Hermite interpolation

Remarks

Such a $p$ exists and is unique. Uniqueness: the difference $q$ of two solutions is $0$ or has degree at most $N - 1$. It has a root of multiplicity at least $k_i$ at each $x_i$, so at least $N$ roots counted with multiplicity, and hence $q = 0$. Existence follows because the conditions form a square linear system whose only solution for zero data is $0$. If all $k_i = 1$ this is polynomial interpolation; if $m = 0$ the solution is the Taylor polynomial of order $k_0 - 1$ at $x_0$.

Example: $m = 1$, nodes $0$ and $1$, $k_0 = k_1 = 2$, data $p(0) = 0$, $p'(0) = 0$, $p(1) = 1$, $p'(1) = 0$. Then $N = 4$ and $p(x) = 3x^2 - 2x^3$: indeed $p(0) = 0$, $p'(x) = 6x - 6x^2$ gives $p'(0) = p'(1) = 0$, and $p(1) = 1$.

Non-example: $3x^2 - 2x^3 + x^2 (x - 1)^2$ satisfies clause 2 for the same data, since the added term has double roots at $0$ and $1$, but its degree is $4 > 3$, so clause 1 fails.

The conditions must cover consecutive derivative orders $0, 1, \dots, k_i - 1$ at each node; dropping this gives Birkhoff interpolation, where uniqueness can fail.