1. Find a polynomial of degree less than 4, such that it satisfies $f(0)=1, f(1)=1, f(2)=5, f(3)=11$.
$$
\begin{array}{l}
f(x)=\frac{(x-1)(x-2)(x-3)}{(0-1)(0-2)(0-3)} f(0)+\frac{(x-0)(x-2)(x-3)}{(1-0)(1-2)(1-3)} f(1) \\
\quad+\frac{(x-0)(x-1)(x-3)}{(2-0)(2-1)(2-3)} f(2)+\frac{(x-0)(x-1)(x-2)}{(3-0)(3-1)(3-2)} f(3) \\
=-\frac{1}{3}\left(x^{3}-9 x^{2}+8 x-3\right) .
\end{array}
$$

Using the Lagrange interpolation formula, we have