Abstract splines in Krein spaces

Abstract

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 z0 ∈ E, we study the existence of solutions of the problems argmin{[ T x, T x ]K : V x = z0} and argmin ˘ [ T x, T x ]K + ρkV x − z0k 2 E : x ∈ H¯ .