This paper is concerned with exponentially ill-posed operator equations with
additive impulsive noise on the right hand side, i.e. the noise is large on a
small part of the domain and small or zero outside. It is well known that
Tikhonov regularization with an L1 data fidelity term outperforms Tikhonov
regularization with an L2 fidelity term in this case. This effect has
recently been explained and quantified for the case of finitely smoothing
operators. Here we extend this analysis to the case of infinitely smoothing
forward operators under standard Sobolev smoothness assumptions on the
solution, i.e. exponentially ill-posed inverse problems. It turns out that high
order polynomial rates of convergence in the size of the support of large noise
can be achieved rather than the poor logarithmic convergence rates typical for
exponentially ill-posed problems. The main tools of our analysis are Banach
spaces of analytic functions and interpolation-type inequalities for such
spaces. We discuss two examples, the (periodic) backwards heat equation and an
inverse problem in gradiometry.Comment: to appear in SIAM J. Numer. Ana