  Tsallis statistics (or q-statistics) in nonextensive statistical mechanics is
a one-parameter description of correlated states. In this paper we use a
translated entropic index: $1 - q \to q$ . The essence of this translation is
to improve the mathematical symmetry of the q-algebra and make q directly
proportional to the nonlinear coupling. A conjugate transformation is defined
$\hat q \equiv \frac{{- 2q}}{{2 + q}}$ which provides a dual mapping between
the heavy-tail q-Gaussian distributions, whose translated q parameter is
between $ - 2 < q < 0$, and the compact-support q-Gaussians, between $0 < q <
\infty $ . This conjugate transformation is used to extend the definition of
the q-Fourier transform to the domain of compact support. A conjugate q-Fourier
transform is proposed which transforms a q-Gaussian into a conjugate $\hat q$
-Gaussian, which has the same exponential decay as the Fourier transform of a
power-law function. The nonlinear statistical coupling is defined such that the
conjugate pair of q-Gaussians have equal strength but either couple
(compact-support) or decouple (heavy-tail) the statistical states. Many of the
nonextensive entropy applications can be shown to have physical parameters
proportional to the nonlinear statistical coupling.
