Multicentric representation of piecewise constant holomorphic functions and Hermite interpolation

TL;DR

This paper explores multicentric representations of piecewise holomorphic functions, focusing on Hermite interpolation, error analysis, and the advantages of multicentric approaches over traditional basis polynomial methods.

Contribution

It reviews various multicentric representation methods, analyzes error accumulation, and demonstrates the benefits of multicentric representations in Hermite interpolation.

Findings

Errors in multicentric representations remain bounded as n increases.

Truncated multicentric representations are advantageous even in simple cases.

Hermite interpolation degree is d(n+1)-1 with truncated power series.

Abstract

In multicentric representation of piecewise holomorphic functions one combines Lagrange interpolation at roots of a polynomial $p$ with convergent power series of $p$ as the "coefficients" multiplying the Lagrange basis polynomials. When these power series are truncated one obtains Hermite interpolation polynomials. In this paper we first review different approaches to obtain multicentric representations with emphasis in piecewise constant holomorphic functions. When the polynomial is of degree $d$ and all power series are truncated after $n^{th}$ power, we formally arrive into a Hermite interpolation polynomial of degree $d(n+1)-1$. The natural way to represent Hermite interpolation is to have for each interpolation condition a basis polynomial which in this case leads to $d(n+1)$ basis polynomials. We then consider the numerical accumulation of errors in the different ways to represent and evaluate the Hermite interpolation. In the multicentric representation due to the convergence of the power series, numerical errors stay bounded as $n$ grows. When we assume that the piecewise constant holomorphic function takes the value $1$ in one of the components and vanishes in the other so that the Hermite interpolation agrees with just one basis polynomial, even then the truncated multicentric representation is favorable. In the general case one would take a linear combination of all $d(n+1)$ basis polynomials.