  Tikhonov regularization with square-norm penalty for linear forward operators
has been studied extensively in the literature. However, the results on
convergence theory are based on technical proofs and difficult to interpret. It
is also often not clear how those results translate into the discrete,
numerical setting. In this paper we present a new strategy to study the
properties of a regularization method on the example of Tikhonov
regularization. The technique is based on the observation that Tikhonov
regularization approximates the unknown exact solution in the range of the
adjoint of the forward operator. This is closely related to the concept of
approximate source conditions, which we generalize to describe not only the
approximation of the unknown solution, but also noise-free and noisy data; all
from the same source space. Combining these three approximation results we
derive the well-known convergence results in a concise way and improve the
understanding by tightening the relation between concepts such as convergence
rates, parameter choice, and saturation. The new technique is not limited to
Tikhonov regularization, it can be applied also to iterative regularization,
which we demonstrate by relating Tikhonov regularization and Landweber
iteration. Because the Tikhonov functional is no longer the centrepiece of the
analysis, we can show that Tikhonov regularization can be used for
oversmoothing regularization. All results are accompanied by numerical
examples.
