  Several convergence results in Hilbert scales under different source
conditions are proved and orders of convergence and optimal orders of
convergence are derived. Also, relations between those source conditions are
proved. The concept of a multiple Hilbert scale on a product space is
introduced, regularization methods on these scales are defined, both for the
case of a single observation and for the case of multiple observations. In the
latter case, it is shown how vector-valued regularization functions in these
multiple Hilbert scales can be used. In all cases convergence is proved and
orders and optimal orders of convergence are shown.
