This is my first question here. I hope I can find an answer to my question. I tried to find the answer in Books, Videos, Scripts and german forums (I'm german). But nobody could help me.
It's about the Hermite - Interpolation.I try to understand a section in my script:
We have given:
$x_{i}, \quad i = 0, \ldots,m$ (different in pairs) $y_{i}^{(k)}, \quad i = 0, \ldots, m\quad k = 0, \ldots, \mu_{i}\quad (\mu_{i} \ge 0)$
We want to find:
A polynomial $p \in P_{n}, \quad n = m + \sum\limits_{i = 0}^{m} \mu_{i}:\quad p^{(k)}(x_{i}) = y_{i}^{(k)}$
For reasons of continuity, the Hermitian interpolation polynomial of a function has the form
$p(x) = \sum\limits_{i = 0}^{m} \sum\limits_{ r = 1}^{\mu_{i} + 1} f[\underbrace{x_{0}, \ldots, x_{0}}_{(\mu_{0} + 1)- times}, \ldots, \underbrace{x_{i - 1}, \ldots, x_{i - 1}}_{(\mu_{i - 1} + 1)- times}, \underbrace{x_{i}, \ldots, x_{i}}_{r - times}] \times \prod\limits_{j = 0}^{i - 1} (x - x_{j})^{\mu_{j} + 1} (x - x_{i})^{r - 1}$.
The divided differences $f[\ldots]$ are based on a recursion scheme analogous to the Lagrangian interpolation, whereby whenever higher-order differences cannot be formed according to the definition formula, the higher-order base values must be used, e.g.
$y[x_{i}, x_{i}] = y_{i}^{(1)}$, $y[x_{i}, x_{i}, x_{i + 1}] = \frac{y[x_{i}, x_{i + 1}] - y_{i}^{(1)}}{x_{i + 1} - x_{i}}$
$y[x_{i}, x_{i},x_{i}] = \frac{1}{2} y_{i}^{(2)}$, $y[x_{i}, x_{i}, x_{i}, x_{i + 1}] = \frac{y[x_{i}, x_{i}, x_{i + 1}] - \frac{1}{2} y_{i}^{(2)}}{x_{i +1} - x_{i}}$
My question now is where the hermite polynomial comes from. I have not found anything on the Internet about this.
I would be happy if someone could help me!
(Sorry for my english, i'll try to improve)
Briefly, the Hermite Polynomials are stated to come from a particular generating function:
$$ g\left(x,t\right)=e^{-t^{2}+2tx}=\sum_{n=0}^{\infty}H_{n}\left(x\right)\frac{t^{n}}{n!} $$ where $H_{n}\left(x\right)$ is the Hermite polynomial.
This was extracted from George Arfken's Mathematical Methods for Physicists - Chap 13.1 Special Functions. Hermite Functions:
https://archive.org/stream/MathematicalMethodsForPhysicistsGeorgeArfken/Mathematical%20methods%20for%20physicists-george%20arfken#page/n731/mode/2up
There are additional details there.
I hope this helps.