均值与Hermite插值
Means and Hermite Interpolation
摘要：
设$m_{2}<m_{1}$是两个给定的非负整数，且$n=m_{1}+m_{2}+1$。对于适当可微的$f$，我们设$P,Q\in π_{n}$为满足$P^{(j)}(a)=f^{(j)}(a),j=0,1,...,m_{1}$和$P^{(j)}(b)=f^{(j)}(b),j=0,1,...,m_{2},$、$Q^{(j)}(a)=f^{(j)}(a),j=0,1,...,m_{2}$和$Q^{(j)}(b)=f^{(j)}(b),j=0,1,...,m_{1}$的$f$的Hermite多项式插值.假设$f\in C^{n+2}(I)$与$f^{(n+1)}(x)\neq 0$为$x\in (a,b)$。如果$m_{1}-m_{2}$是偶数，则存在唯一的$x_{0},a<x_{0}<b,$，使得$P(x_{0})=Q(x_{0})$。如果$m_{1}-m_{2}$是奇数，则存在唯一的$x_{0},a<x_{0}<b,$，使得$f(x_{0})=\tfrac{1}{2}(P(x_{0})+Q(x_{0})) $。$x_{0}$定义了一个严格的对称平均值，我们用$M_{f,m_{1},m_{2}}(a,b)$表示。我们证明了这些手段的各种性质。特别是，我们表明，$f(x)=x^{m_{1}+m_{2}+2}$产生的算术平均值，$f(x)=x^{-1}$产生的调和平均值，和$f(x)=x^{(m_{1}+m_{2}+1)/2}$产生的几何平均值。
英文摘要：
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.
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