  In this article, we interpolate a given real analytic spacelike curve $a$ in
Lorentz-Minkowski space $\mathbb{L}^3$ to another real analytic spacelike curve
$c$, which is "close" enough to $a$ in a certain sense, by a maximal surface
using inverse function theorem for Banach spaces. Using the same method we also
interpolate a given real analytic curve $a$ in Euclidean space $\mathbb{E}^3$
to another real analytic curve $c$, which is "close" enough to $a$ in a certain
sense, by a minimal surface. The Bj\"orling problem and Schwartz's solution to
it play an important role.
