Izvestiya of Saratov University.
ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

Legendre orthogonal polynomials

Adjustment of Functions and Lagrange Interpolation Based on the Nodes Close to the Legendre Nodes

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?