Consider the function
$\displaystyle f(x) = \frac{1}{\alpha (x-\beta)^2 + 1}$
in the interval $I = [-1,1]$. Set $\beta = 0$.
How do I get the expression for the least-squares polynomial, say $\tilde f(x)$ of degree $m = 0$. In terms of the parameter $\alpha$ which could be any. By using the nodes $x_0=-1$, $x_1=0$, $x_2=1$
And how can I deduce which kind of interpolant does a least-squares approximating polynomial of degree $m=1$ corresponds to, when using 2 distinct nodes?
I'm going through some exercises for my exam next week, but I couldn't understand this.
Best Answer
What's the point of mentioning $\beta$ if it's zero ??
It's not very clear what "using the nodes" means. I'll assume it means that error is supposed to be measured only at these points.
A polynomial of degree $m=0$ is a constant; let's call it $k$. Then the squared error is $\left(k-f(-1)\right)^2 + \left(k-f(0)\right)^2 + \left(k-f(1)\right)^2$.
Substitute for $f(-1), f(0), f(1)$. This will give you an expression involving $\alpha$ and $k$. Differentiate wrt to $k$, and find places where this derivative is zero.
The second part of the question is even more fuzzy, but I expect similar techniques will work.