  Digital deblurring of images is an important problem that arises in
multifrequency observations of the Cosmic Microwave Background (CMB) where,
because of the width of the point spread functions (PSF), maps at different
frequencies suffer a different loss of spatial resolution. Deblurring is useful
for various reasons: first, it helps to restore high frequency components lost
through the smoothing effect of the instrument's PSF; second, emissions at
various frequencies observed with different resolutions can be better studied
on a comparable resolution; third, some map-based component separation
algorithms require maps with similar level of degradation. Because of
computational efficiency, deblurring is usually done in the frequency domain.
But this approach has some limitations as it requires spatial invariance of the
PSF, stationarity of the noise, and is not flexible in the selection of more
appropriate boundary conditions. Deblurring in real space is more flexible but
usually not used because of its high computational cost. In this paper (the
first in a series on the subject) we present new algorithms that allow the use
of real space deblurring techniques even for very large images. In particular,
we consider the use of Tikhonov deblurring of noisy maps with applications to
{\it PLANCK}. We provide details for efficient implementations of the
algorithms. Their performance is tested on Gaussian and non-Gaussian simulated
CMB maps, and PSFs with both circular and elliptical symmetry. Matlab code is
made available.
