Improvement on the bound of Hermite matrix polynomials

In this paper, we introduce an improved bound on the 2-norm of Hermite matrix polynomials. As a consequence, this estimate enables us to present and prove a matrix version of the Riemann–Lebesgue lemma for Fourier transforms. Finally, our theoretical results are used to develop a novel procedure for the computation of matrix exponentials with a priori bounds. A numerical example for a test matrix is provided.