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.

翻译：在分段全纯函数的多中心表示中，我们将多项式$p$根处的拉格朗日插值与$p$的收敛幂级数相结合，将幂级数作为'系数'乘以拉格朗日基多项式。当这些幂级数被截断时，即可得到埃尔米特插值多项式。本文首先回顾了获得多中心表示的不同方法，重点讨论分段常数全纯函数。当多项式次数为$d$且所有幂级数在$n$次幂后截断时，我们形式上得到次数为$d(n+1)-1$的埃尔米特插值多项式。表示埃尔米特插值的自然方式是为每个插值条件设置一个基多项式，这会导致$d(n+1)$个基多项式。接着我们研究了不同表示和计算埃尔米特插值方法中误差的数值累积问题。由于幂级数的收敛性，多中心表示中的数值误差随$n$增长保持有界。当我们假设分段常数全纯函数在某一分量取值为$1$而在其他分量为零，使得埃尔米特插值仅与单个基多项式吻合时，截断的多中心表示仍具有优势。在一般情况下，我们将取所有$d(n+1)$个基多项式的线性组合。