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A Faber series approach to cardinal interpolation. (English) Zbl 0751.41002
The paper is concerned with cardinal interpolation based on Faber polynomials. In §2 the authors give a brief introduction to Faber polynomials and in §3 some algorithmic constructions of Faber polynomials in regions \(G\) which are either sectors of disk or Moebius transform of the disk.
For \(\varphi\in C_ 0(R^ d)\) a compactly supported complex/valued function and \(\Phi=(\varphi(j))_{j\in Z^ d}\), one defines the symbol \(\tilde\varphi\) by \(\tilde\varphi(t)=\sum_{j\in\mathbb{Z}^ d}\varphi(j)\centerdot\exp(ij\centerdot t)\), \(t\in R^ d\). Throughout the paper one supposes that \(\tilde\varphi(t)\neq 0\) on \(R^ d\). The fundamental sequence \(\Lambda=(\lambda_ j)_{j\in Z^ d}\) is defined by \(\Lambda*\Phi=(\delta_{0j})\) (the Kronecker symbol) or equivalently \(\tilde\Lambda=1/\tilde\varphi\).
The cardinal interpolation operator is studied as the inverse of Schoenberg operator \(S:\ell_ 2\to\ell_ 2\), \(a\to a*\Phi\) or, in symbol notation, \((Sa)^ \sim=\tilde a\tilde\varphi\). The inverse \(T\) of \(S\) is given by \(Tf=\Lambda*f\). In order to construct \(Tf\) numerically, the authors find approximations \(\lambda^{(n)}\in\ell_ 1\) to \(\Lambda\) such that \(\|\tilde\Lambda-\tilde\Lambda^{(n)}\|_ \infty\to 0\), \(n\to\infty\), namely \(\Lambda^{(n)}=q_ n^{(F)}(\Phi)\), where \(q_ n^{(F)}\) are the partial sums of the Faber series of \(1/z\) in \(G\). For symmetric \(\varphi\), the rate of convergence to cardinal interpolant is superior to the one obtainable from the Neumann series, as given in C. K. Chui, [Multivariate splines, CBMS-NSF Reg. Conf. Ser. Appl. Math. 54, 189 p. (1988; Zbl 0687.41018)].

MSC:
41A05 Interpolation in approximation theory
41A15 Spline approximation
41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)
65D05 Numerical interpolation
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