Spectral and singular value distribution of sequences of block matrices with rectangular Toeplitz blocks. Part I: asymptotically rational block size ratios

Motivated by several applications, especially in the context of numerical discretizations of differential problems, we compute the (asymptotic) spectral and singular value distribution of sequences of block matrices formed by rectangular Toeplitz blocks. We remark that this computation cannot be achieved through the theory of generalized locally Toeplitz (GLT) sequences whenever the Toeplitz blocks have different sizes. In this sense, the present paper paves the way for an enhancement of the GLT apparatus in order to manage block structures formed by rectangular blocks of different sizes. The distribution result is illustrated through numerical experiments, one of which is inspired by the numerical discretization of an interface problem.