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κ -deformed Fourier transform
Accepted Manuscript κ-Deformed Fourier Transform A.M. Scarfone PII: DOI: Reference:
S0378-4371(17)30288-1 http://dx.doi.org/10.1016/j.physa.2017.03.036 PHYSA 18102
To appear in:
Physica A
Received date: 30 November 2016 Revised date: 20 February 2017 Please cite this article as: A.M. Scarfone, κ-Deformed Fourier Transform, Physica A (2017), http://dx.doi.org/10.1016/j.physa.2017.03.036 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
Highlights *We reformulated the standard Fourier transform in a formalism consistent with the k-algebra useful in the study of deformed functions obtained in the framework of the k-deformed statistical mechanics. *We discuss the central limit theorem for the k-sum of n-iterates statistically independent random variables.
*Manuscript Click here to view linked References
κ-Deformed Fourier Transform A.M. Scarfone Istituto dei Sistemi Complessi (ISC-CNR) c/o Politecnico di Torino Corso Duca degli Abruzzi 24, 10129 Torino, Italy.
Abstract We present a new formulation of Fourier transform in the picture of the κalgebra derived in the framework of the κ-generalized statistical mechanics. The κ-Fourier transform is obtained from a κ-Fourier series recently introduced by us [2013 Entropy 15 624]. The kernel of this transform, that reduces to the usual exponential phase in the κ → 0 limit, is composed by a κ-deformed phase and a damping factor that gives a wavelet-like behavior. We show that
the κ-Fourier transform is isomorph to the standard Fourier transform through a changing of time and frequency variables. Nevertheless, the new formalism is useful to study, according to Fourier analysis, those functions defined in the realm of the κ-algebra. As a relevant application, we discuss the central limit theorem for the κ-sum of n-iterate statistically independent random variables. Keywords: Fourier integral transform, log-periodic oscillations, κ-deformed algebra, power-law distribution.
1. Introduction As well known, Fourier transform is a very useful tool widely used in mathematical statistics and physics. It consists of a linear operator acting on a function f (x), with a real argument x, and transforming it into a function Email address: [email protected] (A.M. Scarfone)
Preprint submitted to Physica A
February 20, 2017
5
fb(ω) ≡ F [f (x)](ω), with a complex argument ω: 1 F [f (x)](ω) = √ 2π
+∞ Z f (x) e−i x ω dx .
(1.1)
−∞
More in general, it belongs to the family of integral transforms, defined in +∞ Z H[f (x)](ω) = f (x) h(x, ω) dx ,
(1.2)
−∞
like Laplace, Mellin and Hankel transforms, with the kernel h(x, ω) given by a complex exponential-like form. Fourier transform finds applications in many fields of science, running from 10
mathematics to physics and engineering. In statistical physics, and in general in probability theory, it is applied in the study of random walk, of infinite divisible distributions, in the analysis of distribution stability and in the study of their domains of attraction as well as in the large deviation theory and in the proof of several central limit theorems [1].
15
In quantum mechanics, Fourier transform links the representation of a wave function in the configuration space with its dual representation in the momenta space. In addition, it is a powerful tool widely used in quantum field theory to evaluate the Green function of a quantum propagator thanks to its property of changing a derivative in a multiplicative power.
20
Its discrete version is largely applied in the theory of signals [2], especially after the introduction of fast algorithms, the so called fast Fourier transform, that speeds up the computation process, allowing the elaboration of a large amount of information, which has permitted to develop relevant data compression algorithms and the realization of high-definition image devices.
25
In quantum computing theory, Fourier transform is powerfully used in the generation of quantum protocols for the factorization of large numbers and in solving the discrete logarithm problem [3]. In the past, several versions of Fourier transform have been proposed in literature by using deformed algebraic structures like, for instance, in the framework
30
of quantum groups where the Hopf algebra underlying the braiding groups has 2
been used to define a Fourier transform in the noncommutative space [4]. In [5], within the basic-calculus, it has been advanced a basic-analogous of standard Fourier transform defined in +∞ Z F [f (x); q](ω) = f (x) exp(x|ω; q) dq x ,
(1.3)
−∞
35
that is realized by employing a complex version of the basic-exponential exp(x; q) R [6] and the basic-integral dq x [7]. On a different ground, in the framework of statistical mechanics based on the Tsallis entropy [8], the nonlinear Fourier transform [9] defined in +∞ Z Fq [f (x)](ω) = f (x) ⊗q expq (−i x ω) dx ,
(1.4)
−∞
has been introduced by using the q-exponential [10] and the q-product [11]. This transform was designed with the purpose to investigate the possible gener40
alization of the central limit theorem within the q-statistics, although the final results are still controversial [12, 13]. In this paper, we revisit the standard Fourier transform in a manner that is consistent with the κ-algebra and the κ-calculus derived in the framework of the κ-statistical mechanics. This formalism, together with the κ-entropy, has been
45
introduced in [14] to generalize the Boltzmann-Gibbs entropy and the related theory, with the aim to describe non Gibbsian systems characterized by powerlaw distributions. In the last decade, several papers have been written on the κ-statistical mechanics concerning its foundations, its theoretical consistence and its possible applications to physical and physical-like systems [15, 16, 17,
50
18, 19, 20] (see [21] for an up to date references list). The new formulation of Fourier transform (hereinafter κ-Fourier transform), is obtained starting from a κ-deformed version of Fourier series recently proposed in [22], given by the following linear operator 1 Fκ [f (x)](ω) = √ 2π
+∞ Z f (x) expκ (−x ⊗κ ω)i dκ x ,
−∞
3
(1.5)
and is based on the κ-exponential and the related κ-algebra introduced in [14] 55
(see also [23]), by replacing both the exponential kernel and the integrate operator with their κ-deformed versions. Here and in the following, for the sake of notation, expκ (x)a means (expκ (x))a . Actually, Eq. (1.5) can be rewritten in the form (1.2) of a standard integral transform over the real numbers with the kernel hκ (x, ω) =
60
exp(−i x{κ} ω{κ} ) √ , 1 + κ2 x2
(1.6)
where x{κ} are κ-numbers defined in the remainder [23].
Function hκ (x, ω) contains a deformed phase factor exp(−i x{κ} ω{κ} ) which √ has an asymptotically log-periodic behavior and a damping factor 1/ 1 + κ2 x2 , that confers to the transform a wavelet-like behaviours. Furthermore, transform (1.5) is isomorph to the standard Fourier transform, where the isomorphism is 65
settled by changing the variables in time and frequency. The plan of the paper is as follows. In the next Section 2, we summarize the main aspects of the κ-algebra and its related calculus. We introduce the Euler formula and the cyclic deformed functions in a manner consistent with the κ-formalism. In Section 3, we derive the new formulation of the Fourier
70
transform based on the κ-algebra. A list of its main properties is also given, while a potential application concerning the central limit theorem for κ-sum of n-iterate statistically independent random variables is discussed in Section 4. Our conclusions are reported in Section 5. 2. Mathematical formalism of the κ-algebra
75
In order to embody power-law distributions in statistical physics, in [14] it has been proposed the following generalized entropic form Z Sκ [p] = − p(x) lnκ p(x) dx ,
(2.1)
which mimics the Boltzmann-Gibbs entropy by replacing the standard logarithm with its generalized version lnκ (x)
=
1 xκ − x−κ ≡ sinh κ ln(x) , 2κ κ 4
(2.2)
In agreement with the maximal entropy principle, the corresponding equilibrium 80
distribution reads 1 p(x) = α expκ − (γ + β x) , λ
(2.3)
where expκ (x)
=
κx+
1/κ p 1 ≡ exp 1 + κ2 x2 arcsinh (κ x) , κ
(2.4)
is the κ-exponential, with lnκ (expκ (x)) = expκ (lnκ (x)) = x.
Both these functions reduce to the standard exponential and logarithm in the κ → 0 limit: ln0 (x) ≡ ln(x) and exp0 (x) ≡ exp(x) and consequently, in the
85
same limit, Eqs. (2.1) and (2.3) converge to the Boltzmann-Gibbs entropy and the Gibbs distribution, respectively. We note that the shape of distribution (2.3), is characterized by an asymptotic power law tail, since expκ (x) ∼ x±1/κ , for x → ±∞. Therefore, it differs from the exponential behavior of the Gibbs-distribution, a fact that justify the use
90
of κ-statistical mechanics in the study of free-scale systems. On a mathematical ground, the main features of κ-statistical physics follow from the analytical properties of the κ-exponential and the κ-logarithm [23]. For any |κ| < 1, logκ (x) and expκ (x) are continuous, monotonic, increasing
functions, normalized in lnκ (1) = 0 and expκ (0) = 1, with lnκ (R+ ) ⊆ R and 95
expκ (R) ⊆ R+ .
Another function useful in the formulation of the κ-deformed statistical me-
chanics is given by uκ (x) =
1 xκ + x−κ ≡ cosh κ ln(x) . 2 κ
(2.5)
For any |κ| < 1, the function uκ (x) is continuous, with uκ (R+ ) ∈ R+ , uκ (0) =
uκ (+∞) = +∞ and reaches its minimum at x = 1, where uκ (1) = 1. Function
100
uκ (x), that reduces to a constant in the κ → 0 limit (u0 (x) = 1), is strictly
related to lnκ (x) and expκ (x) and appears recurrently in the study of their analytical properties.
5
These κ-functions, fulfil the following scaling-laws expκ (µ x) = expκ′ (x)µ ,
(2.6)
lnκ (xµ ) = µ lnκ′ (x) ,
(2.7)
uκ (xµ ) = uκ′ (x) ,
(2.8)
where κ′ = µ κ. In particular, for µ = −1 and accounting for the symmetry 105
relations expκ (x) ≡ exp−κ (x), lnκ (x) ≡ ln−κ (x) and uκ (x) ≡ u−κ (x), we obtain expκ (x) expκ (−x) = 1 ,
(2.9)
lnκ (x) + lnκ (1/x) = 0 ,
(2.10)
uκ (x) − uκ (1/x) = 0 .
(2.11)
Equations (2.9) and (2.10) reproduce the well-known relations of standard exponential and logarithm in the κ-formalism. 2.1. κ-Algebra In [24], it has been shown that, starting from a pair of continuous, monotonic 110
increasing functions, inverse each other, we can introduce two different algebraic structures endowed by a generalized sum and product, that form two distinct Abelian fields. By specializing this result to the present case, we can define two deformed algebras on the target space and on the probability space, respectively. Among them, the target space algebra is the relevant one in this work and is
115
revisited in the following, remanding to [24] for a more detailed discussion. To begin with, let us introduce the κ-numbers x{κ} =
1 arcsinh (κ x) , κ
(2.1.1)
1 sinh (κ x) , κ
(2.1.2)
and their dual x{κ} = with x{κ}
{κ}
= x{κ} 6
{κ}
=x.
(2.1.3)
Mapping (2.1.1) [resp. (2.1.2)] defines a non uniform stretching of the real axis 120
so that the space of the κ-numbers Rκ is isomorph to the space of the real numbers, with (+∞){κ} ≡ +∞, (−∞){κ} ≡ −∞ and 0{κ} ≡ 0.
Generalized sum and product are defined on Rκ according to [22] x ⊕κ y = x{κ} + y{κ}
{κ}
x ⊗κ y = x{κ} · y{κ}
{κ}
,
(2.1.4)
,
(2.1.5)
where, in the following, for the sake of notation, we simply mean ⊕ ≡ ⊕κ and ⊗ ≡ ⊗κ unless explicably stated.
125
These operations are associative, commutative and distributed, with the null element ∅ and the identity I, and for any x ∈ Rκ , there exist the opposite (−x)
and the inverse (1/x). Therefore, the algebraic structure (Rκ , ⊕, ⊗) forms a
commutative group. Explicitly, we have
1 sinh arcsinh (κ x) + arcsinh (κ y) , κ 1 1 x ⊗ y = sinh arcsinh (κ x) · arcsinh (κ y) , κ κ x⊕y =
130
(2.1.6) (2.1.7)
with ∅ ≡ 0, I ≡ κ−1 sinh κ, (−x) ≡ −x and (1/x) ≡ κ−1 sinh(κ2 /arcsinh κ x).
In the κ → 0 limit, sum (2.1.6) and product (2.1.7) recover the standard oper-
ations and the κ-algebra reduces to the standard algebra of the real numbers.
Remark also that, for large x and y, the κ-sum approaches asymptotically to the standard product
135
x⊕y ≃x·y ,
x, y ≫ 1 ,
(2.1.8)
x⊕y ≃x+y ,
x, y → 0 .
