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Convergence rates of a Hermite generalization of Floater-Hormann interpolants. (English) Zbl 07169501
Summary: Cirillo and Hormann (2018) introduce an iterative approach to the Hermite interpolation problem, which, starting from the Lagrange interpolant, successively adds \(m\) corrections terms to interpolate the data up to the \(m\) th derivative. The method is general enough to be applied to any interpolant in linear form with a sufficiently continuous set of basis functions, but Cirillo and Hormann focus their attention on Floater-Hormann interpolants, a family of barycentric rational interpolants that are based on a particular blend of local polynomial interpolants of degree \(d\). They show that the resulting iterative rational Hermite interpolants converge at the rate of \(O ( h^{( m + 1 ) ( d + 1 )} )\) as the mesh size \(h\) converges to zero for \(m = 1 , 2\), and their numerical results suggest that the same rate holds for \(m > 2\). In this paper we prove this convergence rate for any \(m \geq 1\).

MSC:
65 Numerical analysis
41 Approximations and expansions
Software:
MPFR
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References:
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