$q$-Generalized representation of the $d$-dimensional Dirac delta and $q$-Fourier transform

TL;DR

This paper introduces a $q$-generalized representation of the Dirac delta in multiple dimensions and explores its application to the $q$-Fourier transform, demonstrating invertibility and effects on the Gibbs phenomenon.

Contribution

It presents a novel $q$-generalized Dirac delta representation and analyzes the invertibility of the $q$-Fourier transform in $d$ dimensions.

Findings

$q$-delta representation is valid in $d$ dimensions

$q$-Fourier transform is invertible for all $d$

$q$-deformation influences the Gibbs phenomenon

Abstract

We discuss a generalized representation of the Dirac delta function in $d$ dimensions in terms of $q$-exponential functions. We apply this new representation to the study of the so-called $q$-Fourier transform, proving its invertibility for any value of $d$. We finally illustrate the effect of the $q$-deformation on the Gibbs phenomenon.

Equations

$$S_{\text{BG}}[\mathbf{p}]:=-\sum _{i=1}^W{p}_i\ln p_i$$

$$S_q^{\text{R}}[\mathbf{p}]=\frac{\ln \left({\sum }_{i=1}^W{p}_i^q\right)}{1-q},q\in \text{mathds}R,$$

$$S_q[\mathbf{p}]=\frac{1-{\sum }_{i=1}^W{p}_i^q}{q-1},q\in \text{mathds}R.$$

$$\frac{{\sum }_{i=1}^W{p}_i^q{ϵ}_i}{{\sum }_{i=1}^W{p}_i^q}=⟨ϵ{⟩}_q,$$

$$p_i({ϵ}_i)=\frac{1}{Z}{\exp }_q\left[-\beta \left({ϵ}_i-⟨ϵ{⟩}_q\right)\right].$$

$${\exp }_q(x):={\left[1+(1-q)x\right]}_{+}^{\frac{1}{1-q}},[x{]}_{+}:=x\theta (x)$$

$$e_q^z:\text{mathds}C\to \text{mathds}C,e_q^z:={\left[1+(1-q)z\right]}^{\frac{1}{1-q}},q\in \text{mathds}R.$$

$$z_q=\frac{1}{q-1}.$$

$$\underset{q\to 1^{\pm }}{\lim }{e}_q^z=e^z.$$

$$e_q^{z+w}\ne e_q^z{e}_q^w\text{ and }{e}_q^z\ne e_q^{z+2\pi i}.$$

$$\begin{gathered} e_q^z={\int }_0^{+\infty }\gamma \left(z_q;t\right)e^{tz}dt,\Re (z)<z_q, \\ \gamma (\eta ;t):=\frac{{\eta }^{\eta }{t}^{\eta -1}}{\Gamma (\eta )}{e}^{-\eta t},\eta >0. \end{gathered}$$

$$\underset{q\to 1^{+}}{\lim }\gamma \left(z_q;t\right)=\delta (t-1).$$

$$e_q^{ix}={\int }_0^{+\infty }\gamma \left(z_q;t\right)e^{itx}dt.$$

$${\omega }_n(q):=\prod _{k=0}^n\left[k(q-1)+1\right]\stackrel{q\to 1}{\to }1,$$

$$e_q^{\pm ix}=1+\sum _{n=1}^{\infty }{\omega }_{n-1}(q)\frac{(\pm ix{)}^n}{n!}\equiv {\cos }_{q}x\pm i{\sin }_{q}x,$$

$${\cos }_{q}x$$

$${\sin }_{q}x$$

$${\delta }_q^{(d)}(\mathbf{x}):=\frac{1}{c(q,d)}{\int }_{\text{mathds}{R}^d}{e}_q^{i\mathbf{k}\cdot \mathbf{x}}{d}^{d}k,q\in \left(1,1+\frac{1}{d}\right).$$

$$\begin{gathered} {\int }_{\text{mathds}{R}^d}\phi (\mathbf{x}){\delta }_q^{(d)}\left(\mathbf{x}\right)d^{d}x= \\ =\frac{1}{c(q,d)}{\int }_{\text{mathds}{R}^d}\left[{\int }_{\text{mathds}{R}^d}\phi (\mathbf{x})e_q^{i\mathbf{k}\cdot \mathbf{x}}{d}^{d}x\right]d^{d}k. \end{gathered}$$

