3. A plan

Because we're starting with given data points, let's look for a solution in the form of a time-varying linear combination

$P(t) = p_0 (t) P_0 + \dots + p_n (t) P_n$.

Here let's use $p_i$ instead of $f_i$ to emphasize that the functions we seek are polynomials in $t$. If each $p_i$ is a polynomial in $t$ of degree at most $n$, then $P (t)$ will be a polynomial curve of degree at most $n$, as required.

How can we arrange to have $P(t_0) = P_0$? Easily: This happens if $p_0 (t_0) = 1$ but $p_1 (t_0)$ $=$ $p_2 (t_0)$ $=$ ...$=$ $p_n (t_0) = 0$, because then at time $t_0$ we get

$P( t_0 ) = 1 \cdot P_0 + 0 \cdot P_1 + \dots + 0 \cdot P_n = P_0$, as desired.

Similarly, we need $p_1 (t)$ to have values at $t_0$, $t_1,\dots, t_n$ of $0, 1, 0,\dots, 0$ respectively. Then we need $p_2 (t)$ with a similar property, and so on.

The desired values of $p_0 (t),\dots, p_n (t)$, more compactly, are

$p_i (t_i) = 1$ for each $i$, and

$p_i (t_k) = 0$ for $k \neq i$.