  The computation of f(A)b, the action of a matrix function on a vector, is a
task arising in many areas of scientific computing. In many applications, the
matrix A is sparse but so large that only a rather small number of Krylov basis
vectors can be stored. Here we discuss a new approach to overcome these
limitations by randomized sketching combined with an integral representation of
f(A)b. Two different approximations are introduced, one based on sketched FOM
and another based on sketched GMRES approximation. The convergence of the
latter method is analyzed for Stieltjes functions of positive real matrices. We
also derive a closed form expression for the sketched FOM approximant and bound
its distance to the full FOM approximant. Numerical experiments demonstrate the
potential of the presented sketching approaches.