$$\begin{gathered} {\int }_{\text{mathds}{R}^d}\phi (\mathbf{x})e_q^{i\mathbf{k}\cdot \mathbf{x}}{d}^{d}x={\int }_0^{+\infty }\left[{\int }_{\text{mathds}{R}^d}\phi (\mathbf{x})e^{it\mathbf{k}\cdot \mathbf{x}}{d}^{d}x\right]\gamma (z_q;t)dt \\ ={\int }_0^{+\infty }\hat{\phi }(t\mathbf{k})\gamma (z_q;t)dt, \end{gathered}$$

$$\hat{\phi }(\mathbf{k}):={\int }_{\text{mathds}{R}^d}\phi (\mathbf{x})e^{i\mathbf{k}\cdot \mathbf{x}}{d}^{d}x$$

$${\int }_{\text{mathds}{R}^d}\phi (\mathbf{x}){\delta }_q^{(d)}\left(\mathbf{x}\right)d^{d}x={\left(\frac{2\pi }{q-1}\right)}^d\frac{\Gamma \left(z_q-d\right)}{c(q,d)\Gamma \left(z_q\right)}\phi (0),$$

$$1<q<1+\frac{1}{d}.$$

$$c(q,d):={\left(\frac{2\pi }{q-1}\right)}^d\frac{\Gamma \left(z_q-d\right)}{\Gamma \left(z_q\right)},z_q:=\frac{1}{q-1},$$

$${\int }_{\text{mathds}{R}^d}\phi (\mathbf{x}){\delta }_q^{(d)}\left(\mathbf{x}\right)d^{d}x=\phi (0),1<q<1+\frac{1}{d},$$

$$\underset{q\to 1^{+}}{\lim }c\left(q,d\right)=(2\pi {)}^d\Rightarrow \underset{q\to 1^{+}}{\lim }{\delta }_q^{(d)}\left(\mathbf{x}\right)={\delta }^{(d)}\left(\mathbf{x}\right).$$

$$\begin{gathered} {\hat{f}}_q(k):={\int }_{-\infty }^{+\infty }f(x){\odot }_q{e}_q^{ikx}dx \\ \equiv {\int }_{-\infty }^{+\infty }f(x)e_q^{ikx{\left[f(x)\right]}^{q-1}}dx,1<q<2. \end{gathered}$$

$$z{\odot }_{q}w:={\left(z^{1-q}+w^{1-q}-1\right)}^{\frac{1}{1-q}},q>1.$$

$$z{\odot }_{q}w\stackrel{q\to 1^{+}}{\to }zw.$$

$$x{\otimes }_{q}y:={\left[x^{1-q}+y^{1-q}-1\right]}_{+}^{\frac{1}{1-q}},x,y>0.$$

$$\begin{gathered} {\hat{f}}_q(k;kx):={\int }_{-\infty }^{+\infty }f(y){\odot }_q{e}_q^{ik(y-x)}dy \\ \equiv {\int }_{-\infty }^{-\infty }f(y)e_q^{ik(y-x){\left[f(y)\right]}^{q-1}}dy\stackrel{q\to 1^{+}}{\to }{e}^{-ikx}\hat{f}(k). \end{gathered}$$

$$f(x)={\left[\frac{2-q}{2\pi }{\int }_{-\infty }^{+\infty }{\hat{f}}_q(k;kx)dk\right]}^{\frac{1}{2-q}},q\in (1,2).$$

Full text

Introduction

Since the foundational works of Clausius, Boltzmann and Gibbs, entropy has had a central role in the theory of thermodynamics and statistical mechanics Callen1985 ; Greven2014 ; Lieb1999 . In the classical Boltzmann–Gibbs (BG) theory of statistical mechanics, the entropy associated to a certain macrostate is expressed in terms of a probability distribution $\mathbf{p}=\{p_{i}\}_{i=1,\dots,W}$ given on the set of $W\in\mathds{N}$ microstates that are compatible with the considered macrostate, through the relation

[TABLE]

