On the nature of the Tsallis-Fourier Transform

By recourse to tempered ultradistributions, we show here that the effect of a q-Fourier transform (qFT) is to map {\it equivalence classes} of functions into other classes in a one-to-one fashion. This suggests that Tsallis' q-statistics may revolve around equivalence classes of distributions and not on individual ones, as orthodox statistics does. We solve here the qFT's non-invertibility issue, but discover a problem that remains open.


Introduction
Non-extensive statistical mechanics (NEXT) [1,2,3], a well known generalization of the Boltzmann-Gibbs (BG) one, is used in many scientific and technological endeavors. NEXT central concept is that of a nonadditive (though extensive [4]) entropic information measure characterized by the real index q (with q = 1 recovering the standard BG entropy). Applications include cold atoms in dissipative optical lattices [5], dusty plasmas [6], trapped ions [7], spin glasses [8], turbulence in the heliosphere [9], selforganized criticality [10], high-energy experiments at LHC/CMS/CERN [11] and RHIC/PHENIX/Brookhaven [12], low-dimensional dissipative maps [13], finance [14], galaxies [15], and Fokker-Planck equation's studies [16], EEG's [17], complex signals [18], Vlasov-Poisson equations [19], etc. q-Fourier transforms, developed by Umarov-Tsallis-Steinberg [20] constitute a central piece in the Tsallis' q-machinery. However, Hilhorst [21], in a lucid study that investigated the feasibility of obtaining an invertible q-Fourier transformation (qFT) by restricting the domain of action of the transform to a suitable subspace of probability distributions. He was able to show that this is invertible transformation does not exist. Even more, by explicit construction, he encountered families of functions, all having the same qFT (the q-Gaussians themselves being part of such families) for which the noninvertibility of the qFT becomes evident.
In the present communication we intend to reconcile the Umarov-Tsallis-Steinberg developments [20] with Hilhorst's findings. We will show below that the qFT does indeed map, in a one-to-one fashion, classes of functions into other classes, not isolated functional instances. Thus, both parts of the controversy are right, but the issue can be resolved by appealing to a higher order of mathematical perspective.

The Complex q-Fourier Transform and its Inverse
Let Ω be the space of functions of the real variable x that are parameterized by a real parameter q: where and Here g(x) is bounded, continuous, and positive-definite.
We will make extensive use in this work of the notion of generalized functions (or distributions), objects extending the notion of functions that are especially useful in making discontinuous functions more like smooth functions, and (going to extremes) describing physical phenomena such as point charges.
Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used to formulate generalized solutions of partial differential equations. Where a classical solution may not exist or be very difficult to establish, a distribution solution to a differential equation is often much easier. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are distributions, such as the Dirac delta function (which is historically called a "function" even though it is not considered a proper function mathematically). In more detail, distributions are a class of linear functionals that map a set of test functions (conventional and well-behaved functions) onto the set of real numbers. In the simplest case, the set of test functions considered is D(R), which is the set of functions {φ : R → R}, having two properties: 2) φ has compact support (is identically zero outside some bounded interval).
Then, a distribution d is a linear mapping D(R) → R.
Here we focus attention on the space of test functions defined in Eq. (3.3) of Appendix. Its dual U is a space of so-called tempered ultradistributions [22,23,24,25], that constitute a generalization of the distributions-set for which the test functions are members of a special space called Schwartz' one S, a function-space in which its members possess derivatives that are rapidly decreasing. S exhibits a notable property: the Fourier transform is an automorphism on S, a property that allows, by duality, to define the Fourier transform for elements in the dual space of S. This dual is the space of tempered distributions. In physics it is not uncommon to face functions that grow exponentially in space or time. In such circumstances Schwartz' space of tempered distributions is too restrictive. Instead, ultradistributions satisfy that need [26], being continuous linear functionals defined on the space of entire functions rapidly decreasing on straight lines parallel to the real axis [26]. Now, following [22], we use the Heaviside step function H to define the q-Fourier transform as (2.7) As has been proved in [22], F is one to one from Ω to U On the real axis: for the real transform, and for its inverse. We define now: Thus, according to (2.7), It has been proved by Hilhorst [21] that F T is NOT one to one from Ω to U .
Let Λ fq the set given by: and We define the equivalence relation and, subsequently, the Umarov-Tsallis-Steinberg q-Fourier transform F U T S [20] F U T S : Λ −→ U (2.17) as: We see that F U T S is an application from equivalence classes into equivalence classes and, as a consequence, one to one from Λ into U! Illustrating our theory we consider the example given by Hilhorst [21] for the function: In Ref. [22] we evaluated the q-Fourier transform on this function and obtained Taking q ′ = q in (2.21) we have for F U T S : and, on the real axis,

Conclusions
We have shown that the q-generalization advanced by Umarov et al. in [20] is to be properly regarded as a transformation between classes of equivalence and thus one-to-one, a finding that reconciles the assertions of Ref. [20] with the lucid observations of Ref. [21].
3 Appendix: Tempered Ultra-distributions and

Distributions of Exponential Type
For the benefit of attentive readers we give here a brief summary of the main properties of distributions of exponential type and tempered ultradistributions.
x p entails x p 1 1 x p 2 2 ...x pn n . We shall denote by | p |= n j=1 p j and call D p the differential operator ∂ p 1 +p 2 +...+pn /∂x 1 For any natural k we define The space H of test functions such that e p|x| |D q φ(x)| is bounded for any p The space of continuous linear functionals defined on H is the space Λ ∞ of the distributions of the exponential type given by ( ref. [24] ).
where k is an integer such that k ≧ 0 and f (x) is a bounded continuous function. In addition we have H ⊂ S ⊂ S ′ ⊂ Λ ∞ , where S is the Schwartz space of rapidly decreasing test functions (ref [25]).
The Fourier transform of a functionφ ∈ H is According to ref. [24], φ(z) is entire analytic and rapidly decreasing on straight lines parallel to the real axis. We shall call H 1 the set of all such functions.
Let Π be the set of all z-dependent pseudo-polynomials, z ∈ C n . Then U is the quotient space By a pseudo-polynomial we understand a function of z of the form s z s j G(z 1 , ..., z j−1 , z j+1 , ..., z n ), with G(z 1 , ..., z j−1 , z j+1 , ..., z n ) ∈ A ω Due to these properties it is possible to represent any ultra-distribution as where the path Γ j runs parallel to the real axis from −∞ to ∞ for Im(z j ) > ζ, ζ > p and back from ∞ to −∞ for Im(z j ) < −ζ, −ζ < −p. (Γ surrounds all the singularities of F (z)).
Eq. (3.6) is our fundamental representation for a tempered ultra-distribution.
Sometimes use will be made of the "Dirac formula" for ultra-distributions where the "density" f (t) is such that While F (z) is analytic on Γ, the density f (t) is in general singular, so that the r.h.s. of (3.8) should be interpreted in the sense of distribution theory.
Another important property of the analytic representation is the fact that on Γ, F (z) is bounded by a power of z [24] |F (z)| ≤ C|z| p , (3.9) where C and p depend on F .
The representation (3.6) implies that the addition of a pseudo-polynomial P (z) to F (z) does not alter the ultra-distribution: However, Γ P (z)φ(z) dz = 0.