Means and Hermite Interpolation
| dc.creator | Horwitz, Alan | |
| dc.date | 2007-11-30 | |
| dc.date | 2008-05-20 | |
| dc.date.accessioned | 2026-07-07T09:39:31Z | |
| dc.date.available | 2026-07-07T09:39:31Z | |
| dc.description | Let $m_{2}<m_{1}$ be two given nonnegative integers with $n=m_{1}+m_{2}+1$. For suitably differentiable $f$, we let $P,Q\in π_{n}$ 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 $f^{(n+1)}(x)\neq 0$ 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})=\tfrac{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. | |
| dc.identifier | https://arxiv.org/abs/0711.4940 | |
| dc.identifier | http://arxiv.org/abs/0711.4940 | |
| dc.identifier | Journal of Mathematical Inequalities, Volume 2, Number 1 (2008), 75-95 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/161201 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | 26E60 | |
| dc.title | Means and Hermite Interpolation | |
| dc.type | text |