Nonextensive Statistical Mechanics

4.7 Central Limit Theorems 137See Fig. 4.18 for illustrations of the interesting closure property (4.72), whichdoes not exist for any other of the presently known linear or nonlinear integraltransforms.Equation (4.74) can be rewritten as follows1√ 2−qβ √ 2−q β1 = K (q) , (4.76)[ 3 − q] √12−qK (q) ≡. (4.77)8A 2(1−q)qSee Fig. 4.19.Through direct derivation we can easily verify another interesting property of theq-Fourier transform, namely the following set of relations (for q ≥ 1) [258]:∫ ∞F q [ f ](0) =dF q [ f ](ξ)dξ ∣ = iξ=0d 2 ∣F q [ f ](ξ) ∣∣∣ξ=0=−qdξ 2−∞∫ ∞dx f(x) , (4.78)−∞∫ ∞dx x [ f (x)] q , (4.79)−∞dx x 2 [ f (x)] 2q−1 , (4.80)d n ∣ { n−1F q [ f ](ξ) ∣∣∣ξ=0 ∏}= (i) n [1 + m(q − 1)]dξ nm=0∫ ∞× dx x n [ f (x)] 1+n(q−1) (n = 1, 2, 3,...) . (4.81)−∞If f (x) is a real, nonnegative, integrable function, we can define a probability distribution,namely p(x) ≡ f (x)/ ∫ ∞−∞dx f(x). We can also define a family of escortdistributions, namely [258]P (n) (x) ≡[ f (x)] 1+n(q−1)∫ ∞−∞ dx [ f (x)]1+n(q−1) [n = 0, 1, 2,...; P(0) (x) = p(x); P (1) (x) = P(x)].With the following definition of associated q-expectation values(4.82)combination of two hypergeometric functions, where we see through inspection that, due to thepresence of x q−q′q−1 , the one-parameter invariance has disappeared for all (q, q ′ ) such that q ≠ q ′ .Inother words, for fixed q,allq ′ -Fourier transforms are invertible, excepting if q ′ = q. Equivalently,for fixed q ′ , all the above functions f (x) = (λ/x) 1q−1 are invertible excepting if q = q ′ .Thisdiscussionappears to suggest that F q [ f ](ξ) is invertible for all admissible q and for all functions f (x)excepting a zero-measure class of them. It is possible that such exceptions could be handled satisfactorilyby using extra information related to q-expectation values, but such discussion is out of thepresent scope.