On the constrained mock-Chebyshev least-squares

Abstract

The algebraic polynomial interpolation on uniformly distributed nodes is affected by the Runge phenomenon, also when the function to be interpolated is analytic. Among all techniques that have been proposed to defeat this phenomenon, there is the mock-Chebyshev interpolation which is an interpolation made on a subset of the given nodes whose elements mimic as well as possible the Chebyshev-Lobatto points. In this work we use the simultaneous approximation theory to combine the previous technique with a polynomial regression in order to increase the accuracy of the approximation of a given analytic function. We give indications on how to select the degree of the simultaneous regression in order to obtain polynomial approximant good in the uniform norm and provide a sufficient condition to improve, in that norm, the accuracy of the mock-Chebyshev interpolation with a simultaneous regression. Numerical results are provided.

A New Approximation Method with High Order Accuracy

“…From the numerical example, we can see that the accuracy is much better than Q d and power interpolation themselves. However, it would be interesting to compare the proposed method with other methods present in literature as a future work, as for example the operators proposed in [26,33], using appropriate test functions.…”

Creating stable quadrature rules with preassigned points by interpolation

“…The idea is to choose a subset of the equidistant points, which best mimic a grid points with such an asymptotic distribution, e.g. the Chebyshev-Lobatto grid points (see [2,4,9,11] and references therein).…”

Approximation Properties of Chebyshev Polynomials in the Legendre Norm

“…And, Chebyshev polynomials also have an important connection with the mock-Chebyshev subset interpolation exploited to cutdown the Runge phenomenon [29,30], which takes advantages of the optimality of the interpolation processes on Chebyshev-Lobatto nodes.…”