In this paper we study the structure of solutions of the one dimensionalweighted total variation regularisation problem, motivated by its applicationin signal recovery tasks. We study in depth the relationship between the weightfunction and the creation of new discontinuities in the solution. A partialsemigroup property relating the weight function and the solution is shown andanalytic solutions for simply data functions are computed. We prove that theweighted total variation minimisation problem is well-posed even in the case ofvanishing weight function, despite the lack of coercivity. This is based on thefact that the total variation of the solution is bounded by the total variationof the data, a result that it also shown here. Finally the relationship to thecorresponding weighted fidelity problem is explored, showing that the twoproblems can produce completely different solutions even for very simple datafunctions.
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