Means and Hermite Interpolation

Abstract

Let $m_2<m_1$ be two given nonnegative integers with $n=m_1+m_2+1$. For suitably differentiable $f$, we let $P,Q\in {\pi }_n$ be the Hermite polynomial interpolants to $f$ which satisfy $P^{(j)}(a)=f^{(j)}(a),j=0,1,...,m_1$ and $P^{(j)}(b)=f^{(j)}(b),j=0,1,...,m_2,$ $Q^{(j)}(a)=f^{(j)}(a),j=0,1,...,m_2$ and $Q^{(j)}(b)=f^{(j)}(b),j=0,1,...,m_1$. Suppose that $f\in C^{n+2}(I)$ with $f^{(n+1)}(x)\ne 0$ for $x\in (a,b)$. If $m_1-m_2$ is even, then there is a unique $x_0,a<x_0<b,$ such that $P(x_0)=Q(x_0)$. If $m_1-m_2$ is odd, then there is a unique $x_0,a<x_0<b,$ such that $f(x_0)=\frac{1}{2}(P(x_0)+Q(x_0))$. $x_0$ defines a strict, symmetric mean, which we denote by $M_{f,m_1,m_2}(a,b)$. We prove various properties of these means. In particular, we show that $f(x)=x^{m_1+m_2+2}$ yields the arithmetic mean, $f(x)=x^{-1}$ yields the harmonic mean, and $f(x)=x^{(m_1+m_2+1)/2}$ yields the geometric mean.