Krylov subspace methods of approximate solving differential equations from the point of view of functional calculus

The paper deals with projection methods of approximate solving the problem
$$ Fx'=Gx+bu(t),\qquad y=\langle x,d\rangle $$
which consist in passage to the reduced-order problem
$$ \widehat F\hat x'=\widehat G\hat x+\hat bu(t),\qquad \hat y=\langle\hat x,\hat d\rangle, $$
where
$$ \widehat F=\Lambda FV,\qquad\widehat G=\Lambda GV,\qquad\hat b=\Lambda b,\qquad\hat d=V^*d. $$
It is shown that, if $V$ and $\Lambda$ are constructed on the basis of Krylov's subspaces, a projection method is equivalent to the replacement in the formula expressing the impulse response via the exponential function of the pencil $\lambda\mapsto\lambda F-G$, of the exponential function by its rational interpolation satisfying some interpolation conditions. Special attention is paid to the case when $F$ is not invertible.