(2.1.9)
for κ 6= 0, as well as
Starting from Eqs. (2.1.6) and (2.1.7), we can introduce other elementary operations like the difference x ⊖ y = x ⊕ (−y) and the quotient x ⊘ y = x ⊗ (1/y).
7
However, the main property of κ-algebra follows from the definitions of κexponential and κ-logarithm expκ (x) = ex{κ} , 140
lnκ (x) = (ln x){κ} .
(2.1.10)
Among the many relations satisfied by expκ (x) and lnκ (x), let us recall that expκ (x ⊕ y) = expκ (x) · expκ (y) ,
(2.1.11)
lnκ (x · y) = lnκ (x) ⊕ lnκ (y) ,
(2.1.12)
as well as the following, that will be used in the remainder expκ (x ⊗ y) = expκ (x)y{κ} = expκ (y)x{κ} , 2
expκ (a x ⊗κ a y) = expκ′ (x ⊗κ′ y)a = expκ′′ (a2 (x ⊗κ′ y)) , (2.1.13) expκ (a x ⊕κ a y) = expκ′ (x ⊕κ′ y)a = expκ (a(x ⊕κ′ y)) , with κ′ = κ a and κ′′ = κ/a. Other relations can be obtained by inspection of [22]. 2.2. κ-Cyclic functions 145
According to the Euler formula, the cyclic functions in the κ-formalism can be introduced starting from the κ-deformed version of the complex exponential. However, within the κ-algebra, we have two substantially different possible definitions [22]. The most natural one is given by expκ (x) → expκ (i x) ≡ exp (i x){κ}
,
(2.2.1)
and coincides with the choice made in [21]. The function expκ (i x) is unitary for 150
|x| ≤ 1/|κ|, while becomes a purely imaginary quantity with increasing modulo for |x| > 1/|κ|, that is
| expκ (i x)| = 1 1/κ p | expκ (i x)| = κ x + κ2 x2 − 1
for |x| ≤ 1/|κ| ,
(2.2.2)
for |x| > 1/|κ| .
(2.2.3)
Consistently, with definition (2.2.1) we can introduce the κ-trigonometric functions according to expκ (i x) = Cosκ (x) + i Sinκ (x) , 8
(2.2.4)
so that expκ (i x) − expκ (−i x) , 2i expκ (i x) + expκ (−i x) . Cosκ (x) = 2
Sinκ (x) =
155
(2.2.5) (2.2.6)
These functions take real values in the range |x| ≤ 1/|κ| and imaginary values
otherwise. This is illustrated in Figure 1, where we plot the Cosκ (x) for κ = 0.05, in the region |x| ≤ 1/|κ| (left panel) and its modulo |Cosκ (x)| (right panel) in the region |x| > 1/|κ|, where the κ-cosine (2.2.6) assumes imaginary values. 1,0 8
10
(x)|
0,0
|Cos
Cos
(x)
0,5
4
10
-0,5
-1,0
0
10 -20
-15
-10
-5
0
x
5
10
15
20
20
25
x
30
Figure 1: Plot of Cosκ (x) defined in Eq. (2.2.6), for κ = 0.05, in the region |x| ≤ 1/|κ| (left panel) and its modulo in the region |x| > 1/|κ| (right panel).
A different definition of the κ-exponential on the complex unit circle is given 160
by expκ (x) → expκ (x)i ≡ exp i x{κ}
.
(2.2.7)
Actually, definitions (2.2.1) and (2.2.7) are related each other in expκ (x)i = expκ′ (i x) ,
(2.2.8)
according to the scaling relation (2.6), with κ′ = −i κ.
However, the complex function expκ (x)i has unitary modulo for any x ∈ R and,
by taking into account that |x| < |x{κ} | and |x − x{κ} | increases for |x| → ∞, it
165
follows that the period of functions expκ (x)i increases as |x| increases. 9
According to Eq. (2.2.7), we introduce a new set of κ-deformed cyclic functions as expκ (x)i = cosκ (x) + i sinκ (x) ,
(2.2.9)
so that sinκ (x) ≡ sin(x{κ} ) ,
(2.2.10)
cosκ (x) ≡ cos(x{κ} ) .
(2.2.11)
We remark that functions (2.2.10) and (2.2.11) are asymptotically log-periodic.
1.0
0,5
0.5
0,0
0.0
cos
cos
(x)
(x)
1,0
-0.5
-0,5
-1.0
-1,0 0
200
x
-1
400
10
0
10
1
10
2
10
x
3
4
10
10
5
10
Figure 2: Plot of cosκ (x) given by Eq. (2.2.11) for k = 0.2 in the linear-linear scale (a) and in the log-linear scale (b) showing its asymptotic log-periodic behavior.
170
Their period increases for |x| → ∞, because sinκ (x) = sinκ (x′ ) ,
when
x′ = x{κ} + 2 n π
and, for large x, we have ∆ ln(x) ≃ 2 n κ π ,
that is
{κ}
,
x′ ≃ x e2 n κ π ,
(2.2.12)
(2.2.13)
where ∆ ln(x) = ln(x′ ) − ln(x) approaches to a constant for x ≫ 1.
This is shown in Figure 2, where we plot the shape of cosκ (x), for κ = 0.2, in the linear-linear scale (left panel) and in the log-linear scale (right panel) that 175
shows its asymptotic log-periodic behavior. 10
As proven in [22], the functions sinκ (x) and cosκ (x) can be derived from the following differential equation d dx
2 p a{κ} d u(x) 2 2 1+κ x u(x) = 0 , +√ dx 1 + κ2 x2
(2.2.14)
with −h ≤ x ≤ h and a{κ} a constant. This is a Sturm-Liouville equation df (x) d p(x) + λ w(x) f (x) = 0 , (2.2.15) dx dx √ √ with p(x) = 1 + κ2 x2 , weight function w(x) = 1/ 1 + κ2 x2 and eigenvalue 180
λ = (a{κ} )2 . A solution of Eq. (2.2.14), with boundary conditions f (−h) = f (h) = 0, corresponds to the κ-sine function
f (x) = A sinκ (a ⊗ x) ,
(2.2.16)
whilst a solution with boundary conditions f ′ (−h) = f ′ (h) = 0, where prime means derivative with respect to its argument, is given by the κ-cosine function f (x) = A cosκ (a ⊗ x) .
(2.2.17)
Similar considerations hold for the function f (x) = A expκ (a ⊗ x)i , 185
(2.2.18)
that is the solution of the same Sturm-Liouville problem, in the h → ∞ limit,
with boundary condition |f (x)| < ∞. 2.3. κ-Calculus
Following [14], we introduce the calculus in the κ-formalism by means of the κ-differential dκ x ≡ dx{κ} , given by the differential of the κ-numbers dx dκ x = √ , 1 + κ2 x2 190
(2.3.1)
that is κ-linear dκ (a ⊗ x) = a{κ} · dκ x ,
(2.3.2)
dκ (x ⊕ y) = dκ x + dκ y ,
(2.3.3)
11
where a is constant. The κ-derivative is defined in
d dx
κ
≡
d , dκ x
(2.3.4)
and is related to the Leibnitz derivative by p d d d = 1 + κ2 x2 ≡ uκ (expκ (x)) , dκ x dx dx
(2.3.5) √ where the last equality follows from the relation uκ (expκ (x)) = 1 + κ2 x2
195
derived in [25]. Within the standard calculus, the derivative of the κ-functions previously introduced read expκ (x) d expκ (x) = , dx uκ (expκ (x)) uκ (x) d lnκ (x) = , dx x d lnκ (x) uκ (x) = κ , dx x cosκ (x) d sinκ (x) = , dx uκ (expκ (x)) sinκ (x) d cosκ (x) = − . dx uκ (expκ (x))
(2.3.6) (2.3.7) (2.3.8) (2.3.9) (2.3.10)
In this way, by using Eq. (2.3.5), we can rewrite these relations in d expκ (x) = expκ (x) , dκ x d sinκ (x) = cosκ (x) , dκ x d cosκ (x) = − sinκ (x) , dκ x
(2.3.11) (2.3.12) (2.3.13)
that are consistent within the κ-formalism. 200
We observe that quantities like expκ (x) dκ x, sinκ (x) dκ x and cosκ (x) dκ x are all exact differentials in the standard calculus, since they correspond, respectively to the following differential forms: d expκ (x), d sinκ (x) and d cosκ (x). In addition, accounting for the κ-linearity, we can verify the relations df (z) d f (x ⊕ y) = , (2.3.14) dκ x dκ z z=x⊕y d df (z) expκ (x ⊗ y) = y{κ} , (2.3.15) dκ x dκ z z=x⊗y 12
and in particular d expκ (x ⊕ y) = expκ (x ⊕ y) , dκ x d expκ (x ⊗ y) = y{κ} expκ (x ⊗ y) , dκ x 205
that will be used in the remainder. Finally, let us introduce the κ-integral
R
(2.3.16) (2.3.17)
f (x) dκ x, as the inverse operator to
the κ-derivative, according to Z d f (x) dκ x + const. = f (x) , dx κ
(2.3.18)
extending, in this way, the fundamental integral theorem to the κ-formalism. We observe that the κ-integral can be written like a weighted ordinary integral.
210
In fact, recalling Eq. (2.3.1), we have Z Z Z f (x) √ dx , f (x) dκ x = f (x) w(x) dx ≡ 1 + κ2 x2
(2.3.19)
where the weight function
w(x) =
1 1 , ≡ √ uκ (expκ (x)) 1 + κ2 x2
(2.3.20)
coincides with that introduced in the Sturm-Liouville problem. In addition, we can also use the following relation Zb
b{κ}
f (x) dκ x =
a
Z
f {κ} (x) dx ,
(2.3.21)
(2.3.22)
a{κ}
where f 215
{κ}
(x) ≡ f (x
{κ}
)=f
1 sinh(κ x) , κ
that links the κ-integral on the κ-numbers to a standard integral on the real numbers. 3. κ-Fourier transform 3.1. The transform In [22], it has been introduced a consistent formulation of Fourier series in the
220
framework of the κ-algebra. There, it has been shown as any square-integrable 13
function f (x) : (−h, h) → R, may be expanded in the κ-Fourier series with
respect to the orthogonal and complete system of functions sinκ (an ⊗ x) and cosκ (an ⊗ x), that is
f (x) = c0 +
∞ X
n=1
(sn sinκ (an ⊗ x) + cn cosκ (an ⊗ x)) ,
(3.1.1)
where an = (n π/h{κ} ){κ} are suitable constants.
225
This series expansion can be written in the complex form f (x) =
∞ X
n=−∞
γn expκ (−an ⊗ x)i ,
(3.1.2)
where the complex Fourier coefficients are given by γn =
r
2 h
Zh
−h
f (x) expκ (an ⊗ x)i dκ x ,
(3.1.3)
and are related to the real Fourier coefficients in γn = (cn −i sn )/2 and γ−n = γn∗ .
Starting from this result, κ-deformed Fourier transform can be formally obtained in the h → ∞ limit, as usually done in the standard theory.
230
Therefore, from Eq. (3.1.3) we derive the following integral transform for a real function f (x) given by 1 Fκ [f (x)](ω) = √ 2π
+∞ Z f (x) expκ (−x ⊗ ω)i dκ x .
(3.1.4)
−∞
Accounting for Eqs. (2.1.5), (2.1.10) and (2.3.1), this formula can be written like a standard integral transform [cfr. Eq. (1.2)], with the kernel hκ (x, ω) =
exp(−i x{κ} ω{κ} ) √ . 1 + κ2 x2
(3.1.5)
In the κ → 0 limit, kernel hκ (x, ω) reduces to the usual exponential phase
235
h(x, ω) = exp(i x ω) and the κ-Fourier transform collapses into the standard Fourier transform. Kernel (3.1.5) is formed by a deformed phase factor, that arises from the generalization of the complex exponential in its κ-deformed version and by a 14
damping factor, that arises from the measure of the κ-integral. As a conse240
quence, the new transform acquires interesting analytical features. For instance, both the real part and the imaginary part of the phase factor have an asymptotically log-periodic behaviour, a fact that may be relevant in the study of log-oscillating phenomena [24]. Otherwise, the damping factor confers to the kernel a wavelet-like behaviour, as shown in Figure 3, where we have plotted
245
both the real and the imaginary part of the kernel hκ (x, ω). However, many properties of the standard Fourier transform are preserved in the generalized version provided that they are reformulated in the κ-formalism.
1,0
0,5
0,5
h (x,
0,0
Im [
0,0
Re [
h (x,
)]
)]
1,0
-0,5
-0,5
-1,0
-1,0 -150
-100
-50
0
x
50
100
-150
150
-100
-50
0
x
50
100
150
Figure 3: Real (left panel) and imaginary (right panel) part of the kernel hκ (x, ω), for k = 0.1 and ω = 1.
