  A relation between matrix-valued complete Bernstein functions and
matrix-valued Stieltjes functions is applied to prove that the solutions of
matricial convolution equations with extended LICM kernels belong to special
classes of functions. In particular the cases of the solutions of the
viscoelastic duality relation and the solutions of the matricial Sonine
equation are discussed, with applications in anisotropic linear viscoelasticity
and a generalization of fractional calculus.
  In the first case it is in particular shown that duality of completely
monotone relaxation functions and Bernstein creep functions in general requires
inclusion in the relaxation function of a Newtonian viscosity term in addition
to the memory effects represented by the completely monotone kernel.
  We define anisotropic generalized fractional derivatives (GFD) by replacing
the kernel $t^{-\alpha}/\Gamma(1-\alpha)$ of the Caputo derivatives with
completely monotone matrix-valued kernels which are weakly singular at 0.
