  We provide a representation of the $C^*$-algebra generated by
multidimensional integral operators with piecewise constant kernels and
discrete ergodic operators. This representation allows us to find the spectrum
and to construct the explicit functional calculus on this algebra. The method
can be useful in various applications since many discrete approximations of
integral and differential operators belong to this algebra. Some examples are
also presented: 1) we construct an explicit functional calculus for extended
Fredholm integral operators with piecewise constant kernels, 2) we find a wave
function and spectral estimates for 3D discrete Schr\"odinger equation with
planar, guided, local potential defects, and point sources. The accuracy of
approximation of continuous multi-kernel integral operators by the operators
with piecewise constant kernels is also discussed.
