Holomorphic mappings from strongly pseudoconvex domains to complex manifolds.

Let D be a strongly pseudoconvex domain with smooth boundary in C^N. It follows from Henkin-Ramirez integral kernel representation that every holomorphic function on D continuous up to the boundary can be uniformly approximated on the closure of D by functions holomorphic in a neighbourhood of D.
In the talk we discuss some approximation results for mappings from
strongly pseudoconvex domains to complex manifolds and for sections of certain holomorphic fiber bundles on strongly pseudoconvex
domains.
The main analytic technique which we use is a method of gluing holomorphic sprays over Cartan pairs in Stein manifolds, with control up to the boundary of the domain.
This is joint work with Franc Forstneric.