  In this paper we describe a self-contained method for performing the
spectral-imaging deconvolution of X-ray data on clusters of galaxies observed
by the ASCA satellite. Spatially-resolved spectral studies of data from this
satellite require such a correction because its optics redistribute photons
over regions which are of comparable size to the angular scales of interest in
clusters. This scattering is a function not only of spatial position but also
energy. To perform a correction for these effects we employ Maximum-Likelihood
deconvolution of the image (within energy bands of 1 keV) to determine the
spatial redistribution, followed by a Monte-Carlo energy reassignment of photon
energies with position to determine the spectral redistribution. We present
tests on simulated cluster data, convolved with the various instrumental
characteristics and the X-ray background, which show that our methodology can
successfully recover a variety of intrinsic temperature profiles in typical
observational circumstances. In Paper-II we apply our spectral-imaging
deconvolution procedure to a large sample of galaxy clusters to determine
temperature profiles, some of which will be used in subsequent mass
determinations, presented in Paper-III.
