  Regularization is a long-standing challenge for ill-posed linear inverse
problems, and a prototype is the Fredholm integral equation of the first kind
with additive Gaussian measurement noise. We introduce a new RKHS
regularization adaptive to measurement data and the underlying linear operator.
This RKHS arises naturally in a variational approach, and its closure is the
function space in which we can identify the true solution. Also, we introduce a
small noise analysis to compare regularization norms by sharp convergence rates
in the small noise limit. Our analysis shows that the RKHS- and
$L^2$-regularizers yield the same convergence rate when their optimal
hyper-parameters are selected using the true solution, and the RKHS-regularizer
has a smaller multiplicative constant. However, in computational practice, the
RKHS regularizer significantly outperforms the $L^2$-and $l^2$-regularizers in
producing consistently converging estimators when the noise level decays or the
observation mesh refines.
