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

Sequences of block matrices with rectangular Toeplitz blocks arise in several applications, including the numerical discretization of differential problems. We compute the (asymptotic) spectral and singular value distribution of these sequences in the case where the asymptotic block size ratios are not necessarily rational as in [A. Adriani, I. Furci, C. Garoni, and S. Serra-Capizzano, Spectral and singular value distribution of sequences of block matrices with rectangular Toeplitz blocks. Part I: Asymptotically rational block size ratios. J. Numer. Math. DOI: 10.1515/jnma-2025-0091]. Our derivation relies on the recent notion of generalized approximating classes of sequences as well as on a functional-based approach to describe the spectral and singular value distributions. The distribution result is illustrated through numerical experiments, one of which is inspired by the finite element approximation of a system of differential equations.