Shepard-Bernoulli operators

We introduce the Shepard–Bernoulli operator as a combination of the Shepard operator with a new univariate interpolation operator: the generalized Taylor polynomial. Some properties and the rate of convergence of the new combined operator are studied and compared with those given for classical combined Shepard operators. An application to the interpolation of discrete solutions of initial value problems is given. 1. The problem Let X = {x1, . . . , xN} be a set of N distinct points of R, s ∈ N, and let f be a function defined on a domain D containing X. The classical Shepard operators (first introduced in [24] in the particular case s = 2) are defined by (1.1) SN,μ [f ] (x) = N ∑ i=1 Aμ,i (x) f (xi) , μ > 0, where the weight functions Aμ,i (x) in barycentric form are (1.2) Aμ,i (x) = |x− xi| N ∑ k=1 |x− xk| and |·| denotes the Euclidean norm in R. The interpolation operator SN,μ [f ] is stable, in the sense that min i f (xi) ≤ SN,μ [f ] (x) ≤ max i f (xi) , but for μ > 1 the interpolating function SN,μ [f ] (x) has flat spots in the neighborhood of all data points. Also, the degree of exactness of the operator SN,μ [·] is 0, in the sense that if it is restricted to the polynomial space P := {p : deg (p) ≤ m}, then SN,μ [·]|Pm = IdPm (the identity function on P) only for m = 0. These drawbacks can be avoided by replacing each value f (xi) in (1.1) with an interpolation operator in xi, applied to f , having a certain degree of exactness m > 0. More precisely, if for each i = 1, . . . , N P [·, xi] denotes such an interpolation operator in xi, then the related combined Shepard operator is (1.3) SN,μP [f ](x) = N ∑ i=1 Aμ,i (x)P [f, xi] (x) . Received by the editor November 4, 2004 and, in revised form, June 3, 2005. 2000 Mathematics Subject Classification. Primary 41A05, 41A25; Secondary 65D05.