  In this paper, we investigate Strichartz estimates for discrete linear
Schr\"odinger and discrete linear Klein-Gordon equations on a lattice
$h\mathbb{Z}^d$ with $h>0$, where $h$ is the distance between two adjacent
lattice points. As for fixed $h>0$, Strichartz estimates for discrete
Schr\"odinger and one-dimensional discrete Klein-Gordon equations are
established by Stefanov-Kevrekidis \cite{SK2005}. Our main result shows that
such inequalities hold uniformly in $h\in(0,1]$ with additional fractional
derivatives on the right hand side. As an application, we obtain local
well-posedness of a discrete nonlinear Schr\"odinger equation with a priori
bounds independent of $h$. The theorems and the harmonic analysis tools
developed in this paper would be useful in the study of the continuum limit
$h\to 0$ for discrete models, including our forthcoming work \cite{HY} where
strong convergence for a discrete nonlinear Schr\"odinger equation is
addressed.
