Abstract
Background. We investigate the relationship between the boundedness of Lebesgue constants for the Lagrange polynomial interpolation on a compact subset of \[\mathbb R\] and the existence of a Faber basis in the space of continuous functions on this compact set. Objective. The aim of the paper is to describe the conditions on the matrix of interpolation nodes under which the interpolation of any continuous function coincides with the decomposition of this function in a series on the Faber basis. Methods. The methods of general theory of Schauder bases and the results which describe the convergence of interpolating Lagrange processes are used. Results. The structure of matrices of interpolation nodes which generate the interpolating Faber bases is described. Conclusions. Every interpolating Faber basis is generated by the interpolating Lagrange process with the interpolating matrix of a special kind and bounded Lebesgue constants.
Highlights
LetThe polynomial Lagrange interpolation is an important and widely used method for approximation of continuous functions
It is well known that if the domain of the function is massive enough, e.g. with nonnegative Lebesgue measure, even under unlimited interpolation node increase the uniform convergence of the interpolation process can be guaranteed only for sufficiently smooth functions
The divergence of Lagrange interpolation processes is studied in details
Summary
We investigate the relationship between the boundedness of Lebesgue constants for the Lagrange polynomial interpolation on a compact subset of and the existence of a Faber basis in the space of continuous functions on this compact set. The aim of the paper is to describe the conditions on the matrix of interpolation nodes under which the interpolation of any continuous function coincides with the decomposition of this function in a series on the Faber basis. The methods of general theory of Schauder bases and the results which describe the convergence of interpolating Lagrange processes are used. The structure of matrices of interpolation nodes which generate the interpolating Faber bases is described. Every interpolating Faber basis is generated by the interpolating Lagrange process with the interpolating matrix of a special kind and bounded Lebesgue constants
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