We are going to consider the general problem whether the sum of
two closed operators on a Banach space is closed on the
intersection of their domains. We introduce absolute functional
calculus for sectorial operators, which is stronger than
$H^\infty$-calculus. Using this technique, we prove a theorem of
Dore-Venni type for sums of closed operators. There, we are able
to remove any assumptions such as R-boundedness or BIP on one of
the operators given that the second operator has absolute
calculus. Moreover, we show that any sectorial operator has
absolute calculus on the real interpolation spaces between its
domain and the space itself.

In many areas of applied sciences, engineering and technology there are three problems dealing with data and signals: (i) data compression; (ii) signal representations; and (iii) recovery of signals from partial or indirect information about the signals, often contaminated by noise. Major advances in these problems have been achieved in recent years where wavelets, multiresolution analysis, and kernel methods have played key roles. We consider problem (iii) and give an overview of specific contributions to inverse and ill-posed problems where reproducing kernel Hilbert spaces provide a natural setting.