  The concept of qualification for spectral regularization methods for inverse
ill-posed problems is strongly associated to the optimal order of convergence
of the regularization error. In this article, the definition of qualification
is extended and three different levels are introduced: weak, strong and
optimal. It is shown that the weak qualification extends the definition
introduced by Mathe and Pereverzev in 2003, mainly in the sense that the
functions associated to orders of convergence and source sets need not be the
same. It is shown that certain methods possessing infinite classical
qualification, e.g. truncated singular value decomposition (TSVD), Landweber's
method and Showalter's method, also have generalized qualification leading to
an optimal order of convergence of the regularization error. Sufficient
conditions for a SRM to have weak qualification are provided and necessary and
sufficient conditions for a given order of convergence to be strong or optimal
qualification are found. Examples of all three qualification levels are
provided and the relationships between them as well as with the classical
concept of qualification and the qualification introduced by Mathe and
Perevezev are shown. In particular, spectral regularization methods having
extended qualification in each one of the three levels and having zero or
infinite classical qualification are presented. Finally several implications of
this theory in the context of orders of convergence, converse results and
maximal source sets for inverse ill-posed problems, are shown.
