  In this article, we study the problem of best $L_1$ approximation of
Heaviside-type functions in Chebyshev and weak-Chebyshev spaces. We extend the
Hobby-Rice theorem into an appropriate framework and prove the unicity of best
$L_1$ approximation of Heaviside-type functions in an even-dimensional
Chebyshev space under the condition that the dimension of the subspace composed
of the even functions is half the dimension of the whole space. We also apply
the results to compute best $L_1$ approximations of Heaviside-type functions by
polynomials and Hermite polynomial splines with fixed knots.
