Matrix representations of multidimensional integral and ergodic operators

We provide a representation of the $C^{\ast }$-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.