Function correction and Lagrange – Jacobi type interpolation

Abstract

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. In the present paper, we prove that there exists the matrix of nodes $\mathfrak{M}_\gamma$ arbitrarily close to the Jacoby matrix $\mathfrak{M}^{(\alpha,\beta)}$, $\alpha,\beta>-1$ with the following property: any function $f\in{C[-1,1]}$ can be adjusted in a set of an arbitrarily small measure such that interpolation process of adjusted continuous function $g$ based on the nodes $\mathfrak{M}_\gamma$ will be uniformly convergent to $g$ on $[a,b]\subset(-1,1)$.

Keywords

• lagrange interpolation
• jacobi orthogonal polynomials
• adjustment of functions