Boundedness of Lebesgue Constants and Interpolating Faber Bases

Abstract

Background. 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. 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.

Lebesgue Constants For Cantor Sets

“…2. Our definition of the class BL C is different from the analogous definition in [1], where any arrays of points from [a, b] are allowed, where [a, b] is the minimal interval containing K. We consider a more restricted class, since our main interest lies in the existence of Faber interpolation bases in various spaces of functions defined on K.…”

“…It remains to consider the cases n ≤ 2. For n = 2, by (5.4) 1) , and for α > 2 (recall that R ≥ s), the right hand side is as large as we wish for large enough s ∈ N.…”

“…This result was extended by Erdös and P. Vértesi [3] (see also Cor. 7 in [1]): If K belongs to BL C , then K is nowhere dense and its one-dimensional Lebesgue measure is zero.…”

“…[14], p. 161), but does not guarantee the convergence of the Newton interpolation process, which includes polynomials of all degrees and, accordingly, the existence of Faber interpolating bases (for details, see e.g. [1]).…”