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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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##### References:
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