On the asymptotics of polynomial interpolation to <mml:math altimg="si1.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:msup><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mi>x</mml:mi><mml:mo>|</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msup></mml:math> at the Chebyshev nodes

Polynomial interpolation with equidistant nodes is notoriously unreliable due to the Runge phenomenon, and is also numerically ill-conditioned. By taking advantage of the optimality of the interpolation processes on Chebyshev nodes, one of the best strategies to defeat the Runge phenomenon is to use the mock-Chebyshev points, which are selected from a satisfactory uniform grid, for polynomial interpolation. Yet, little literature exists on the computation of these points. In this study, we investigate the properties of the mock-Chebyshev nodes and propose a subsetting method for constructing mock-Chebyshev grids. Moreover, we provide a precise formula for the cardinality of a satisfactory uniform grid. Some numerical experiments using the points obtained by the method are given to show the effectiveness of the proposed method and numerical results are also provided.

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Discrete Orthogonality of Bivariate Polynomials of A2, C2 and G2

Symmetry ◽

10.3390/sym11060751 ◽

2019 ◽

Vol 11 (6) ◽

pp. 751

Author(s):

Jiří Hrivnák ◽

Jiří Patera ◽

Marzena Szajewska

Keyword(s):

Polynomial Interpolation ◽

Interpolation Method ◽

Root Systems ◽

Practical Implementation ◽

Recursive Construction ◽

Finite Sets ◽

Bivariate Polynomials ◽

Fourier Methods ◽

Chebyshev Nodes ◽

Orthogonality Relations

We develop discrete orthogonality relations on the finite sets of the generalized Chebyshev nodes related to the root systems A 2 , C 2 and G 2 . The orthogonality relations are consequences of orthogonality of four types of Weyl orbit functions on the fragments of the dual weight lattices. A uniform recursive construction of the polynomials as well as explicit presentation of all data needed for the discrete orthogonality relations allow practical implementation of the related Fourier methods. The polynomial interpolation method is developed and exemplified.

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Rational interpolation to |x| at the Chebyshev nodes

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700030756 ◽

1997 ◽

Vol 56 (1) ◽

pp. 81-86 ◽

Cited By ~ 7

Author(s):

Lev Brutman ◽

Eli Passow

Keyword(s):

Polynomial Interpolation ◽

Rational Interpolation ◽

Chebyshev Nodes ◽

Rational Interpolants ◽

Equidistant Points ◽

Special Case ◽

Interpolation Nodes

Recently the authors considered Newman-type rational interpolation to |x| induced by arbitrary sets of interpolation nodes and showed that under mild restrictions on the location of the interpolation nodes, the corresponding sequence of rational interpolants converges to |x|. In the present paper we consider the special case of the Chebyshev nodes which are known to be very efficient for polynomial interpolation. It is shown that, in contrast to the polynomial case, the approximation of |x| induced by rational interpolation at the Chebyshev nodes has the same order as rational interpolation at equidistant points.

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