Szegő-Type Limit Theorems for Generalized Discrete Convolution Operators

We study the asymptotic behavior of the averaged $f$-trace of a truncated generalized multidimensional discrete convolution operator as the truncation domain expands. By definition, the averaged $f$-trace of a finite-dimensional operator $A$ is equal to $n^{-1}\sum_{k=1}^nf(\lambda_k)$, where $n$ is the dimension of the space in which the operator $A$ acts, the set of numbers $\lambda_k$, $k=1,\dots,n$, is the complete collection of eigenvalues of the operator $A$, counting multiplicity; a generalized discrete convolution is an operator from the closure of the algebra generated by discrete convolution operators and by operators of multiplication by functions admitting a continuous continuation onto the sphere at infinity.