It is duty to note that, in addition to Eq. (3.1.5), there are other possible 250
generalizations of the kernel. They should satisfy the following reasonable conditions: 1) the kernel must reduce to the standard complex exponential exp(i ω x) in a suitable limit; 2) the kernel should be an unimodular function on the whole real axis. These requests exclude the function h(0) κ (x, ω) = expκ (−i x ω) ,
(3.1.6)
since condition 2) is violated in the far region of the real axis, |x ω| ≥ 1/|κ|.
255
Otherwise, the following functions are suitable candidates iω h(1) , κ (x, ω) = expκ (−x)
15
(3.1.7)
ix h(2) , κ (x, ω) = expκ (−ω)
(3.1.8)
i h(3) κ (x, ω) = expκ (−x ω) ,
(3.1.9)
since they have all unitary modulus on R and converge to exp(i ω x) in the κ → 0 limit. However, although some of these functions are equivalent each others, like, for instance
x{κ} , ω , x, ω{κ} = h(2) hκ (x, ω) = h(1) κ κ
(3.1.10)
they define different integral transforms with different analytical properties. 260
By inspection, definition (3.1.4), with the kernel (3.1.5) is the most consistent choice. In fact, it turns out to be isomorph to the standard transform, so that all the analytical properties of the Fourier transform are preserved when rewritten in the κ-formalism. Isomorphism follows from a changing of time and frequency variables according
265
to X = x{κ} and Ω = ω{κ} , so that transform (3.1.4) can be related to a standard Fourier transform of a deformed function Fκ [f (x)](ω) ≡ F [f
{κ}
1 (X)] (Ω) = √ 2π
+∞ Z f {κ} (X) e−i X Ω dX .
(3.1.11)
−∞
It is worthy to note that, if the Fourier transform of a function f (x) exists, then certainly exists its κ-Fourier transform since +∞ +∞ Z Z f (x) |f (x)| dκ x = Fκ [f (x)](ω) ≤ √1 + κ2 x2 dx ≤ kf (x)k , −∞
(3.1.12)
−∞
where the norm of f (x) is defined in the Banach space L1 (R) as usual kf (x)k = 270
+∞ Z |f (x)| dx .
(3.1.13)
−∞
Therefore, the functions space of convergent κ-Fourier transform is wider than the functions space of convergent standard Fourier transform, thanks to the √ factor 1 + κ2 x2 that enforces the convergence of the integral.
16
Finally, when the function f (x) has a well defined parity, Eq. (3.1.4) can be rewritten in the form of a κ-cosine transform Z∞
1 Fκ [f (x)](ω) = √ 2π 275
0
f (x) cosκ (ω ⊗ x) dκ x ,
(3.1.14)
for even functions or a κ-sine transform i Fκ [f (x)](ω) = √ 2π
Z∞ 0
f (x) sinκ (ω ⊗ x) dκ x ,
(3.1.15)
for odd functions. For the sake of illustration, we present in Table 1 the κ-Fourier transform of several functions. Clearly, the new formulation here proposed has its main advantage in the analysis of κ-deformed functions. Thus, certain κ-deformed 280
functions, that can be hardly handled with the standard transform, are easily solved in a closed form with the present formalism, and vice versa. For this reason, we have considered in Table 1 the transform of some deformed functions like expκ (x) or cosκ (x) instead of the corresponding un-deformed versions that, otherwise, are well processed with the standard analysis.
285
Table 1. κ-Fourier transform of several simple functions. f (x) Step function
θ(x)
Modulation
cosκ (a ⊗ x)
Causal κ-exponential Symmetric κ-exponential
θ(x) expκ (−a ⊗ x) 1
κ-Phasor
expκ (a ⊗ x)i
Signum Rectangular
2
1 √1 q2 π a{κ} +i ω{κ} a{κ} 2 2 π a2{κ} +ω{κ}
√ 2 π δ(ω)
√ 2 π uκ (expκ (a)) δ(ω − a) i
δ(x − a) Sgn(x) Π xa 17
{κ}
uκ (expκ (a)) (δ(ω + a) + δ(ω − a))
expκ (−a ⊗ |x|)
Constant Impuslse
pπ
Fκ [f (x)](ω) √ 2 π δ(ω) + √2 π 1i ω
q
expκ (ω⊗a) √1 2q π uκ (expκ (a)) 2 1 π i ω{κ}
2 π
a{κ} sincκ (ω ⊗ a)
As an example, we have plotted in Figure 4, the shape of the sincκ (x) func290
tion, corresponding to the κ-Fourier transform of the rectangular function, for several values of the deformation parameter κ. The stretching effect produced by the κ-deformation is clearly observable. 1,0
1,0
0,0
0,0
-100
-50
0
50
100
= 0.10
0,5
sin
c (x )
0,5
sin
c (x )
0,5
sin
c (x )
1,0
= 0.05
= 0.00
-100
0,0
-50
0
x
x
50
100
-100
-50
0
x
50
100
Figure 4: κ-sinc function for several values of the deformation parameter κ.
3.2. The inverse transform As expected, the original function f (x) can be reconstructed starting from 295
the transformed function fbκ (ω) by means of the inverse integral transform, given
by
Fκ−1
+∞ Z h i 1 b fbκ (ω) expκ (ω ⊗ x)i dκ ω . fκ (ω) (x) = √ 2π
(3.2.1)
−∞
This can be shown by using standard arguments, as follows Fκ−1
[Fκ [f (x)]]
=
1 2π
=
1 2π
=
Z∞
−∞
+∞ Z f (y) expκ (ω ⊗ y)i expκ (−ω ⊗ x)i dκ ω dκ y
−∞ Z∞
f (y) ei ω{κ} (y{κ} −x{κ} ) dω{κ} dy{κ}
−∞
f (y) δ y{κ} − x{κ} dy{κ} = f (x) ,
(3.2.2)
where we used the relations p δ y{κ} − x{κ} = δ(y − x) 1 + κ2 x2 , 18
(3.2.3)
and δ y{κ} − x{κ} dκ x = δ(y − x) dx .
300
(3.2.4)
Otherwise, plugging Eq. (3.2.1) in Eq. (3.1.4), we obtain h i Fκ Fκ−1 [fbκ (ω)] = =
=
1 2π 1 2π
+∞ Z fbκ (ω ′ ) expκ (−ω ′ ⊗ x)i expκ (ω ⊗ x)i dκ ω ′ dκ x
−∞
+∞ Z ′ fbκ (ω) e−i (ω{κ} −ω{κ} ) x{κ} dω ′
{κ}
−∞
dx{κ}
+∞ Z ′ ′ fbκ (ω ′ ) δ(ω{κ} − ω{κ} ) dω{κ} = fbκ (ω) .
(3.2.5)
−∞
Many properties of standard Fourier transforms can be reformulated in the κ-formalism. For instance, let us consider the κ-version of the multiplication theorem +∞ +∞ Z Z ∗ fbκ (ω) b gκ∗ (ω) dκ ω , f (x) g (x) dκ x =
−∞
(3.2.6)
−∞
that, rewritten in
+∞ Z
−∞ 305
f (x) g ∗ (x) √ dx = 1 + κ2 x2
+∞ Z fbκ (ω) b gκ∗ (ω) √ dω , 1 + κ2 ω 2
(3.2.7)
−∞
states a relation between the product of the functions f (x) and g(x) with the product of their κ-deformed Fourier transforms fbκ (ω) and gbκ (ω). Equation
(3.2.6) follows from standard arguments, according to +∞ Z fbκ (ω)b gκ∗ (ω) dκ ω
=
+∞ Z f (x) expκ (−ω ⊗ x)i b g ∗ (ω) dκ x dκ ω
−∞
−∞
=
+∞ Z f (x) gbκ∗ (ω) expκ (ω ⊗ x)i dκ ω dκ x
−∞
=
+∞ Z f (x) g ∗ (x) dκ x ,
−∞
19
(3.2.8)
and, in particular, for f (x) = g(x), it gives the Plancherel theorem in the κformalism +∞ +∞ Z Z 2 |f (x)| dκ x = |fbκ (ω)|2 dκ ω .
−∞ 310
(3.2.9)
−∞
This relation can be rewritten by means of standard integrals in +∞ Z
−∞
|f (x)|2 √ dx = 1 + κ2 x2
+∞ Z
−∞
|fbκ (ω)|2 √ dω 1 + κ2 ω 2
(3.2.10)
that differs clearly from the well known Plancherel theorem formulation. Actually, Eq. (3.2.9) states a relationships between κ-Fourier transformed functions and corresponds to the Plancherel relation only in the κ → 0 limit. 3.3. Main properties 315
Since transform (3.1.4) can be mapped into a standard Fourier transform, it is not a surprise, has already stated, that many properties of Fourier transform still hold in the deformed version if opportunely rephrased in the κ-formalism. This is shown in Table 2, where we report the most relevant relations, leaving the reader to verify easily.
20
Table 2. Main κ-Fourier properties.
320
Linearity
Fκ [α f (x) + β g(x)](ω) = α Fκ [f (x)](ω) + β Fκ [g(x)](ω)
Scaling
Fκ [f (α x)] (ω) =
κ-Scaling Complex conjugation Duality Reverse κ-Frequency shift κ-Time shift Transform of κ-derivative κ-Derivative of transform Transform of integral
Fκ′ [f (x)] (ω ′ )
1 α
where κ′ = κ/α and ω ′ = (a/κ) sinh arcsinh(κ ω)/a2 Fκ [f (α ⊗ x)] (ω) = α 1 Fκ [f (x)] α1 ⊗ ω {κ} h i∗ h i Fκ f (x) (ω) = Fκ f (x) (−ω) h i Fκ Fκ f (x) (ν) (ω) = f (−ω) Fκ [f (−x)] (ω) = Fκ [f (x)](−ω) Fκ expκ (ω0 ⊗ x)i f (x) (ω) = Fκ [f (x)](ω ⊖ ω0 )
Fκ [f (x ⊕ x0 )] (ω) = expκ (ω ⊗ x0 )i Fκ [f (x)](ω) i h (ω) = i ω{κ} Fκ [f (x)](ω) Fκ ddfκ(x) x d Fκ [f (x)](ω) = −i ω{κ} Fκ x{κ} f (x) (ω) dκ ω " # Rx Fκ f (y) dy (ω) = i ω1 Fκ [f (x)](ω) + 2 π Fκ [f (x)](0) δ(ω) {κ}
−∞
κ-Convolution
Modulation
√
Fκ [(f ∗ g)(x)] (ω) = 2 π Fκ [f (x)](ω) Fκ [g(x)](ω) +∞ R where (f ∗ g)(x) = f (y) g(x ⊖ y) dκ y −∞
Fκ [f (x) g(x)] (ω) =
√1 2π
(Fκ [f (x)] ∗ Fκ [g(x)]) (ω)
4. Limiting distribution of κ-sum of n-iterate statistically independent random variables As known, Fourier transform has many interesting applications both in 325
mathematical statistics and in physics. Among them, the problem of searching for stable distributions of a large number of iterates of independent random variables can be easily handled within the canonical Fourier transform theory. Differently, the problem of searching for stable distributions of a large number of iterates of random variables with a specific statistics dependence or statisti-
330
cally independent random variables with a specific iteration, has been a topic of investigation in the recent years. It can be studied by introducing opportunely defined Fourier transforms [9]. In this section we are looking for stable distributions of κ-sum of n-iterate 21
statistically independent random variables. 335
In order to introduce this question in the framework of κ-statistics, let us consider a possible generalization of the characteristic function ϕ (f (x), ω) = hh(x, ω)i ≡
√ 2 π F [f (x)](ω) .
(4.1)
For a given normalized probability distribution f (x) of a random variable X, is natural to define, in the picture of κ-algebra, the quantity ϕκ (f (x), ω) =
√ 2 π Fκ [f (x)](ω) ,
(4.2)
that mimics Eq. (4.1) recovered in the κ → 0 limit.
340
Like in the standard case, Eq. (4.2) coincides with the linear average of the kernel (3.1.5) in the space of the real numbers Z ϕκ (f (x), ω) = hhκ (x, ω)i ≡ f (x) hκ (x, ω) dx .
(4.3)
Equivalently, ϕκ (f (x), ω) can be written as the κ-average of the deformed phase factor ϕκ (f (x), ω) = N hexpκ (−x ⊗ ω)i iκ ,
(4.4)
where the κ-average is defined in hO(x)iκ = 345
with N =
R
1 N
Z
O(x) f (x) dκ x ,
(4.5)
f (x) dκ x the normalization constant. Both Eqs. (4.3) and (4.4)
reduce to the standard characteristic function ϕ(f (x), ω) = hexp(−i x ω)i in the
κ → 0 limit.
