= Solution
Define the <Lagrange interpolation polynomial>[cardinal polynomials]
$$
\ell_k(x)=\prod_{\substack{0\le j\le n\\j\ne k}}
\frac{x-x_j}{x_k-x_j}.
$$
They satisfy $\ell_k(x_i)=\delta_{ik}$. Hence evaluating the proposed <polynomial interpolation> formula at $x_i$ gives $p_n(x_i)=a_i$, so the required coefficients are
$$
\boxed{a_k=f_k}.
$$
The polynomial $\sum_kf_k\ell_k$ has degree at most $n$ and takes every prescribed value. Its uniqueness follows because the difference of two such polynomials would have $n+1$ distinct roots while having degree at most $n$.
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