A Faber series approach to cardinal interpolation

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 $\varphi {|_{{\mathbb {Z}^d}}}$, and then convolving with the data $f{|_{{\mathbb {Z}^d}}}$. 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.