As an example, let us consider the deformed Gibbs distribution fκG (x) = θ(x) expκ (−β x) ,
(4.6)
where characteristic function ϕκ′ (f (x), ω), with deformation parameter κ′ = 350
β κ, can be easily calculated in ϕκ′ fκG (x), ω =
κ′
κ′ , β − i arcsinh(κ′ ω) 22
(4.7)
and recovers, in the κ → 0 limit, the well-known result ϕ f G (x), ω = 1/(β −
i ω).
Otherwise, the standard characteristic function of distribution (4.6), ϕ fκG (x), ω , assumes a rather complicate expression by means of special functions which are
355
somewhat cumbersome in the analytical manipulation. Remark that, ϕκ (f (x), ω) carries similar information of ϕ (f (x), ω) since both these functions are related to the starting density f (x) by an analytic mapping. In fact, as like as f (x) and ϕ (f (x), ω) are Fourier transform each other, f (x) and ϕκ (f (x), ω) are κFourier transforms each other. This observation justify the use of the κ-Fourier
360
transform instead of the classical Fourier transform in the analysis based on the κ-formalism. Following standard arguments, the phase factor in Eq. (4.4) can be expanded in powers of ω{κ} , such that ϕκ (f (x), ω) = N
∞ X n (−i ω{κ} )n
. x{κ} κ n! n=0
Taking into account the inequality Z Z D n n E N x{κ} f (x) dκ x < xn f (x) dx = hxn iκ=0 , x{κ} = κ
365
(4.8)
(4.9)
it follows that the n-order of the κ-linear momentum certainly exists if the
standard momentum of the same order exists. On the other hand, it is easy to find distributions with only the few first finite standard momenta in spite of the κ-linear momenta that exist for any order. Again, distribution (4.6) is a paradigm. In fact, in this case, it is easy to verify 370
that only the few standard momenta with n < 1/κ − 1 exist [26], while the κ-momenta are finite for any order.
Finally, from Eq. (4.8) we can obtain the relation D
x{κ}
n E
κ
in dn ϕκ (f (x), ω) = N dκ ω n
.
(4.10)
ω=0
provided that the left hand side exist, a relation that can be also derived straightforward by using the κ-derived of transform given in Table 2. 23
375
Definition (4.2) is useful to study the limit distribution of the κ-sum of n statistically independent random variables, that is, the limit distribution f (Sn ), for large n, of κ-summed random variables Sn = X 1 ⊕ X 2 ⊕ . . . ⊕ X n ,
(4.11)
f (Sn ) = Πni=1 f (xi ) .
(4.12)
given by
In this case, function ϕ (f (Sn ), ω) coincides with the product of the character380
istic functions of f (xi ), i.e. ϕκ (f (Sn ), ω) = ϕκ (f (x1 ), ω) · ϕκ (f (x2 ), ω) · . . . ϕκ (f (xn ), ω) ,
(4.13)
as it follows from the distributive property of κ-sum and κ-product. In particular, if the quantities xi are also identically distributed, then ϕκ (f (Sn ), ω) = n
ϕκ (f (x), ω) . To study the problem of searching families of stable distributions of κ-sum 385
of n-iterate of statistically independent and identically distributed (iid) random variables, we consider a pair of iid random variables X, with density distribution f1 (x). The density distribution f2 (x) of S2 = X ⊕ X, can be obtain from the relation
f2 (y) =
Z
f1 (x) f1 (y ⊖ x) dx .
(4.14)
Stable distributions fulfill the condition f1 (x) = f2 (x) = . . . = fn (x), where 390
fi (x), with i = 1, . . . , n, refers to the pdf of the i-iterate Si given in Eq. (4.11). They can be derived easily by using the property of κ-Fourier transform under κ-convolution (cfr. Table 2), since the characteristic function of stable distributions is invariant under κ-convolution. Let us consider the following ansatz
395
x x fκ (x; σ) = Cκ expκ − √ , (4.15) ⊗κ √ 2σ 2σ √ 2 where Cκ = e−κ /4 / 2 π σ is the normalization constant. Equation (4.15) represents a possible κ-generalization of Gaussian distribution. Standard Gaussian 24
is recovered in the κ → 0 limit.
Note that, Eq. (4.15) differs from other versions of κ-Gaussian proposed in the
400
literature [16, 20]. For instance, the following function x2 e fκ (x) = Aκ expκ − 2 , 2σ
(4.16)
that corresponds to the asymptotic solution of a diffusive process studied in
[16], has a power law tail different from that of distribution (4.15) that rather decays with a log-normal tail, being 1 fκ (x; σ) ≈ exp − 2 ln2 κ
!! √ 2κ x , σ
(4.17)
for x ≫ 1.
405
However, in spite of Eq. (4.16), Eq. (4.15) is invariant under κ-Fourier trans√ form, since its κ-characteristic function ϕκ′ (fκ (x; σ), ω), with κ′ = κ/( 2 σ), is yet a κ-Gaussian given by ϕκ′ (fκ (x; σ), ω) = ≡
σ σ σ Cκ expκ′′ − √ ω ⊗κ′′ √ ω 2 2 σ fκ′′ (ω; 1/σ) ,
(4.18)
where κ′′ = κ/σ 2 . More important, function (4.15) is invariant after n-iterates of κ-convolution, defined in +∞ Z (f ∗ κ g)(x) = f (y) g(x ⊖κ y) dκ y ,
(4.19)
−∞
410
that is commutative (f ∗ κ g)(x) = (g ∗ κ f )(x), associative ((f ∗ κ g) ∗ κ h)(x) =
(f ∗ κ (g ∗ κ h))(x) and bilinear ((c1 f +c2 g) ∗ κ h)(x) = c1 (f ∗ κ h)+c2 (g ∗ κ h). In fact, posing
∗ κ′ fκ )(x; σ) , ∗ κ′ . . . F (n) (x; σ) = (fκ ∗ κ′ f κ
(4.20)
its κ-Fourier transform is related to Eq. (4.18) by n ϕκ′ F (n) (x; σ), ω = (ϕκ′ (fκ (x; σ), ω)) . 25
(4.21)
On the other hand, accounting for relations (2.1.13), the n-power of Eq. (4.18) 415
can be written in n
(ϕκ′ (fκ (x; σ), ω)) = σn Cκn where σn = σ
σn σn expκ′′n − √ ω ⊗κ′′n √ ω , 2 2
(4.22)
√ √ n and κn = k n, that corresponds to the characteristic of the
function (x, fκ(n) n
σn ) = Cκn expκn
x x ⊗ κn √ −√ 2 σn 2 σn
.
(4.23)
Remark that, the structure of Eq. (4.19) used in this proof differs from Eq. (4.14) since, the former, contains a κ-integral instead of a standard inte420
gral. Nevertheless, accounting for relation (2.3.19), it is easy to verify that the function
√x ⊗κ √x exp − κ 1 2σ 2σ r fκ (x, σ) = √ , 2 2πσ 1 + κ2 √x2 σ
(4.24)
properly normalized, is stable under composition (4.14). Therefore, we can confirm that
√x √x − exp ⊗ κ n κ n 1 2 σn 2 σn r , (x, σn ) = √ fκ(n) n 2 2 π σn x 2 1 + κn √ 2 σ
(4.25)
n
is the density distribution of a random variable corresponding to the κ-sum of 425
n-iterate random variables independent and identically distributed according to Eq. (4.24). In analogy with the log-normal distribution and the sinh-normal distribution, Eq. (4.24) defines a family of arcsinh-normal distributions parameterized by the deformation parameter κ and belongs to the family of the Johnson SU
430
distributions introduced in [27] and given by 2
x−ξ 1 δ e− 2 (γ+δ arcsinh( λ )) r f (x) = √ . 2 2πλ x−ξ 1+ λ
Family (4.26) coincides with distribution (4.24) for γ = ξ = 0, δ = √ λ = 2 σn /κn . 26
(4.26) √ 2/κn and
In [14] it has been shown that the κ-sum is substantially equivalent to the relativistic addition of momenta and there it was conjectured a possible relation 435
between the κ-statistics and the theory of the special relativity. In this sense, the κ parameter plays the role of a speed limit according to the relation κ ∝ 1/c. Consequently, in the κ → 0 limit, corresponding to the Galilean relativistic limit
(c → ∞), the κ-sum reduces to the standard sum and consistently, the arcsinhnormal distribution (4.25) recovers the Gaussian distribution that, as stated by
440
the standard central limit theorem, is the stable limiting distribution of the sum of iid random variables. It is fair to note that a similar result has been derived recently in [28] by using a different approach. We observe also that, like distribution (4.15), dis0.6 = 0.00 = 0.30 -2
10
= 0.60
x x) exp (-
0.3
exp (-
x x)
= 0.90
-6
10
= 0.00 = 0.30 = 0.60 = 0.90 -10
0.0
10 -2
-1
0
x
1
-1
2
10
0
10
1
x|
10
2
10
|
Figure 5: Plot of κ-Gaussian (4.24) in the linear-linear scale (left panel) and in the log-linear scale (right panel) for several values of κ. The full-line coincides with the standard Gaussian function.
tribution (4.25) has a log-normal asymptotic behaviour rather than a power-law 445
tail. This is shown in Figure 5 (left panel), where we have depicted, for the sake of illustration, the shape of the κ-Gaussian (4.24) for several values of the deformation parameter. In the right panel of Figure 5, the same curves are reported in a log-linear scale, showing the log-normal asymptotic behaviour. Finally, like as like the standard Gaussian is the limiting distribution of iid
450
summed random variables, we expect the same holds for distribution (4.25) when iid random variables are κ-summed. We show the reliability of such statement by means of a numerical computation reported in Figure 6 (left panel) 27
n =1 n=3 n=7 n = 15
0.5
Numerical 1
1
= 0.5 = 0.0
0.2
(X ;
0.3
f
f
(n)
(X ;
)
)
0.4
0.2
0.1
0.1
0.0
-15
0.0
-10
-5
0
x
5
10
-15
15
-10
-5
0
x
5
10
15
Figure 6: κ-convolution of rectangular-function after several iterates, with κ1 = 0.5 (left panel). The comparison of the numerical curve after n = 15 iterates (full line) with the standard Gaussian (dotted line) and the κ-Gaussian (4.25) (dashed line), are depicted in the right panel.
where we plot the distribution of the random variable Sn , with κ1 = 0.5, after several n-iterations. The starting distribution for the single random variable 455
is assumed to be a rectangular-function. As expected, iterated distribution quickly approaches to a bell-shape. In the same Figure 6 (right panel), this limiting distribution, obtained by a numerical computation after n = 15 iterates (full line), is compared with the standard Gaussian (κ1 = 0, dotted line) and the κ-Gaussian (4.25) (κ1 = 0.5, dashed line). It is evident the good fit between
460
the numerical curve and the κ-Gaussian with respect to the standard Gaussian. A further consistency of this result is supported by recalling that the κ-sum for large values reduces to a standard product [cfr. Eq. (2.1.8)]. This means that, in the far region of large x values, the random variable Sn corresponds to the product of n iid random variables that, according to the central limit theorem,
465
has a log-normal limit distribution, in agreement with the tail of Eq. (4.25). 5. Conclusions In this paper we have reformulated the standard Fourier transform in a formalism consistent with the κ-algebra and the κ-calculus. The new formulation has been derived starting from a κ-deformed Fourier series recently introduced
470
by us in [22]. 28
The κ-Fourier transform fbκ (ω) ≡ Fκ [f (x)](ω), belongs to the integral trans-
forms (1.2) and is characterized by a kernel hκ (x, ω) composed by a deformed phase and a damping factor that confer to hκ (x, ω) a wavelet-like shape. In
addition, both the real part and the imaginary part of the phase factor have an 475
asymptotical log-periodic behaviour. We have shown that the κ-deformed transform of a function f (x) is isomorph to the canonical transform since fbκ (ω) is equivalent to the standard Fourier
transform of the function f {κ} (x) ≡ f (x{κ} ), that is
Fκ [f (x)](ω) ≡ F [f {κ} (x)](ω{κ} ) .
(5.1)
However, in spite of this equivalence, the κ-Fourier transform turns out to be 480
more appropriate to handle functions defined in the realm of the κ-algebra. As a relevant example, we have applied our formalism to the study of the limit distribution of κ-summed statistically independent random variables, that is the distribution given in Eq. (4.25) of the random variables Sn = X ⊕X ⊕. . .⊕X by
using similar arguments employed in the derivation of the central limit theorem. 485
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[18] Silva R 2006 Phys. Lett A 352 17; Pereira F I M, Silva R and Alcaniz J S 2009 Nucl Phys A 828 136; Casas G A, Nobre F D and Curado E M F 2012 Phys. Rev. E 86 061136 [19] Cravero M et al. 2004 Physica A 340 410; Macedo-Filho A et al. 2013 Phys. Lett. A 377 842; Clementi F, Gallegati M and Kaniadakis G 2009 J. Stat.
