A Faber series approach to cardinal interpolation
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For a compactly supported function φ in ℝd we study quasiinterpolants based on point evaluations at the integer lattice. We restrict ourselves to the case where the coefficient sequence λf, for given data f, is computed by applying a univariate polynomial q to the sequence φ Zd, and then convolving with the data fZd. Such operators appear in the wellknown Neumann series formulation of quasiinterpolation. A criterion for the polynomial q is given such that the corresponding operator defines a quasiinterpolant. Since our main application is cardinal interpolation, which is well defined if the symbol of φ 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 φ 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 φ, we prove that the rate of convergence to the cardinal interpolant is superior to the one obtainable from the Neumann series. © 1992 American Mathematical Society.
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Chui, C. K., Stöckler, J., & Ward, J. D.
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