(we have fixed the Boltzmann constant equal to one). The expression above, that is valid for classical mixing/ergodic systems, provides a connection between the macroscopic and the microscopic description of the system. Shannon Shannon1948 , and later Khinchin Khinchin1957 , showed that the concept of entropy, and its expression in Eq. (1), play also a fundamental role in information theory, and it is indeed at the heart of the theory of communication. Due to the relevance of the concept of entropy both in physics and in information theory, during the last decades many generalizations of the BG entropy have been proposed Beck1995 ; Tempesta2015 ; Enciso2017 ; Tsallis2009B . In particular, the Rényi entropy Renyi1961

[TABLE]

Both entropies generalize the BG one, which is recovered for $q\to 1$. The nonadditive entropy $S_{q}$ has been profusely investigated and applied to the study of a wide spectrum of physical properties of complex systems Beck1995 ; Tsallis2009B . The parameter $q$ measures the deviation from the classical case. In the spirit of the maximum entropy principle introduced by Jaynes Jaynes1957 , it can be shown that the generalized entropies in Eqs. (2) are extremized by the same family of distributions under the same kind of constraints Lenzi2000 ; Tsallis1998 . Indeed, denoting by $\{\epsilon_{i}\}_{i=1,\dots,W}$ the energies corresponding to the $W$ microstates appearing in the sums in Eqs. (2), imposing a fixed average energy constraint111Observe that, under the assumption that the value of $\beta$ is kept fixed, the constraint must be imposed using escort distributions (see below) to guarantee the invariance of the maximizing distribution under energy shifts. See also Refs. Lenzi2000 ; Tsallis1998 for additional details. as

[TABLE]

the maximizing distribution of the entropies in Eqs. (2) is

[TABLE]

where $Z$ is a proper normalization factor, $\beta>0$ is a Lagrange multiplier and, for $q\in\mathds{R}$,

[TABLE]

is the so called $q$-exponential function. The $q$-exponential function generalizes the usual exponential, which is recovered in the $q\to 1$ limit.

Probability distributions, as well as other physical quantities, having a $q$-exponential shape are found in the analysis of data obtained in high-energy experiments HEP2010 , finance Borland2002 , dusty plasmas Liu2008 , and theoretical investigations on optical lattices Renzoni2006 ; Lutz2013 , low-dimensional dissipative maps Ananos2004 ; Robledo2004 ; Lyra1998 , diffusion processes in superconductors Andrade2010 , among many others. The ubiquity of distributions with power-law tails in the form of Eq. (4) has suggested the existence of a $q$-generalized Central Limit Theorem (CLT) for some classes of correlated random variables, in analogy with the connection between Maxwell distribution in BG statistical mechanics and the usual CLT for uncorrelated (or weakly correlated) random variables. This possibility has been investigated, for example, by the authors of Ref. uts2008 and led, as by-product, to a generalization of many standard mathematical concepts Jauregui2010 , on the basis of a new deformed algebra previously introduced by Borges Borges2004 . In particular, a possible generalization of the usual Fourier transform, called $q$-Fourier transform ($q$-FT) was proposed in Ref. uts2008 . The new integral transform was defined in formal analogy with the usual Fourier transform and expressed in terms of the analytic prolongation of the deformed exponential function given in Eq. (5). Hilhorst Hilhorst2010 observed however that the $q$-FT as defined in Ref. uts2008 cannot be inverted. For this reason, a modified definition of $q$-FT, that is invertible, has been proposed in Ref. Jauregui2011 .

Inspired by the results discussed above, in the present paper we analyze a generalization of the $q$-FT, in the form adopted in Ref. Jauregui2011 , to the $d$-dimensional case. The invertibility of this expression is proven using a new representation of the Dirac delta function in $d$ dimensions, based again on $q$-exponentials. We will finally give some numerical examples, discussing a series representation of the inverse $q$-FT and comparing it with the classical Fourier series.

Preliminaries: the $q$-exponential

In the present paper we will use the $q$-exponential function defined as

[TABLE]

The previous definition is not, strictly speaking, the usual one adopted in the literature, given in Eq. (5), but instead its analytic continuation to the complex plane. Indeed, the $q$-exponential in Eq. (5) is a real function of a real variable, and a cutoff appears in Eq. (5) that is absent in Eq. (6). If $z\in\mathds{R}$, the two definitions coincide for $z\lessgtr z_{q}$ if $q\gtrless 1$, where

[TABLE]