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31
S0378-4371(17)30288-1 http://dx.doi.org/10.1016/j.physa.2017.03.036 PHYSA 18102
To appear in:
Physica A
Received date: 30 November 2016 Revised date: 20 February 2017 Please cite this article as: A.M. Scarfone, κ-Deformed Fourier Transform, Physica A (2017), http://dx.doi.org/10.1016/j.physa.2017.03.036 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
Highlights *We reformulated the standard Fourier transform in a formalism consistent with the k-algebra useful in the study of deformed functions obtained in the framework of the k-deformed statistical mechanics. *We discuss the central limit theorem for the k-sum of n-iterates statistically independent random variables.
*Manuscript Click here to view linked References
κ-Deformed Fourier Transform A.M. Scarfone Istituto dei Sistemi Complessi (ISC-CNR) c/o Politecnico di Torino Corso Duca degli Abruzzi 24, 10129 Torino, Italy.
Abstract We present a new formulation of Fourier transform in the picture of the κalgebra derived in the framework of the κ-generalized statistical mechanics. The κ-Fourier transform is obtained from a κ-Fourier series recently introduced by us [2013 Entropy 15 624]. The kernel of this transform, that reduces to the usual exponential phase in the κ → 0 limit, is composed by a κ-deformed phase and a damping factor that gives a wavelet-like behavior. We show that
the κ-Fourier transform is isomorph to the standard Fourier transform through a changing of time and frequency variables. Nevertheless, the new formalism is useful to study, according to Fourier analysis, those functions defined in the realm of the κ-algebra. As a relevant application, we discuss the central limit theorem for the κ-sum of n-iterate statistically independent random variables. Keywords: Fourier integral transform, log-periodic oscillations, κ-deformed algebra, power-law distribution.
1. Introduction As well known, Fourier transform is a very useful tool widely used in mathematical statistics and physics. It consists of a linear operator acting on a function f (x), with a real argument x, and transforming it into a function Email address: [email protected] (A.M. Scarfone)
Preprint submitted to Physica A
February 20, 2017
5
fb(ω) ≡ F [f (x)](ω), with a complex argument ω: 1 F [f (x)](ω) = √ 2π
+∞ Z f (x) e−i x ω dx .
(1.1)
−∞
More in general, it belongs to the family of integral transforms, defined in +∞ Z H[f (x)](ω) = f (x) h(x, ω) dx ,
(1.2)
−∞
like Laplace, Mellin and Hankel transforms, with the kernel h(x, ω) given by a complex exponential-like form. Fourier transform finds applications in many fields of science, running from 10
mathematics to physics and engineering. In statistical physics, and in general in probability theory, it is applied in the study of random walk, of infinite divisible distributions, in the analysis of distribution stability and in the study of their domains of attraction as well as in the large deviation theory and in the proof of several central limit theorems [1].
15
In quantum mechanics, Fourier transform links the representation of a wave function in the configuration space with its dual representation in the momenta space. In addition, it is a powerful tool widely used in quantum field theory to evaluate the Green function of a quantum propagator thanks to its property of changing a derivative in a multiplicative power.
20
Its discrete version is largely applied in the theory of signals [2], especially after the introduction of fast algorithms, the so called fast Fourier transform, that speeds up the computation process, allowing the elaboration of a large amount of information, which has permitted to develop relevant data compression algorithms and the realization of high-definition image devices.
25
In quantum computing theory, Fourier transform is powerfully used in the generation of quantum protocols for the factorization of large numbers and in solving the discrete logarithm problem [3]. In the past, several versions of Fourier transform have been proposed in literature by using deformed algebraic structures like, for instance, in the framework
30
of quantum groups where the Hopf algebra underlying the braiding groups has 2
been used to define a Fourier transform in the noncommutative space [4]. In [5], within the basic-calculus, it has been advanced a basic-analogous of standard Fourier transform defined in +∞ Z F [f (x); q](ω) = f (x) exp(x|ω; q) dq x ,
(1.3)
−∞
35
that is realized by employing a complex version of the basic-exponential exp(x; q) R [6] and the basic-integral dq x [7]. On a different ground, in the framework of statistical mechanics based on the Tsallis entropy [8], the nonlinear Fourier transform [9] defined in +∞ Z Fq [f (x)](ω) = f (x) ⊗q expq (−i x ω) dx ,
(1.4)
−∞
has been introduced by using the q-exponential [10] and the q-product [11]. This transform was designed with the purpose to investigate the possible gener40
alization of the central limit theorem within the q-statistics, although the final results are still controversial [12, 13]. In this paper, we revisit the standard Fourier transform in a manner that is consistent with the κ-algebra and the κ-calculus derived in the framework of the κ-statistical mechanics. This formalism, together with the κ-entropy, has been
45
introduced in [14] to generalize the Boltzmann-Gibbs entropy and the related theory, with the aim to describe non Gibbsian systems characterized by powerlaw distributions. In the last decade, several papers have been written on the κ-statistical mechanics concerning its foundations, its theoretical consistence and its possible applications to physical and physical-like systems [15, 16, 17,
50
18, 19, 20] (see [21] for an up to date references list). The new formulation of Fourier transform (hereinafter κ-Fourier transform), is obtained starting from a κ-deformed version of Fourier series recently proposed in [22], given by the following linear operator 1 Fκ [f (x)](ω) = √ 2π
+∞ Z f (x) expκ (−x ⊗κ ω)i dκ x ,
−∞
3
(1.5)
and is based on the κ-exponential and the related κ-algebra introduced in [14] 55
(see also [23]), by replacing both the exponential kernel and the integrate operator with their κ-deformed versions. Here and in the following, for the sake of notation, expκ (x)a means (expκ (x))a . Actually, Eq. (1.5) can be rewritten in the form (1.2) of a standard integral transform over the real numbers with the kernel hκ (x, ω) =
60
exp(−i x{κ} ω{κ} ) √ , 1 + κ2 x2
(1.6)
where x{κ} are κ-numbers defined in the remainder [23].
Function hκ (x, ω) contains a deformed phase factor exp(−i x{κ} ω{κ} ) which √ has an asymptotically log-periodic behavior and a damping factor 1/ 1 + κ2 x2 , that confers to the transform a wavelet-like behaviours. Furthermore, transform (1.5) is isomorph to the standard Fourier transform, where the isomorphism is 65
settled by changing the variables in time and frequency. The plan of the paper is as follows. In the next Section 2, we summarize the main aspects of the κ-algebra and its related calculus. We introduce the Euler formula and the cyclic deformed functions in a manner consistent with the κ-formalism. In Section 3, we derive the new formulation of the Fourier
70
transform based on the κ-algebra. A list of its main properties is also given, while a potential application concerning the central limit theorem for κ-sum of n-iterate statistically independent random variables is discussed in Section 4. Our conclusions are reported in Section 5. 2. Mathematical formalism of the κ-algebra
75
In order to embody power-law distributions in statistical physics, in [14] it has been proposed the following generalized entropic form Z Sκ [p] = − p(x) lnκ p(x) dx ,
(2.1)
which mimics the Boltzmann-Gibbs entropy by replacing the standard logarithm with its generalized version lnκ (x)
=
1 xκ − x−κ ≡ sinh κ ln(x) , 2κ κ 4
(2.2)
In agreement with the maximal entropy principle, the corresponding equilibrium 80
distribution reads 1 p(x) = α expκ − (γ + β x) , λ
(2.3)
where expκ (x)
=
κx+
1/κ p 1 ≡ exp 1 + κ2 x2 arcsinh (κ x) , κ
(2.4)
is the κ-exponential, with lnκ (expκ (x)) = expκ (lnκ (x)) = x.
Both these functions reduce to the standard exponential and logarithm in the κ → 0 limit: ln0 (x) ≡ ln(x) and exp0 (x) ≡ exp(x) and consequently, in the
85
same limit, Eqs. (2.1) and (2.3) converge to the Boltzmann-Gibbs entropy and the Gibbs distribution, respectively. We note that the shape of distribution (2.3), is characterized by an asymptotic power law tail, since expκ (x) ∼ x±1/κ , for x → ±∞. Therefore, it differs from the exponential behavior of the Gibbs-distribution, a fact that justify the use
90
of κ-statistical mechanics in the study of free-scale systems. On a mathematical ground, the main features of κ-statistical physics follow from the analytical properties of the κ-exponential and the κ-logarithm [23]. For any |κ| < 1, logκ (x) and expκ (x) are continuous, monotonic, increasing
functions, normalized in lnκ (1) = 0 and expκ (0) = 1, with lnκ (R+ ) ⊆ R and 95
expκ (R) ⊆ R+ .
Another function useful in the formulation of the κ-deformed statistical me-
chanics is given by uκ (x) =
1 xκ + x−κ ≡ cosh κ ln(x) . 2 κ
(2.5)
For any |κ| < 1, the function uκ (x) is continuous, with uκ (R+ ) ∈ R+ , uκ (0) =
uκ (+∞) = +∞ and reaches its minimum at x = 1, where uκ (1) = 1. Function
100
uκ (x), that reduces to a constant in the κ → 0 limit (u0 (x) = 1), is strictly
related to lnκ (x) and expκ (x) and appears recurrently in the study of their analytical properties.
5
These κ-functions, fulfil the following scaling-laws expκ (µ x) = expκ′ (x)µ ,
(2.6)
lnκ (xµ ) = µ lnκ′ (x) ,
(2.7)
uκ (xµ ) = uκ′ (x) ,
(2.8)
where κ′ = µ κ. In particular, for µ = −1 and accounting for the symmetry 105
relations expκ (x) ≡ exp−κ (x), lnκ (x) ≡ ln−κ (x) and uκ (x) ≡ u−κ (x), we obtain expκ (x) expκ (−x) = 1 ,
(2.9)
lnκ (x) + lnκ (1/x) = 0 ,
(2.10)
uκ (x) − uκ (1/x) = 0 .
(2.11)
Equations (2.9) and (2.10) reproduce the well-known relations of standard exponential and logarithm in the κ-formalism. 2.1. κ-Algebra In [24], it has been shown that, starting from a pair of continuous, monotonic 110
increasing functions, inverse each other, we can introduce two different algebraic structures endowed by a generalized sum and product, that form two distinct Abelian fields. By specializing this result to the present case, we can define two deformed algebras on the target space and on the probability space, respectively. Among them, the target space algebra is the relevant one in this work and is
115
revisited in the following, remanding to [24] for a more detailed discussion. To begin with, let us introduce the κ-numbers x{κ} =
1 arcsinh (κ x) , κ
(2.1.1)
1 sinh (κ x) , κ
(2.1.2)
and their dual x{κ} = with x{κ}
{κ}
= x{κ} 6
{κ}
=x.
(2.1.3)
Mapping (2.1.1) [resp. (2.1.2)] defines a non uniform stretching of the real axis 120
so that the space of the κ-numbers Rκ is isomorph to the space of the real numbers, with (+∞){κ} ≡ +∞, (−∞){κ} ≡ −∞ and 0{κ} ≡ 0.
Generalized sum and product are defined on Rκ according to [22] x ⊕κ y = x{κ} + y{κ}
{κ}
x ⊗κ y = x{κ} · y{κ}
{κ}
,
(2.1.4)
,
(2.1.5)
where, in the following, for the sake of notation, we simply mean ⊕ ≡ ⊕κ and ⊗ ≡ ⊗κ unless explicably stated.
125
These operations are associative, commutative and distributed, with the null element ∅ and the identity I, and for any x ∈ Rκ , there exist the opposite (−x)
and the inverse (1/x). Therefore, the algebraic structure (Rκ , ⊕, ⊗) forms a
commutative group. Explicitly, we have
1 sinh arcsinh (κ x) + arcsinh (κ y) , κ 1 1 x ⊗ y = sinh arcsinh (κ x) · arcsinh (κ y) , κ κ x⊕y =
130
(2.1.6) (2.1.7)
with ∅ ≡ 0, I ≡ κ−1 sinh κ, (−x) ≡ −x and (1/x) ≡ κ−1 sinh(κ2 /arcsinh κ x).
In the κ → 0 limit, sum (2.1.6) and product (2.1.7) recover the standard oper-
ations and the κ-algebra reduces to the standard algebra of the real numbers.
Remark also that, for large x and y, the κ-sum approaches asymptotically to the standard product
135
x⊕y ≃x·y ,
x, y ≫ 1 ,
(2.1.8)
x⊕y ≃x+y ,
x, y → 0 .
