Exact minimizer for the couple (L∞,BV)(L∞,BV) and the one-dimensional analogue of the Rudin–Osher–Fatemi model

We provide a simple algorithm that constructs an exact minimizer for the EE-functional E(t,f;L∞,BV)=inf‖g‖L∞≤t‖f−g‖BV. Here L∞,BVL∞,BV stand for the space of bounded functions and the space of functions with bounded variation on the interval [a,b][a,b], respectively. As a corollary we obtain the following formula for the KK-functional K(N,f;BV,L∞)∽supa≤x0≤⋯≤xN≤b∑i=1N|f(xi)−f(xi+1)|. We also discussed the connection between the results and the Rudin–Osher–Fatemi denoising model.