On the constrained mock-Chebyshev least-squares

The algebraic polynomial interpolation on n+1n+1 uniformly distributed nodes can be affected by the Runge phenomenon, also when the function ff 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 PP that interpolates ff on a subset of m+1m+1 of the given nodes whose elements mimic as well as possible   the Chebyshev–Lobatto points of order mm. In this work we use the simultaneous approximation theory to produce a polynomial Pˆ of degree rr, greater than mm, which still interpolates ff on the m+1m+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 rr 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.