Lebesgue constant minimizing linear rational interpolation of continuous functions over the interval

• Published in: Computers & mathematics with applications (1987) Vol. 33; no. 6; pp. 77 - 86
• Main Authors: Berrut, J.-P., Mittelmann, H.D.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 1997
• Subjects: Interpolation; Lebesgue constant; Linear interpolation; Rational interpolation; Lebesgue constant; Interpolation; Linear interpolation; Rational interpolation
• ISSN: 0898-1221; 1873-7668
• DOI: 10.1016/S0898-1221(97)00034-5

Polynomial interpolation between large numbers of arbitrary nodes does notoriously not, in general, yield useful approximations of continuous functions. Following [1], we suggest to determine for each set of
n + 1 nodes another denominator of degree
n, and then to interpolate every continuous function by a rational function with that same denominator, so that the resulting interpolation process remains a linear projection. The optimal denominator is chosen so as to minimize the Lebesgue constant for the given nodes. It has to be computed numerically. For that purpose, the barycentric representation of rational interpolants, which displays the linearity of the interpolation and reduces the determination of the denominator to that of the barycentric weights, is used. The optimal weights can then be computed by solving an optimization problem with simple bounds which could not be solved accurately by the first author in [1]. We show here how to do so, and we present numerical results.