  In this paper we study the structure of solutions of the one dimensional
weighted total variation regularisation problem, motivated by its application
in signal recovery tasks. We study in depth the relationship between the weight
function and the creation of new discontinuities in the solution. A partial
semigroup property relating the weight function and the solution is shown and
analytic solutions for simply data functions are computed. We prove that the
weighted total variation minimisation problem is well-posed even in the case of
vanishing weight function, despite the lack of coercivity. This is based on the
fact that the total variation of the solution is bounded by the total variation
of the data, a result that it also shown here. Finally the relationship to the
corresponding weighted fidelity problem is explored, showing that the two
problems can produce completely different solutions even for very simple data
functions.
