On the constrained mock-Chebyshev least-squares

Abstract

The algebraic polynomial interpolation on n+1 uniformly distributed nodes can be affected by the Runge phenomenon, also when the function f to be interpolated is analytic. Among all techniques that have been proposed to defeat this phenomenon, there is the mock-Chebyshev interpolation which produces a polynomial P that interpolates f on a subset of m+1 of the given nodes whose elements mimic as well as possible the Chebyshev–Lobatto points of order m. In this work we use the simultaneous approximation theory to produce a polynomial Pˆ of degree r, greater than m, which still interpolates f on the m+1 mock-Chebyshev nodes minimizing, at the same time, the approximation error in a least-squares sense on the other points of the sampling grid. We give indications on how to select the degree r in order to obtain polynomial approximant good in the uniform norm. Furthermore, we provide a sufficient condition under which the accuracy of the mock-Chebyshev interpolation in the uniform norm is improved. Numerical results are provided.