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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: Birkhäuser. 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
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