 Multidimensional problems
42B35 Function spaces arising in harmonic analysis
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