I am stucked at this problem:
If $f\in C^1[a,b]$ and $x_0,...,x_n$ are $n+1$ distinct points in $[a,b]$, Then there exist unique polynomial $H_{2n+1}$ of degree at most $2n+1$ that satisfies the conditions $H_{2n+1}(x_k)=f(x_k)$ and $H_{2n+1}'(x_k)=f'(x_k)$ for each $k\in\{0,...,n\}$ (This is the Hermite interpolation polynomial for $f$).
We can prove it as follows:
Suppose that there is another polynomial with $P(x_k)=f(x_k)$ and $P'(x_k)=f'(x_k)$ for $k = 0,...,n$ and that the degree of $P$ is at most $2n+1$.
Let $D(x)=H_{2n+1}(x)-P(x)$
Then $D$ is a polynomial of degree at most $2n+1$ with $D(x_k)=0$ and $D'(x_k)=0$ for each $k=0,1,...,n$. Thus $D$ has zeros of multiplicity $2$ at each $x_k$, so
$D(x)=(x-x_0)^2(x-x_1)^2...(x-x_n)^2 Q(x)$
Either, $D(x)$ is of degree $2n$ or more which should be a contradiction, or $Q(x)=0$, which implies that $D(x)=0$, This implies that $P(x)$ is $H_{2n+1}(x)$, so this polynomial is unique. Q.E.D.
Now suppose that instead of requiring $f\in C^1[a,b]$ we require that $f\in C^2[a,b]$ and instead of requiring $H_{2n+1}'(x_k)=f'(x_k)$ we require that $H_{2n+1}''(x_k)= f''(x_k)$ for each $k=0,1,...,n$. Explain why the uniqueness proof fails in this case. (Do not give counter-examples, just explain why proving the uniqueness fails)
I've tried to explain it using systems of linear equations but I am not sure how good it was. (We got $n$ linear equations in $n$ variables and additional $n$ linear equations in $n-2$ variables of the $n$ variables from the first system, I am not sure how to continue)
Thanks for any hint/help.
From the given data you can approximate f'(x_k). Now you make a new paired data for f'(X). We know the pairs (f'(x_k),f"(x_k))=(g(x_k),g'(x_k)). Using Hermite again we can find g.