What would be an appropriate interpolation method that does some smoothing (similar to splines), allows for unknown (relatively big) discontinuities, can be run on a dataset in an irregular grid and with an equally irregular density of points (ie: if X is the date, we can have 20 points on one day and only 5 or even none on the next day).
Solutions such as a simple spline do not work very well, as the discontinuities really affect in a bad way the interpolation (Splines on interpolation with discontinuities).
Total variation based interpolation may be a good solution, but I have the feeling that random googling will not give me the optimal solution of my problem: an expert opinion would be more efficient than the hours I have spent looking at interpolation research that does not exactly fit my problem.
Cheers