# Lagrange interpolation

## Function correction and Lagrange – Jacobi type interpolation

It is well-known that the Lagrange interpolation 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 an arbitrarily small measure such that its Fourier series will be uniformly convergent. The question arises whether the class of continuous functions has a similar property with respect to any interpolation process.

## Interpolation of Continuous in Ordered H-variation Functions

In 1972 D. Vaterman introduced a class of functions of Λ-bounded variation (in particular, a harmonic variation or an H-variation). Later he introduced also the class of functions of ordered ¤-bounded variation and the class of continuous in Λ-variation functions. These classes have been used by many authors in studies on the convergence and summability of the Fourier series.

## 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?