(2.1.9)
for κ 6= 0, as well as
Starting from Eqs. (2.1.6) and (2.1.7), we can introduce other elementary operations like the difference x ⊖ y = x ⊕ (−y) and the quotient x ⊘ y = x ⊗ (1/y).
7
However, the main property of κ-algebra follows from the definitions of κexponential and κ-logarithm expκ (x) = ex{κ} , 140
lnκ (x) = (ln x){κ} .
(2.1.10)
Among the many relations satisfied by expκ (x) and lnκ (x), let us recall that expκ (x ⊕ y) = expκ (x) · expκ (y) ,
(2.1.11)
lnκ (x · y) = lnκ (x) ⊕ lnκ (y) ,
(2.1.12)
as well as the following, that will be used in the remainder expκ (x ⊗ y) = expκ (x)y{κ} = expκ (y)x{κ} , 2
expκ (a x ⊗κ a y) = expκ′ (x ⊗κ′ y)a = expκ′′ (a2 (x ⊗κ′ y)) , (2.1.13) expκ (a x ⊕κ a y) = expκ′ (x ⊕κ′ y)a = expκ (a(x ⊕κ′ y)) , with κ′ = κ a and κ′′ = κ/a. Other relations can be obtained by inspection of [22]. 2.2. κ-Cyclic functions 145
According to the Euler formula, the cyclic functions in the κ-formalism can be introduced starting from the κ-deformed version of the complex exponential. However, within the κ-algebra, we have two substantially different possible definitions [22]. The most natural one is given by expκ (x) → expκ (i x) ≡ exp (i x){κ}
,
(2.2.1)
and coincides with the choice made in [21]. The function expκ (i x) is unitary for 150
|x| ≤ 1/|κ|, while becomes a purely imaginary quantity with increasing modulo for |x| > 1/|κ|, that is
| expκ (i x)| = 1 1/κ p | expκ (i x)| = κ x + κ2 x2 − 1
for |x| ≤ 1/|κ| ,
(2.2.2)
for |x| > 1/|κ| .
(2.2.3)
Consistently, with definition (2.2.1) we can introduce the κ-trigonometric functions according to expκ (i x) = Cosκ (x) + i Sinκ (x) , 8
(2.2.4)
so that expκ (i x) − expκ (−i x) , 2i expκ (i x) + expκ (−i x) . Cosκ (x) = 2
Sinκ (x) =
155
(2.2.5) (2.2.6)
These functions take real values in the range |x| ≤ 1/|κ| and imaginary values
otherwise. This is illustrated in Figure 1, where we plot the Cosκ (x) for κ = 0.05, in the region |x| ≤ 1/|κ| (left panel) and its modulo |Cosκ (x)| (right panel) in the region |x| > 1/|κ|, where the κ-cosine (2.2.6) assumes imaginary values. 1,0 8
10
(x)|
0,0
|Cos
Cos
(x)
0,5
4
10
-0,5
-1,0
0
10 -20
-15
-10
-5
0
x
5
10
15
20
20
25
x
30
Figure 1: Plot of Cosκ (x) defined in Eq. (2.2.6), for κ = 0.05, in the region |x| ≤ 1/|κ| (left panel) and its modulo in the region |x| > 1/|κ| (right panel).
A different definition of the κ-exponential on the complex unit circle is given 160
by expκ (x) → expκ (x)i ≡ exp i x{κ}
.
(2.2.7)
Actually, definitions (2.2.1) and (2.2.7) are related each other in expκ (x)i = expκ′ (i x) ,
(2.2.8)
according to the scaling relation (2.6), with κ′ = −i κ.
However, the complex function expκ (x)i has unitary modulo for any x ∈ R and,
by taking into account that |x| < |x{κ} | and |x − x{κ} | increases for |x| → ∞, it
165
follows that the period of functions expκ (x)i increases as |x| increases. 9
According to Eq. (2.2.7), we introduce a new set of κ-deformed cyclic functions as expκ (x)i = cosκ (x) + i sinκ (x) ,
(2.2.9)
so that sinκ (x) ≡ sin(x{κ} ) ,
(2.2.10)
cosκ (x) ≡ cos(x{κ} ) .
(2.2.11)
We remark that functions (2.2.10) and (2.2.11) are asymptotically log-periodic.
1.0
0,5
0.5
0,0
0.0
cos
cos
(x)
(x)
1,0
-0.5
-0,5
-1.0
-1,0 0
200
x
-1
400
10
0
10
1
10
2
10
x
3
4
10
10
5
10
Figure 2: Plot of cosκ (x) given by Eq. (2.2.11) for k = 0.2 in the linear-linear scale (a) and in the log-linear scale (b) showing its asymptotic log-periodic behavior.
170
Their period increases for |x| → ∞, because sinκ (x) = sinκ (x′ ) ,
when
x′ = x{κ} + 2 n π
and, for large x, we have ∆ ln(x) ≃ 2 n κ π ,
that is
{κ}
,
x′ ≃ x e2 n κ π ,
(2.2.12)
(2.2.13)
where ∆ ln(x) = ln(x′ ) − ln(x) approaches to a constant for x ≫ 1.
This is shown in Figure 2, where we plot the shape of cosκ (x), for κ = 0.2, in the linear-linear scale (left panel) and in the log-linear scale (right panel) that 175
shows its asymptotic log-periodic behavior. 10
As proven in [22], the functions sinκ (x) and cosκ (x) can be derived from the following differential equation d dx
2 p a{κ} d u(x) 2 2 1+κ x u(x) = 0 , +√ dx 1 + κ2 x2
(2.2.14)
with −h ≤ x ≤ h and a{κ} a constant. This is a Sturm-Liouville equation df (x) d p(x) + λ w(x) f (x) = 0 , (2.2.15) dx dx √ √ with p(x) = 1 + κ2 x2 , weight function w(x) = 1/ 1 + κ2 x2 and eigenvalue 180
λ = (a{κ} )2 . A solution of Eq. (2.2.14), with boundary conditions f (−h) = f (h) = 0, corresponds to the κ-sine function
f (x) = A sinκ (a ⊗ x) ,
(2.2.16)
whilst a solution with boundary conditions f ′ (−h) = f ′ (h) = 0, where prime means derivative with respect to its argument, is given by the κ-cosine function f (x) = A cosκ (a ⊗ x) .
(2.2.17)
Similar considerations hold for the function f (x) = A expκ (a ⊗ x)i , 185
(2.2.18)
that is the solution of the same Sturm-Liouville problem, in the h → ∞ limit,
with boundary condition |f (x)| < ∞. 2.3. κ-Calculus
Following [14], we introduce the calculus in the κ-formalism by means of the κ-differential dκ x ≡ dx{κ} , given by the differential of the κ-numbers dx dκ x = √ , 1 + κ2 x2 190
(2.3.1)
that is κ-linear dκ (a ⊗ x) = a{κ} · dκ x ,
(2.3.2)
dκ (x ⊕ y) = dκ x + dκ y ,
(2.3.3)
11
where a is constant. The κ-derivative is defined in
d dx
κ
≡
d , dκ x
(2.3.4)
and is related to the Leibnitz derivative by p d d d = 1 + κ2 x2 ≡ uκ (expκ (x)) , dκ x dx dx
(2.3.5) √ where the last equality follows from the relation uκ (expκ (x)) = 1 + κ2 x2
195
derived in [25]. Within the standard calculus, the derivative of the κ-functions previously introduced read expκ (x) d expκ (x) = , dx uκ (expκ (x)) uκ (x) d lnκ (x) = , dx x d lnκ (x) uκ (x) = κ , dx x cosκ (x) d sinκ (x) = , dx uκ (expκ (x)) sinκ (x) d cosκ (x) = − . dx uκ (expκ (x))
(2.3.6) (2.3.7) (2.3.8) (2.3.9) (2.3.10)
In this way, by using Eq. (2.3.5), we can rewrite these relations in d expκ (x) = expκ (x) , dκ x d sinκ (x) = cosκ (x) , dκ x d cosκ (x) = − sinκ (x) , dκ x
(2.3.11) (2.3.12) (2.3.13)
that are consistent within the κ-formalism. 200
We observe that quantities like expκ (x) dκ x, sinκ (x) dκ x and cosκ (x) dκ x are all exact differentials in the standard calculus, since they correspond, respectively to the following differential forms: d expκ (x), d sinκ (x) and d cosκ (x). In addition, accounting for the κ-linearity, we can verify the relations df (z) d f (x ⊕ y) = , (2.3.14) dκ x dκ z z=x⊕y d df (z) expκ (x ⊗ y) = y{κ} , (2.3.15) dκ x dκ z z=x⊗y 12
and in particular d expκ (x ⊕ y) = expκ (x ⊕ y) , dκ x d expκ (x ⊗ y) = y{κ} expκ (x ⊗ y) , dκ x 205
that will be used in the remainder. Finally, let us introduce the κ-integral
R
(2.3.16) (2.3.17)
f (x) dκ x, as the inverse operator to
the κ-derivative, according to Z d f (x) dκ x + const. = f (x) , dx κ
(2.3.18)
extending, in this way, the fundamental integral theorem to the κ-formalism. We observe that the κ-integral can be written like a weighted ordinary integral.
210
In fact, recalling Eq. (2.3.1), we have Z Z Z f (x) √ dx , f (x) dκ x = f (x) w(x) dx ≡ 1 + κ2 x2
(2.3.19)
where the weight function
w(x) =
1 1 , ≡ √ uκ (expκ (x)) 1 + κ2 x2
(2.3.20)
coincides with that introduced in the Sturm-Liouville problem. In addition, we can also use the following relation Zb
b{κ}
f (x) dκ x =
a
Z
f {κ} (x) dx ,
(2.3.21)
(2.3.22)
a{κ}
where f 215
{κ}
(x) ≡ f (x
{κ}
)=f
1 sinh(κ x) , κ
that links the κ-integral on the κ-numbers to a standard integral on the real numbers. 3. κ-Fourier transform 3.1. The transform In [22], it has been introduced a consistent formulation of Fourier series in the
220
framework of the κ-algebra. There, it has been shown as any square-integrable 13
function f (x) : (−h, h) → R, may be expanded in the κ-Fourier series with
respect to the orthogonal and complete system of functions sinκ (an ⊗ x) and cosκ (an ⊗ x), that is
f (x) = c0 +
∞ X
n=1
(sn sinκ (an ⊗ x) + cn cosκ (an ⊗ x)) ,
(3.1.1)
where an = (n π/h{κ} ){κ} are suitable constants.
225
This series expansion can be written in the complex form f (x) =
∞ X
n=−∞
γn expκ (−an ⊗ x)i ,
(3.1.2)
where the complex Fourier coefficients are given by γn =
r
2 h
Zh
−h
f (x) expκ (an ⊗ x)i dκ x ,
(3.1.3)
and are related to the real Fourier coefficients in γn = (cn −i sn )/2 and γ−n = γn∗ .
Starting from this result, κ-deformed Fourier transform can be formally obtained in the h → ∞ limit, as usually done in the standard theory.
230
Therefore, from Eq. (3.1.3) we derive the following integral transform for a real function f (x) given by 1 Fκ [f (x)](ω) = √ 2π
+∞ Z f (x) expκ (−x ⊗ ω)i dκ x .
(3.1.4)
−∞
Accounting for Eqs. (2.1.5), (2.1.10) and (2.3.1), this formula can be written like a standard integral transform [cfr. Eq. (1.2)], with the kernel hκ (x, ω) =
exp(−i x{κ} ω{κ} ) √ . 1 + κ2 x2
(3.1.5)
In the κ → 0 limit, kernel hκ (x, ω) reduces to the usual exponential phase
235
h(x, ω) = exp(i x ω) and the κ-Fourier transform collapses into the standard Fourier transform. Kernel (3.1.5) is formed by a deformed phase factor, that arises from the generalization of the complex exponential in its κ-deformed version and by a 14
damping factor, that arises from the measure of the κ-integral. As a conse240
quence, the new transform acquires interesting analytical features. For instance, both the real part and the imaginary part of the phase factor have an asymptotically log-periodic behaviour, a fact that may be relevant in the study of log-oscillating phenomena [24]. Otherwise, the damping factor confers to the kernel a wavelet-like behaviour, as shown in Figure 3, where we have plotted
245
both the real and the imaginary part of the kernel hκ (x, ω). However, many properties of the standard Fourier transform are preserved in the generalized version provided that they are reformulated in the κ-formalism.
1,0
0,5
0,5
h (x,
0,0
Im [
0,0
Re [
h (x,
)]
)]
1,0
-0,5
-0,5
-1,0
-1,0 -150
-100
-50
0
x
50
100
-150
150
-100
-50
0
x
50
100
150
Figure 3: Real (left panel) and imaginary (right panel) part of the kernel hκ (x, ω), for k = 0.1 and ω = 1.
