  This is the first in a series of articles about recovering the full algebraic
structure of a boundary conformal field theory (CFT) from the scaling limit of
the critical Ising model in slit-strip geometry. Here, we introduce spaces of
holomorphic functions in continuum domains as well as corresponding spaces of
discrete holomorphic functions in lattice domains. We find distinguished sets
of functions characterized by their singular behavior in the three infinite
directions in the slit-strip domains. We prove convergence results of the
distinguished discrete holomorphic functions to the continuum ones. In the
subsequent articles, the discrete holomorphic functions will be used for the
calculation of the Ising model fusion coefficients (as well as for the
diagonalization of the Ising transfer matrix), and the convergence of the
functions is used to prove the convergence of the fusion coefficients. It will
also be shown that the vertex operator algebra of the boundary conformal field
theory can be recovered from the limit of the fusion coefficients via geometric
transformations involving the distinguished continuum functions.
