Optimal multilevel matrix algebra operators

We study the optimal Frobenius operator in a general matrix vector space an d in particular in the multilevel trigonometric matrix vector spaces, by em phasizing both the algebraic and geometric properties. These general result s are used to extend the Korovkin matrix theory for the approximation of bl ock Toeplitz matrices via trigonometric vector spaces. The abstract theory is then applied to the analysis of the approximation properties of several sine and cosine based vector spaces. Few numerical experiments are performe d to give evidence of the theoretical results.