  This is the first in a series of papers in which we investigate the resolvent
and spectral measure on non-trapping asymptotically hyperbolic manifolds with
applications to the restriction theorem, spectral multiplier results and
Strichartz estimates. In this first paper, we use semiclassical Lagrangian
distributions and semiclassical intersecting Lagrangian distributions, along
with Mazzeo-Melrose 0-calculus, to construct the high energy resolvent on
general non- trapping asymptotically hyperbolic manifolds, generalizing the
work due to Melrose, Sa Barreto and Vasy. We note that there is an independent
work by Y. Wang which also constructs the high-energy resolvent.
