A method for uniform approximation under interpolation constraints

Abstract

A method for uniform approximation of a real function f(x) under a finite number of auxiliary interpolation conditions is proposed. The method is obtained by introducing the Shepard basis functions in a well-known Walsh theorem. The approximation properties for f(x) and for its derivatives, whenever they exist, are given. Two meaningful applications are provided: the first is in the area of numerical methods for evaluating Cauchy principal value intergals; the second is obtained by imposing a finite number of interpolation constraints to a Bernstein polynomial.

Keywords:

• Uniform approximation
• interpolation
• Shepard basis functions
• Walsh-type theorem

C.R. Categories:

• 5.13-5.16

A.M.S. Classification:

• 41A29
• 41A05
• 65D05
• 65D30

^*Work sponsored by“Ministero deH'Universitá e della Ricerca Scientifica”and“Consiglio Nazionale delle Ricerche”of Italy.

^*Work sponsored by“Ministero deH'Universitá e della Ricerca Scientifica”and“Consiglio Nazionale delle Ricerche”of Italy.

Notes

^*Work sponsored by“Ministero deH'Universitá e della Ricerca Scientifica”and“Consiglio Nazionale delle Ricerche”of Italy.