  In theoretical physics, we sometimes have two perturbative expansions of
physical quantity around different two points in parameter space. In terms of
the two perturbative expansions, we introduce a new type of smooth
interpolating function consistent with the both expansions, which includes the
standard Pad\'e approximant and fractional power of polynomial method
constructed by Sen as special cases. We point out that we can construct
enormous number of such interpolating functions in principle while the "best"
approximation for the exact answer of the physical quantity should be unique
among the interpolating functions. We propose a criterion to determine the
"best" interpolating function, which is applicable except some situations even
if we do not know the exact answer. It turns out that our criterion works for
various examples including specific heat in two-dimensional Ising model,
average plaquette in four-dimensional SU(3) pure Yang-Mills theory on lattice
and free energy in c=1 string theory at self-dual radius. We also mention
possible applications of the interpolating functions to system with phase
transition.
