Abstract splines in Krein spaces

We present generalizations to Krein spaces of the abstract interpolation and smoothing problems proposed by Atteia in Hilbert spaces: given a Krein space K and Hilbert spaces H and E (bounded) surjective operators T : H → K and V : H → E , ρ > 0 and a fixed z 0 ∈ E , we study the existence of solutions of the problems argmin { [ T x , T x ] K : V x = z 0 } and argmin { [ T x , T x ] K + ρ ‖ V x − z 0 ‖ E 2 : x ∈ H } .