In the following, we will consider $q>1$ only. For $q>1$ the analytic properties of the $q$-exponential are analogous to the ones of the function $f(z)=z^{-a}$, $a>0$, on the complex plane. In particular, the $q$-exponential has a pole for $z=z_{q}$. We choose as branch cut the half line $[z_{q},+\infty)$ along the positive real axis, see Fig. 1. If $z_{q}\in\mathds{Q}^{+}$, we can construct a Riemann surface for the $q$-exponential with a finite number of branches. If otherwise $z_{q}\in\mathds{R}^{+}\setminus\mathds{Q}$, the number of branches is infinite. Analyticity is recovered for $z_{q}\to+\infty$, i.e., for $q\to 1$, when

[TABLE]

The $q$-exponential function is therefore a deformation of the usual exponential. Many properties of the usual exponential are, however, lost for $q\neq 1$. For example, for $q\neq 1$ and $z,w\in\mathds{C}$,

[TABLE]

The $q$-exponential function in Eq. (6) can be also written, for $q>1$ and $\Re(z)<z_{q}$, as a linear superposition of exponentials, in the form Jeffrey2007

[TABLE]

Observe that, as expected,

[TABLE]

Eq. (10) has been extensively discussed by Beck Beck2003 in the context of superstatistics. Eq. (10) implies that, given $x\in\mathds{R}$ and $q>1$,

[TABLE]

The function $\operatorname{e}_{q}^{ikx}$, with $x\in\mathds{R}$ and $k\in\mathds{R}$, is sometimes called $q$-plane wave of momentum $k$ Jauregui2010 . Its real part and imaginary part are a deformation of the cosine function and the sine function respectively. In particular, denoting by

[TABLE]

A representation of the Dirac delta function

In Ref. Jauregui2010 a representation of the Dirac delta function on the real line in terms of $q$-exponentials has been proposed. The proof of this new representation has been obtained by different authors Chevreuil2010 ; Mamode2010 ; Plastino2011 using different approaches. In the following, we will generalize these results, showing that an integral representation for the Dirac delta function on $\mathds{R}^{d}$ can be obtained using again $q$-exponentials. Let us introduce the following distribution

[TABLE]

In the previous expression, $c(q,d)$ is a numerical prefactor depending on $q$ and $d$ that we will suitably fix later on. We will show now that the function $\delta_{q}^{(d)}(\mathbf{x})$ behaves like a Dirac delta function in $d$ dimensions. Indeed, given a test function $\varphi(\mathbf{x})$ that is infinitely differentiable and rapidly decreasing at infinity, we have that

[TABLE]

Using Eq. (12) we can write

[TABLE]

where

[TABLE]

is the Fourier transform of $\varphi$. Using the fact that $\int_{\mathds{R}^{d}}\hat{\varphi}(\mathbf{k})\operatorname{d}^{d}k=\varphi(\mathbf{0})$, we finally have

[TABLE]

under the condition

[TABLE]

If we impose now

[TABLE]

we obtain the result

[TABLE]

i.e., the function $\delta_{q}^{(d)}$ acts as a Dirac distribution function on $\mathds{R}^{d}$. Observe that, as expected,

[TABLE]

The results above recover the $d=1$ case analyzed in Refs. Chevreuil2010 ; Jauregui2010 ; Mamode2010 ; Plastino2011 as particular case.

The $q$-Fourier transform in $d$ dimensions, and its inversion

The interest of the authors of Ref. Jauregui2010 in the function $\delta_{q}^{(1)}(x)$ was motivated by the problem of the inversion of the so called $q$-Fourier transform ($q$-FT). The $q$-FT of a real, nonnegative integrable function $f(x)$ is given by uts2008

[TABLE]

In the previous expression, we have introduced the binary operation $\odot_{q}$ between two complex numbers $z,w\in\mathds{C}$, defined as

[TABLE]

Observe that the operation above is a generalization of the usual product, being

[TABLE]

The operation $\odot_{q}$ is abelian, $z\odot_{q}w=w\odot_{q}z$, and such that $z\odot_{q}1=z$ and $\lim_{w\to 0}z\odot_{\eta}w=0$ $\forall z\neq 0$. Moreover, the operation introduced in Eq. (24) generalizes the so-called $q$-product between two real positive quantities, as defined in Ref. Borges2004 , namely

[TABLE]

