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

The Relationship between Tsallis Statistics, the Fourier Transform, and Nonlinear Coupling