Adjustment of functions and Lagrange interpolation based on the nodes close to the Legendre nodes

V. V. Novikov

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 17 (2017) no. 4, pp. 394-401

Résumé

It is well known that the Lagrange interpolation of a continuous function based on the Chebyshev nodes may be divergent everywhere (for arbitrary nodes, almost everywhere) like the Fourier series of a summable function. On the other hand any measurable almost everywhere finite function can be “adjusted” in a set of arbitrarily small measure such that its Fourier series will be uniformly convergent. The question arises: does the class of continuous functions have a similar property with respect to any interpolation process? In the present paper we prove that there exists a matrix of nodes $\mathfrak{M}_\gamma$ arbitrarily close to the Legendre matrix with the following property: any function $f\in{C[-1,1]}$ can be adjusted in a set of arbitrarily small measure such that the interpolation process of adjusted continuous function $g$ based on the nodes $\mathfrak{M}_\gamma$ will be uniformly convergent to $g$ on $[a,b]\subset(-1,1)$.