## 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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