Saddle-point problems, generalized abstract splines, and compatibility

M. L. ARIAS; G. CORACH; M. C. GONZALEZ

Conferencia; 25th Conference on Operator Theory; 2014

Saddle-point problems and splines appear frequently in numerical analysis, for instance in the study of nite element methods. In this talk, we shall consider the notion of generalized abstract spline, which is a generalization of the concept of abstract spline, introduced by M. Atteia in order to unify the treatment of many spline-like functions.We shall show that, under certain compatibility conditions, some results regarding saddle-point problems and abstract splines, which are known in the finite dimensional case, or under a closed range condition, can be extended to a more general scenario. By compatibility we mean a geometrical condition between certain subspaces in which a positive operator and a closed subspace are involved. Finally, we shall relate this notion to generalized Bott-Duffin inverses.