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 P^ 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.

On the constrained mock-Chebyshev least-squares

Mazza M.
2015-01-01

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 P^ 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.
2015
2015
Chebyshev-Lobatto nodes; Constrained least-squares; Mock-Chebyshev interpolation; Runge phenomenon
De Marchi, S.; Dell'Accio, F.; Mazza, M.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/2086530
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