Saddle point problems, Bott–Duffin inverses, abstract splines and oblique projections

In this note, we present links between several different areas in numerical analysis and operator theory. More precisely, we show that several saddle point problems are solvable in the presence of a property (called compatibility) between a positive operator and a closed subspace of a Hilbert space, and the same happens with certain abstract spline problems. These notions are related to the generalized Bott–Duffin inverse.