I am trying to solve the following problem:
Given the conditions on a curve $c(u)$ of degree $4$ at the points $-1$, $0$, $1$ as:
$c(-1) = 4$; $c'(-1) = 4$; $c(0) = 6$; $c(1) = -4$; $c'(1) = -6$;
find the generalized Hermite polynomials and express the curve $c$ using them.
What I have so far is that we have only up to the first derivative to interpolate the end points which gives me:
$H^4_0(-1) = H^4_4=1$
$H'^ 4_1(-1)= H'^4_3(1) = 1$
Now I don't have enough conditions to determine $H^4_0, H^4_1, H^4_2, H^4_0, H^4_3, H^4_4$ so I should use the interior point $0$ as well using a Lagrange type condition for this.
Here is where I am having trouble understanding how to do this: first of all I don't have any derivatives so I will just have conditions of the type $H^4_i(u_j)= 1 $ or $0$. Iam not sure for which $i$, $j$ the last equality is $1$ and for which is $0$ because I only have $3$ interpolation points $(u_0 = -1, u_1= 0, u_2=1)$ and $0$ is interior point but I have $5$ polynomials Hermite.
Could someone enlighten me? Thank you very much for your help!