On the interpolation of discontinuous functions

TL;DR

This paper introduces an index of convergence based on subsequence densities, providing a comprehensive framework for understanding interpolation of discontinuous functions, with applications to Lagrange and Shepard operators.

Contribution

It proposes a novel index of convergence that captures the behavior of subsequences at discontinuities, enhancing interpolation theory analysis.

Findings

The index effectively describes convergence phenomena at discontinuities.

Applications demonstrate improved understanding of Lagrange and Shepard operators.

Provides a unified approach to interpolation at points of discontinuity.

Abstract

Given a sequence of real numbers, we consider its subsequences converging to possibly different limits and associate to each of them an index of convergence which depends on the density of the associated subsequences. This index turns out to be useful for a complete description of some phenomena in interpolation theory at points of discontinuity of the first kind. In particular we give some applications to Lagrange and Shepard operators.