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The $$L_ 2$$-approximation orders of principal shift-invariant spaces generated by a radial basis function. (English) Zbl 0820.41014
Braess, D. (ed.) et al., Numerical methods in approximation theory. Vol. 9: Proceedings of the conference held in Oberwolfach, Germany, November 24-30, 1991. Basel: BirkhĂ¤user. ISNM, Int. Ser. Numer. Math. 105, 245-268 (1992).
For $$f\in L_ 2(\mathbb{R}^ d)$$ denote by $$S(f)$$ the $$L_ 2$$-closure of the finite linear combinations of the integer translations of $$f$$. For $$\{S_ h= S(\phi_ n)\}_ h$$, $$h\in (0, 1]$$, define $$S^ h_ h= S^ h(\phi_ h)= \{f(\cdot/h): f\in S_ h\}$$. $$\{S_ h\}_ h$$ (or $$\{\phi_ h\}_ h$$) provides an approximation order $$k> 0$$ if $$E(f, S^ h_ h)= O(h^ k)$$ for every $$f\in W$$, where $$E(f, S)= \min\{\| f- g\|: g\in S\}$$ and $$W$$ is some smooth subspace of $$L_ 2(\mathbb{R}^ d)$$, and $$\|\cdot\|$$ is the $$L_ 2(\mathbb{R}^ d)$$-norm. In this paper the author discusses the $$L_ 2$$-approximation order of principal shift-invariant spaces. For the stationary case (i.e. $$\phi_ h= \phi$$ for all $$h$$), he proves the following results:
(1) Let $$\phi$$ be a fundamental solution of a constant coefficient homogeneous elliptic operator of order $$m> d/2$$, then $$\phi$$ provides an approximation order $$m$$ for every $$f\in W^ m_ 2(\mathbb{R}^ d)$$, and for any such non-trivial $$f$$, $$E(f, S^ h)\neq o(h^ m)$$.
(2) Assume that $$\phi$$ satisfies the following two conditions: (a) $$M_ \phi$$ is essentially bounded below and above by positive constants on some O-neighborhood, where $$M^ 2_ \phi= \sum_{\beta\in 2\pi\mathbb{Z}^ d\backslash 0} |\widehat\phi(\cdot+ \beta)|^ 2$$ and $$\widehat\phi$$ is the Fourier transform of $$\phi$$. (b) $$\widehat\phi(w)\sim\log | w|$$ around the origin. Then: (i) $$\phi$$ provides no positive approximation order $$k$$ for any $$f\in W^ k_ 2(\mathbb{R}^ d)$$ and any $$k> 0$$. (ii) $$E(f, S^ h)= o(1)$$ for all $$f\in L_ 2(\mathbb{R}^ d)$$.
(iii) For every $$k> 0$$ and $$f\in W^ k_ 2(\mathbb{R}^ d)$$, $$E(f, S^ h)\leq \text{const}|\log h|^{-1}\| f\|_{W^ k_ 2(\mathbb{R}^ d)}$$ for all $$h< h_ 0$$, where const and $$h_ 0$$ depend on $$k$$ but not on $$f$$. For the non-stationary case he mainly proves the result: Assume that $$\widehat\phi(w)\sim | w|^{- j} e^{- n}| w|^ r$$, $$q_ 1(w)$$ on $$\mathbb{R}^ d$$ for some positive $$j$$, $$n$$, $$r$$ and real $$l\leq j$$, where $$q_ l(w)= | w|^ l$$ when $$| w|\geq 1$$ and $$q_ 1(w)= 1$$ when $$| w|\leq 1$$. Let $$\phi_ h= \lambda(h)^ d\phi(\lambda(h)\cdot)$$. Then for $$0< c< (2\pi)^ r$$ and for every $$f\in W^ k_ 2(\mathbb{R}^ d)$$, $$E(f, S^ h_ h)\leq o(h^ k)+ \text{const} \lambda(h)^{-(j+ l- r)} e^{- nc\lambda(h)^{-r}}F$$, where $$F= \| f\|_{W^ k_ 2} h^ k$$ when $$k\leq j$$ and $$F= \| f\|_{W^ j_ 2} h^ j$$ when $$k\geq j$$. Applications of these results also appear.
For the entire collection see [Zbl 0807.00018].

MSC:
 41A30 Approximation by other special function classes