It is duty to note that, in addition to Eq. (3.1.5), there are other possible 250
generalizations of the kernel. They should satisfy the following reasonable conditions: 1) the kernel must reduce to the standard complex exponential exp(i ω x) in a suitable limit; 2) the kernel should be an unimodular function on the whole real axis. These requests exclude the function h(0) κ (x, ω) = expκ (−i x ω) ,
(3.1.6)
since condition 2) is violated in the far region of the real axis, |x ω| ≥ 1/|κ|.
255
Otherwise, the following functions are suitable candidates iω h(1) , κ (x, ω) = expκ (−x)
15
(3.1.7)
ix h(2) , κ (x, ω) = expκ (−ω)
(3.1.8)
i h(3) κ (x, ω) = expκ (−x ω) ,
(3.1.9)
since they have all unitary modulus on R and converge to exp(i ω x) in the κ → 0 limit. However, although some of these functions are equivalent each others, like, for instance
x{κ} , ω , x, ω{κ} = h(2) hκ (x, ω) = h(1) κ κ
(3.1.10)
they define different integral transforms with different analytical properties. 260
By inspection, definition (3.1.4), with the kernel (3.1.5) is the most consistent choice. In fact, it turns out to be isomorph to the standard transform, so that all the analytical properties of the Fourier transform are preserved when rewritten in the κ-formalism. Isomorphism follows from a changing of time and frequency variables according
265
to X = x{κ} and Ω = ω{κ} , so that transform (3.1.4) can be related to a standard Fourier transform of a deformed function Fκ [f (x)](ω) ≡ F [f
{κ}
1 (X)] (Ω) = √ 2π
+∞ Z f {κ} (X) e−i X Ω dX .
(3.1.11)
−∞
It is worthy to note that, if the Fourier transform of a function f (x) exists, then certainly exists its κ-Fourier transform since +∞ +∞ Z Z f (x) |f (x)| dκ x = Fκ [f (x)](ω) ≤ √1 + κ2 x2 dx ≤ kf (x)k , −∞
(3.1.12)
−∞
where the norm of f (x) is defined in the Banach space L1 (R) as usual kf (x)k = 270
+∞ Z |f (x)| dx .
(3.1.13)
−∞
Therefore, the functions space of convergent κ-Fourier transform is wider than the functions space of convergent standard Fourier transform, thanks to the √ factor 1 + κ2 x2 that enforces the convergence of the integral.
16
Finally, when the function f (x) has a well defined parity, Eq. (3.1.4) can be rewritten in the form of a κ-cosine transform Z∞
1 Fκ [f (x)](ω) = √ 2π 275
0
f (x) cosκ (ω ⊗ x) dκ x ,
(3.1.14)
for even functions or a κ-sine transform i Fκ [f (x)](ω) = √ 2π
Z∞ 0
f (x) sinκ (ω ⊗ x) dκ x ,
(3.1.15)
for odd functions. For the sake of illustration, we present in Table 1 the κ-Fourier transform of several functions. Clearly, the new formulation here proposed has its main advantage in the analysis of κ-deformed functions. Thus, certain κ-deformed 280
functions, that can be hardly handled with the standard transform, are easily solved in a closed form with the present formalism, and vice versa. For this reason, we have considered in Table 1 the transform of some deformed functions like expκ (x) or cosκ (x) instead of the corresponding un-deformed versions that, otherwise, are well processed with the standard analysis.
285
Table 1. κ-Fourier transform of several simple functions. f (x) Step function
θ(x)
Modulation
cosκ (a ⊗ x)
Causal κ-exponential Symmetric κ-exponential
θ(x) expκ (−a ⊗ x) 1
κ-Phasor
expκ (a ⊗ x)i
Signum Rectangular
2
1 √1 q2 π a{κ} +i ω{κ} a{κ} 2 2 π a2{κ} +ω{κ}
√ 2 π δ(ω)
√ 2 π uκ (expκ (a)) δ(ω − a) i
δ(x − a) Sgn(x) Π xa 17
{κ}
uκ (expκ (a)) (δ(ω + a) + δ(ω − a))
expκ (−a ⊗ |x|)
Constant Impuslse
pπ
Fκ [f (x)](ω) √ 2 π δ(ω) + √2 π 1i ω
q
expκ (ω⊗a) √1 2q π uκ (expκ (a)) 2 1 π i ω{κ}
2 π
a{κ} sincκ (ω ⊗ a)
As an example, we have plotted in Figure 4, the shape of the sincκ (x) func290
tion, corresponding to the κ-Fourier transform of the rectangular function, for several values of the deformation parameter κ. The stretching effect produced by the κ-deformation is clearly observable. 1,0
1,0
0,0
0,0
-100
-50
0
50
100
= 0.10
0,5
sin
c (x )
0,5
sin
c (x )
0,5
sin
c (x )
1,0
= 0.05
= 0.00
-100
0,0
-50
0
x
x
50
100
-100
-50
0
x
50
100
Figure 4: κ-sinc function for several values of the deformation parameter κ.
3.2. The inverse transform As expected, the original function f (x) can be reconstructed starting from 295
the transformed function fbκ (ω) by means of the inverse integral transform, given
by
Fκ−1
+∞ Z h i 1 b fbκ (ω) expκ (ω ⊗ x)i dκ ω . fκ (ω) (x) = √ 2π
(3.2.1)
−∞
This can be shown by using standard arguments, as follows Fκ−1
[Fκ [f (x)]]
=
1 2π
=
1 2π
=
Z∞
−∞
+∞ Z f (y) expκ (ω ⊗ y)i expκ (−ω ⊗ x)i dκ ω dκ y
−∞ Z∞
f (y) ei ω{κ} (y{κ} −x{κ} ) dω{κ} dy{κ}
−∞
f (y) δ y{κ} − x{κ} dy{κ} = f (x) ,
(3.2.2)
where we used the relations p δ y{κ} − x{κ} = δ(y − x) 1 + κ2 x2 , 18
(3.2.3)
and δ y{κ} − x{κ} dκ x = δ(y − x) dx .
300
(3.2.4)
Otherwise, plugging Eq. (3.2.1) in Eq. (3.1.4), we obtain h i Fκ Fκ−1 [fbκ (ω)] = =
=
1 2π 1 2π
+∞ Z fbκ (ω ′ ) expκ (−ω ′ ⊗ x)i expκ (ω ⊗ x)i dκ ω ′ dκ x
−∞
+∞ Z ′ fbκ (ω) e−i (ω{κ} −ω{κ} ) x{κ} dω ′
{κ}
−∞
dx{κ}
+∞ Z ′ ′ fbκ (ω ′ ) δ(ω{κ} − ω{κ} ) dω{κ} = fbκ (ω) .
(3.2.5)
−∞
Many properties of standard Fourier transforms can be reformulated in the κ-formalism. For instance, let us consider the κ-version of the multiplication theorem +∞ +∞ Z Z ∗ fbκ (ω) b gκ∗ (ω) dκ ω , f (x) g (x) dκ x =
−∞
(3.2.6)
−∞
that, rewritten in
+∞ Z
−∞ 305
f (x) g ∗ (x) √ dx = 1 + κ2 x2
+∞ Z fbκ (ω) b gκ∗ (ω) √ dω , 1 + κ2 ω 2
(3.2.7)
−∞
states a relation between the product of the functions f (x) and g(x) with the product of their κ-deformed Fourier transforms fbκ (ω) and gbκ (ω). Equation
(3.2.6) follows from standard arguments, according to +∞ Z fbκ (ω)b gκ∗ (ω) dκ ω
=
+∞ Z f (x) expκ (−ω ⊗ x)i b g ∗ (ω) dκ x dκ ω
−∞
−∞
=
+∞ Z f (x) gbκ∗ (ω) expκ (ω ⊗ x)i dκ ω dκ x
−∞
=
+∞ Z f (x) g ∗ (x) dκ x ,
−∞
19
(3.2.8)
and, in particular, for f (x) = g(x), it gives the Plancherel theorem in the κformalism +∞ +∞ Z Z 2 |f (x)| dκ x = |fbκ (ω)|2 dκ ω .
−∞ 310
(3.2.9)
−∞
This relation can be rewritten by means of standard integrals in +∞ Z
−∞
|f (x)|2 √ dx = 1 + κ2 x2
+∞ Z
−∞
|fbκ (ω)|2 √ dω 1 + κ2 ω 2
(3.2.10)
that differs clearly from the well known Plancherel theorem formulation. Actually, Eq. (3.2.9) states a relationships between κ-Fourier transformed functions and corresponds to the Plancherel relation only in the κ → 0 limit. 3.3. Main properties 315
Since transform (3.1.4) can be mapped into a standard Fourier transform, it is not a surprise, has already stated, that many properties of Fourier transform still hold in the deformed version if opportunely rephrased in the κ-formalism. This is shown in Table 2, where we report the most relevant relations, leaving the reader to verify easily.
20
Table 2. Main κ-Fourier properties.
320
Linearity
Fκ [α f (x) + β g(x)](ω) = α Fκ [f (x)](ω) + β Fκ [g(x)](ω)
Scaling
Fκ [f (α x)] (ω) =
κ-Scaling Complex conjugation Duality Reverse κ-Frequency shift κ-Time shift Transform of κ-derivative κ-Derivative of transform Transform of integral
Fκ′ [f (x)] (ω ′ )
1 α
where κ′ = κ/α and ω ′ = (a/κ) sinh arcsinh(κ ω)/a2 Fκ [f (α ⊗ x)] (ω) = α 1 Fκ [f (x)] α1 ⊗ ω {κ} h i∗ h i Fκ f (x) (ω) = Fκ f (x) (−ω) h i Fκ Fκ f (x) (ν) (ω) = f (−ω) Fκ [f (−x)] (ω) = Fκ [f (x)](−ω) Fκ expκ (ω0 ⊗ x)i f (x) (ω) = Fκ [f (x)](ω ⊖ ω0 )
Fκ [f (x ⊕ x0 )] (ω) = expκ (ω ⊗ x0 )i Fκ [f (x)](ω) i h (ω) = i ω{κ} Fκ [f (x)](ω) Fκ ddfκ(x) x d Fκ [f (x)](ω) = −i ω{κ} Fκ x{κ} f (x) (ω) dκ ω " # Rx Fκ f (y) dy (ω) = i ω1 Fκ [f (x)](ω) + 2 π Fκ [f (x)](0) δ(ω) {κ}
−∞
κ-Convolution
Modulation
√
Fκ [(f ∗ g)(x)] (ω) = 2 π Fκ [f (x)](ω) Fκ [g(x)](ω) +∞ R where (f ∗ g)(x) = f (y) g(x ⊖ y) dκ y −∞
Fκ [f (x) g(x)] (ω) =
√1 2π
(Fκ [f (x)] ∗ Fκ [g(x)]) (ω)
4. Limiting distribution of κ-sum of n-iterate statistically independent random variables As known, Fourier transform has many interesting applications both in 325
mathematical statistics and in physics. Among them, the problem of searching for stable distributions of a large number of iterates of independent random variables can be easily handled within the canonical Fourier transform theory. Differently, the problem of searching for stable distributions of a large number of iterates of random variables with a specific statistics dependence or statisti-
330
cally independent random variables with a specific iteration, has been a topic of investigation in the recent years. It can be studied by introducing opportunely defined Fourier transforms [9]. In this section we are looking for stable distributions of κ-sum of n-iterate 21
statistically independent random variables. 335
In order to introduce this question in the framework of κ-statistics, let us consider a possible generalization of the characteristic function ϕ (f (x), ω) = hh(x, ω)i ≡
√ 2 π F [f (x)](ω) .
(4.1)
For a given normalized probability distribution f (x) of a random variable X, is natural to define, in the picture of κ-algebra, the quantity ϕκ (f (x), ω) =
√ 2 π Fκ [f (x)](ω) ,
(4.2)
that mimics Eq. (4.1) recovered in the κ → 0 limit.
340
Like in the standard case, Eq. (4.2) coincides with the linear average of the kernel (3.1.5) in the space of the real numbers Z ϕκ (f (x), ω) = hhκ (x, ω)i ≡ f (x) hκ (x, ω) dx .
(4.3)
Equivalently, ϕκ (f (x), ω) can be written as the κ-average of the deformed phase factor ϕκ (f (x), ω) = N hexpκ (−x ⊗ ω)i iκ ,
(4.4)
where the κ-average is defined in hO(x)iκ = 345
with N =
R
1 N
Z
O(x) f (x) dκ x ,
(4.5)
f (x) dκ x the normalization constant. Both Eqs. (4.3) and (4.4)
reduce to the standard characteristic function ϕ(f (x), ω) = hexp(−i x ω)i in the
κ → 0 limit.
