Best $L_1$ approximation of Heaviside-type functions in Chebyshev and weak-Chebyshev subspaces of $C^0[a,b]$

In this article, we study the problem of the best $L_1$ approximation of jump functions - piecewise continuous functions with a jump discontinuity - in Chebyshev and weak-Chebyshev subspaces of $C^0[a,b]$. We extend the Hobby-Rice theorem \cite{HobbyRice1965} into an appropriate framework and prove the unicity of best $L_1$ approximation of jump functions in an even-dimensional Chebyshev subspace. We also apply the results to polynomials and Hermite polynomial splines with fixed knots.