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A Faber series approach to cardinal interpolation


Authors: C. K. Chui, J. Stöckler and J. D. Ward
Journal: Math. Comp. 58 (1992), 255-273
MSC: Primary 41A05; Secondary 41A58
DOI: https://doi.org/10.1090/S0025-5718-1992-1106961-3
MathSciNet review: 1106961
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Abstract: For a compactly supported function $ \varphi $ in $ {\mathbb{R}^d}$ we study quasiinterpolants based on point evaluations at the integer lattice. We restrict ourselves to the case where the coefficient sequence $ \lambda f$, for given data f, is computed by applying a univariate polynomial q to the sequence $ \varphi {\vert _{{\mathbb{Z}^d}}}$, and then convolving with the data $ f{\vert _{{\mathbb{Z}^d}}}$. Such operators appear in the well-known Neumann series formulation of quasi-interpolation. A criterion for the polynomial q is given such that the corresponding operator defines a quasi-interpolant.

Since our main application is cardinal interpolation, which is well defined if the symbol of $ \varphi $ does not vanish, we choose q as the partial sum of a certain Faber series. This series can be computed recursively. By this approach, we avoid the restriction that the range of the symbol of $ \varphi $ must be contained in a disk of the complex plane excluding the origin, which is necessary for convergence of the Neumann series. Furthermore, for symmetric $ \varphi $, we prove that the rate of convergence to the cardinal interpolant is superior to the one obtainable from the Neumann series.


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DOI: https://doi.org/10.1090/S0025-5718-1992-1106961-3
Keywords: Cardinal interpolation, symbols, Faber polynomials
Article copyright: © Copyright 1992 American Mathematical Society