(using $f[x_1, ... , x_n]$ to denote the forward difference operator)
I have a polynomial $P(x)$ interpolating $5$ points $x_0, ... , x_4$ and $2$ derivative values $x_0, x_3$ across an evenly spaced grid such that $x_i = x_0 + ih$. I constructed a Hermite polynomial to interpolate the data, and obtained a polynomial of degree $9$ from $f[z_0, z_1, ... , z_n]$ where $z_1 = z_0 = x_0$ and $n = 10$, however, I also have the constraint that the polynomial must be of degree $6$.
My initial idea was to alter the Lagrangian basis s.t. the kronecker delta only interpolates the $2$ derivative points. If I were to take this approach, I would in effect be doing the forward difference similar to $f[z_0, z_1, x_1, x_2, z_3, z_4, x_4]$ (where $z_3 = z_4 = x_3$, etc.) correct?
Is this a valid operation? While looking for a result over the past few days I've been finding a lot of sites referencing Birkhoff interpolation which is a concept that I do not fully understand, and while I don't think it's relevant, perhaps I'm incorrect? Thanks in advance! (thank you for cleaning this up)