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.
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