Putnam 1977 A3

Problem A-3 Let $u, f$, and $g$ be functions, defined for all real numbers $x$, such that $$\frac{u(x+1)+u(x-1)}{2}=f(x) \text { and } \frac{u(x+4)+u(x-4)}{2}=g(x) .$$

Determine $u(x)$ in terms of $f$ and $g$.