具有分段常数核的混合多维积分算子及其表示
Mixed multidimensional integral operators with piecewise constant kernels and their representations
摘要：
考虑混合多维积分算子代数。特别是，Fredholm积分算子的第一和第二类属于这个代数。对于分段常数核，我们提供了一个明确的表示代数作为一个简单的矩阵代数的产品。这种表示允许我们计算逆算子（或求解相应的积分方程）并显式地找到谱。此外，还构造了显式迹和行列式。因此，粗略地说，对积分算子的分析归结为对矩阵的分析。由于仅对核进行了近似，因此保留了频谱的所有定性特征。
英文摘要：
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.
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