Fractal Polynomial Interpolation

TLDR

The study introduces a general framework for constructing non‑smooth versions of classical approximants via fractal interpolation, extending polynomial interpolation and enabling real‑data interpolation with fractal functions derived from any classical approximant. This framework generalizes the Lagrange polynomial basis to a basis of fractal polynomials, providing a systematic method for building fractal interpolation functions that replace smooth approximants. The authors obtain explicit Fourier transforms of these fractal interpolants, prove that they are dense in the space of continuous functions on a compact interval, and demonstrate convergence of the interpolation process as the partition is refined, assuming convergence of the underlying smooth interpolant.

Abstract

A general procedure to define non-smooth versions of classical approximants by means of fractal interpolation functions is proposed. A complete and explicit description in the frequency domain of the functions constructed is obtained through their exact Fourier transforms. In particular, the generalization of the polynomial interpolation is developed. The Lagrange basis of the space of polynomials of degree lower or equal than N is generalized to a basis of fractal polynomials. As a consequence of the process, the density of the polynomial fractal interpolation functions with non-null scale vector in the space of continuous functions in a compact interval is deduced. Furthermore, a method for the interpolation of real data is proposed, by the construction of a fractal function coming from any classical approximant. The convergence of the process when the partition is refined is proved, supposing the convergence of the smooth interpolant.