As an example, let us consider the deformed Gibbs distribution fκG (x) = θ(x) expκ (−β x) ,
(4.6)
where characteristic function ϕκ′ (f (x), ω), with deformation parameter κ′ = 350
β κ, can be easily calculated in ϕκ′ fκG (x), ω =
κ′
κ′ , β − i arcsinh(κ′ ω) 22
(4.7)
and recovers, in the κ → 0 limit, the well-known result ϕ f G (x), ω = 1/(β −
i ω).
Otherwise, the standard characteristic function of distribution (4.6), ϕ fκG (x), ω , assumes a rather complicate expression by means of special functions which are
355
somewhat cumbersome in the analytical manipulation. Remark that, ϕκ (f (x), ω) carries similar information of ϕ (f (x), ω) since both these functions are related to the starting density f (x) by an analytic mapping. In fact, as like as f (x) and ϕ (f (x), ω) are Fourier transform each other, f (x) and ϕκ (f (x), ω) are κFourier transforms each other. This observation justify the use of the κ-Fourier
360
transform instead of the classical Fourier transform in the analysis based on the κ-formalism. Following standard arguments, the phase factor in Eq. (4.4) can be expanded in powers of ω{κ} , such that ϕκ (f (x), ω) = N
∞ X n (−i ω{κ} )n
. x{κ} κ n! n=0
Taking into account the inequality Z Z D n n E N x{κ} f (x) dκ x < xn f (x) dx = hxn iκ=0 , x{κ} = κ
365
(4.8)
(4.9)
it follows that the n-order of the κ-linear momentum certainly exists if the
standard momentum of the same order exists. On the other hand, it is easy to find distributions with only the few first finite standard momenta in spite of the κ-linear momenta that exist for any order. Again, distribution (4.6) is a paradigm. In fact, in this case, it is easy to verify 370
that only the few standard momenta with n < 1/κ − 1 exist [26], while the κ-momenta are finite for any order.
Finally, from Eq. (4.8) we can obtain the relation D
x{κ}
n E
κ
in dn ϕκ (f (x), ω) = N dκ ω n
.
(4.10)
ω=0
provided that the left hand side exist, a relation that can be also derived straightforward by using the κ-derived of transform given in Table 2. 23
375
Definition (4.2) is useful to study the limit distribution of the κ-sum of n statistically independent random variables, that is, the limit distribution f (Sn ), for large n, of κ-summed random variables Sn = X 1 ⊕ X 2 ⊕ . . . ⊕ X n ,
(4.11)
f (Sn ) = Πni=1 f (xi ) .
(4.12)
given by
In this case, function ϕ (f (Sn ), ω) coincides with the product of the character380
istic functions of f (xi ), i.e. ϕκ (f (Sn ), ω) = ϕκ (f (x1 ), ω) · ϕκ (f (x2 ), ω) · . . . ϕκ (f (xn ), ω) ,
(4.13)
as it follows from the distributive property of κ-sum and κ-product. In particular, if the quantities xi are also identically distributed, then ϕκ (f (Sn ), ω) = n
ϕκ (f (x), ω) . To study the problem of searching families of stable distributions of κ-sum 385
of n-iterate of statistically independent and identically distributed (iid) random variables, we consider a pair of iid random variables X, with density distribution f1 (x). The density distribution f2 (x) of S2 = X ⊕ X, can be obtain from the relation
f2 (y) =
Z
f1 (x) f1 (y ⊖ x) dx .
(4.14)
Stable distributions fulfill the condition f1 (x) = f2 (x) = . . . = fn (x), where 390
fi (x), with i = 1, . . . , n, refers to the pdf of the i-iterate Si given in Eq. (4.11). They can be derived easily by using the property of κ-Fourier transform under κ-convolution (cfr. Table 2), since the characteristic function of stable distributions is invariant under κ-convolution. Let us consider the following ansatz
395
x x fκ (x; σ) = Cκ expκ − √ , (4.15) ⊗κ √ 2σ 2σ √ 2 where Cκ = e−κ /4 / 2 π σ is the normalization constant. Equation (4.15) represents a possible κ-generalization of Gaussian distribution. Standard Gaussian 24
is recovered in the κ → 0 limit.
Note that, Eq. (4.15) differs from other versions of κ-Gaussian proposed in the
400
literature [16, 20]. For instance, the following function x2 e fκ (x) = Aκ expκ − 2 , 2σ
(4.16)
that corresponds to the asymptotic solution of a diffusive process studied in
[16], has a power law tail different from that of distribution (4.15) that rather decays with a log-normal tail, being 1 fκ (x; σ) ≈ exp − 2 ln2 κ
!! √ 2κ x , σ
(4.17)
for x ≫ 1.
405
However, in spite of Eq. (4.16), Eq. (4.15) is invariant under κ-Fourier trans√ form, since its κ-characteristic function ϕκ′ (fκ (x; σ), ω), with κ′ = κ/( 2 σ), is yet a κ-Gaussian given by ϕκ′ (fκ (x; σ), ω) = ≡
σ σ σ Cκ expκ′′ − √ ω ⊗κ′′ √ ω 2 2 σ fκ′′ (ω; 1/σ) ,
(4.18)
where κ′′ = κ/σ 2 . More important, function (4.15) is invariant after n-iterates of κ-convolution, defined in +∞ Z (f ∗ κ g)(x) = f (y) g(x ⊖κ y) dκ y ,
(4.19)
−∞
410
that is commutative (f ∗ κ g)(x) = (g ∗ κ f )(x), associative ((f ∗ κ g) ∗ κ h)(x) =
(f ∗ κ (g ∗ κ h))(x) and bilinear ((c1 f +c2 g) ∗ κ h)(x) = c1 (f ∗ κ h)+c2 (g ∗ κ h). In fact, posing
∗ κ′ fκ )(x; σ) , ∗ κ′ . . . F (n) (x; σ) = (fκ ∗ κ′ f κ
(4.20)
its κ-Fourier transform is related to Eq. (4.18) by n ϕκ′ F (n) (x; σ), ω = (ϕκ′ (fκ (x; σ), ω)) . 25
(4.21)
On the other hand, accounting for relations (2.1.13), the n-power of Eq. (4.18) 415
can be written in n
(ϕκ′ (fκ (x; σ), ω)) = σn Cκn where σn = σ
σn σn expκ′′n − √ ω ⊗κ′′n √ ω , 2 2
(4.22)
√ √ n and κn = k n, that corresponds to the characteristic of the
function (x, fκ(n) n
σn ) = Cκn expκn
x x ⊗ κn √ −√ 2 σn 2 σn
.
(4.23)
Remark that, the structure of Eq. (4.19) used in this proof differs from Eq. (4.14) since, the former, contains a κ-integral instead of a standard inte420
gral. Nevertheless, accounting for relation (2.3.19), it is easy to verify that the function
√x ⊗κ √x exp − κ 1 2σ 2σ r fκ (x, σ) = √ , 2 2πσ 1 + κ2 √x2 σ
(4.24)
properly normalized, is stable under composition (4.14). Therefore, we can confirm that
√x √x − exp ⊗ κ n κ n 1 2 σn 2 σn r , (x, σn ) = √ fκ(n) n 2 2 π σn x 2 1 + κn √ 2 σ
(4.25)
n
is the density distribution of a random variable corresponding to the κ-sum of 425
n-iterate random variables independent and identically distributed according to Eq. (4.24). In analogy with the log-normal distribution and the sinh-normal distribution, Eq. (4.24) defines a family of arcsinh-normal distributions parameterized by the deformation parameter κ and belongs to the family of the Johnson SU
430
distributions introduced in [27] and given by 2
x−ξ 1 δ e− 2 (γ+δ arcsinh( λ )) r f (x) = √ . 2 2πλ x−ξ 1+ λ
Family (4.26) coincides with distribution (4.24) for γ = ξ = 0, δ = √ λ = 2 σn /κn . 26
(4.26) √ 2/κn and
In [14] it has been shown that the κ-sum is substantially equivalent to the relativistic addition of momenta and there it was conjectured a possible relation 435
between the κ-statistics and the theory of the special relativity. In this sense, the κ parameter plays the role of a speed limit according to the relation κ ∝ 1/c. Consequently, in the κ → 0 limit, corresponding to the Galilean relativistic limit
(c → ∞), the κ-sum reduces to the standard sum and consistently, the arcsinhnormal distribution (4.25) recovers the Gaussian distribution that, as stated by
440
the standard central limit theorem, is the stable limiting distribution of the sum of iid random variables. It is fair to note that a similar result has been derived recently in [28] by using a different approach. We observe also that, like distribution (4.15), dis0.6 = 0.00 = 0.30 -2
10
= 0.60
x x) exp (-
0.3
exp (-
x x)
= 0.90
-6
10
= 0.00 = 0.30 = 0.60 = 0.90 -10
0.0
10 -2
-1
0
x
1
-1
2
10
0
10
1
x|
10
2
10
|
Figure 5: Plot of κ-Gaussian (4.24) in the linear-linear scale (left panel) and in the log-linear scale (right panel) for several values of κ. The full-line coincides with the standard Gaussian function.
tribution (4.25) has a log-normal asymptotic behaviour rather than a power-law 445
tail. This is shown in Figure 5 (left panel), where we have depicted, for the sake of illustration, the shape of the κ-Gaussian (4.24) for several values of the deformation parameter. In the right panel of Figure 5, the same curves are reported in a log-linear scale, showing the log-normal asymptotic behaviour. Finally, like as like the standard Gaussian is the limiting distribution of iid
450
summed random variables, we expect the same holds for distribution (4.25) when iid random variables are κ-summed. We show the reliability of such statement by means of a numerical computation reported in Figure 6 (left panel) 27
n =1 n=3 n=7 n = 15
0.5
Numerical 1
1
= 0.5 = 0.0
0.2
(X ;
0.3
f
f
(n)
(X ;
)
)
0.4
0.2
0.1
0.1
0.0
-15
0.0
-10
-5
0
x
5
10
-15
15
-10
-5
0
x
5
10
15
Figure 6: κ-convolution of rectangular-function after several iterates, with κ1 = 0.5 (left panel). The comparison of the numerical curve after n = 15 iterates (full line) with the standard Gaussian (dotted line) and the κ-Gaussian (4.25) (dashed line), are depicted in the right panel.
where we plot the distribution of the random variable Sn , with κ1 = 0.5, after several n-iterations. The starting distribution for the single random variable 455
is assumed to be a rectangular-function. As expected, iterated distribution quickly approaches to a bell-shape. In the same Figure 6 (right panel), this limiting distribution, obtained by a numerical computation after n = 15 iterates (full line), is compared with the standard Gaussian (κ1 = 0, dotted line) and the κ-Gaussian (4.25) (κ1 = 0.5, dashed line). It is evident the good fit between
460
the numerical curve and the κ-Gaussian with respect to the standard Gaussian. A further consistency of this result is supported by recalling that the κ-sum for large values reduces to a standard product [cfr. Eq. (2.1.8)]. This means that, in the far region of large x values, the random variable Sn corresponds to the product of n iid random variables that, according to the central limit theorem,
465
has a log-normal limit distribution, in agreement with the tail of Eq. (4.25). 5. Conclusions In this paper we have reformulated the standard Fourier transform in a formalism consistent with the κ-algebra and the κ-calculus. The new formulation has been derived starting from a κ-deformed Fourier series recently introduced
470
by us in [22]. 28
The κ-Fourier transform fbκ (ω) ≡ Fκ [f (x)](ω), belongs to the integral trans-
forms (1.2) and is characterized by a kernel hκ (x, ω) composed by a deformed phase and a damping factor that confer to hκ (x, ω) a wavelet-like shape. In
addition, both the real part and the imaginary part of the phase factor have an 475
asymptotical log-periodic behaviour. We have shown that the κ-deformed transform of a function f (x) is isomorph to the canonical transform since fbκ (ω) is equivalent to the standard Fourier
transform of the function f {κ} (x) ≡ f (x{κ} ), that is
Fκ [f (x)](ω) ≡ F [f {κ} (x)](ω{κ} ) .
(5.1)
However, in spite of this equivalence, the κ-Fourier transform turns out to be 480
more appropriate to handle functions defined in the realm of the κ-algebra. As a relevant example, we have applied our formalism to the study of the limit distribution of κ-summed statistically independent random variables, that is the distribution given in Eq. (4.25) of the random variables Sn = X ⊕X ⊕. . .⊕X by
using similar arguments employed in the derivation of the central limit theorem. 485
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