  We consider the Adler $D$-function, which is defined in the spacelike region,
and the $e^+e^-$ annihilation ratio smeared, according to the Poggio, Quinn,
and Weinberg method, which is determined for timelike argument. We show that
the method of the nonperturbative $a$-expansion allows one to describe these
Euclidean and Minkowskian characteristics of the process of $e^+e^-$
annihilation down to the lowest energy scales.
