# Mixed multidimensional integral operators with piecewise constant kernels and their representations

## TL;DR

This paper develops an explicit matrix algebra representation for mixed multidimensional integral operators with piecewise constant kernels, enabling straightforward computation of inverses, spectra, traces, and determinants.

## Contribution

It introduces a novel explicit matrix algebra representation for these integral operators, simplifying their analysis and computation.

## Key findings

- Explicit inverse operators can be computed.
- Spectra of the operators are determined explicitly.
- Traces and determinants are explicitly constructed.

## Abstract

We consider the algebra of mixed multidimensional integral operators. In particular, Fredholm integral operators of the first and second kind belongs to this algebra. For the piecewise constant kernels we provide an explicit representation of the algebra as a product of simple matrix algebras. This representation allows us to compute the inverse operators (or to solve the corresponding integral equations) and to find the spectrum explicitly. Moreover, explicit traces and determinants are also constructed. So, roughly speaking, the analysis of integral operators is reduced to the analysis of matrices. All the qualitative characteristics of the spectrum are preserved since only the kernels are approximated.