  The connection between ghost-free formulations of RG-invariant perturbation
theory in the both Euclidean and Minkowskian regions is studied. Our basic tool
is the "double spectral representation", similar to definition of Adler
function, that stems from first principles of local QFT. It relates real
functions defined in the Euclidean and Minkowskian regions. On this base we
establish a simple relation between -- The trick of resummation of the
$\pi^2$-terms (known from early 80s) for the invariant QCD coupling and
observables in the time-like region and -- Invariant Analytic Approach (devised
a few years ago) with the "analyticized" coupling $\alpha_{\rm an}(Q^2)$ and
nonpower perturbative expansion for observables in the space-like domain which
are free of unphysical singularities . As a result, we formulate a
self-consistent scheme, Analytic Perturbation Theory (APT), that relates a
renorm-invariant, effective coupling functions $\alpha_{an}(Q^2) $ and
$\tilda{alpha}(s),$ as well as non-power perturbation expansions for
observables in both space- and time-like domains, that are free of extra
singularities and obey better convergence in the infrared region. Then we
consider the quark threshold crossing and devise a global APT scheme for the
data analysis in the whole accessible space-like and time-like domain with
various numbers of active quarks. Preliminary estimates indicate that this
global scheme produces results a bit different, sometimes even in the
five-flavour region, on a few per cent level for $\bar{\alpha}_s$ -- from the
usual one, thus influencing the total picture of the QCD parameter correlation.