Eq. (26) coincides with Eq. (24) for $z,w\in\mathds{R}^{+}$ and $z^{1-q}+w^{1-q}\geq 1$. The $q$-FT defined in Eq. (23) recovers therefore the usual Fourier transform for $q\to 1^{+}$. However, the expression in Eq. (23) requires some further discussion. Indeed, as first noted by Hilhorst Hilhorst2010 , the introduced integral transform cannot be, in general, inverted. This observation affects possible applications of Eq. (23), and, for this reason, an extended definition of $q$-FT has been introduced in Ref. Jauregui2011 , namely

[TABLE]

The introduction of the shift variable $x$ allows us to invert the integral transform for $1<q<2$. Observe that the variable $x$ appears in the product $kx$ only: we have stressed this fact in our notation. It can be proved that Jauregui2011

[TABLE]

The proof of Eq. (28) strongly relies on the properties of the function $\delta_{q}^{(1)}(x)$ discussed above. Moreover, the standard formula for the inversion of the Fourier transform is obtained for $q\to 1^{+}$.

We present here a derivation of the result in Eq. (28) in a more general setting, namely considering a generalization of Eq. (27) to $d$ dimensions. Given a nonnegative function $f\colon\mathds{R}^{d}\to\mathds{R}^{+}$, we define its $q$-FT in $d$ dimensions as

[TABLE]

The previous expression straightforwardly generalizes Eq. (27), apart from the nontrivial constraint $q\in\left(1,1+\frac{1}{d}\right)$, which guarantees invertibility. The inversion of the integral transform in Eq. (29) can be obtained using the function $\delta_{q}^{(d)}$ discussed in the previous section. Indeed, assuming that $q\in\left(1,1+\frac{1}{d}\right)$,

[TABLE]

In the last step we have supposed that $\mathbf{x}$ belongs to the interior of the support of $f$ (otherwise an additional factor will appear, due to the fact that the Dirac delta function is evaluated on a boundary point Jauregui2011 ). It follows that

[TABLE]

Eq. (31) generalizes the result in Eq. (28), which is indeed recovered for $d=1$. Moreover, the standard relation between $f$ and its Fourier transform is obtained for $q\to 1^{+}$.

On series expansion and $q$-FT

Let us now consider a positive real function $f\colon[-\nicefrac{{T}}{{2}},\nicefrac{{T}}{{2}}]\to\mathds{R}^{+}$ that is square integrable on its domain. It is well known that $f$ can be represented in terms of a Fourier series, that in the notation introduced in Eq. (29) reads

[TABLE]

In the $T\to\infty$ limit we formally recover the expression of the Fourier transform and its inverse.

It is tempting to generalize the previous standard result deforming the circular functions according to Eqs. (13). However, it is easily verified that the orthogonality condition among the $q$-deformed functions in Eq. (13) is not satisfied, and therefore the quantities in Eqs. (13) cannot be considered a basis in a Hilbert space. Despite this important fact, for $q\in(1,2)$, we have that, for a given positive function $f$ on the real line,

[TABLE]

If we consider therefore a function $f$ with compact support in $[-\nicefrac{{T}}{{2}},\nicefrac{{T}}{{2}}]$, it is expected that, although $f^{(q)}\neq f$ for $q\neq 1$, the sum

[TABLE]

provides a good approximation of $f$ for $N\gg 1$ and, if $q\neq 1$, for $T\gg 1$. In particular, we expect that for $q\approx 1$, the condition $T\gg 1$ can be relaxed and the expression $S_{N}^{(q)}[f]$ still provides a good approximation of $f$ for $N\gg 1$, despite the fact that no specific periodicity can be associated to a $q$-plane wave for $q\neq 1$.

To numerically illustrate this fact, let us consider, for example, $q=1+\nicefrac{{1}}{{10}}$, and two different distribution densities. Let us first start with a smooth distribution, namely a Gaussian distribution

[TABLE]

and let us apply Eq. (34) to it on the interval $[-2,2]$. In Fig. 3(a) we show that both $S_{50}^{(1)}[p]$ and $S_{50}^{(1.1)}[p]$ approximate very well the function $p$ on the considered domain.

Let us now consider the uniform distribution

[TABLE]

on the interval $[-\nicefrac{{T}}{{2}},\nicefrac{{T}}{{2}}]$. In Fig. 3(b) we show the approximation for $p(x)$ in the case $T=4$ obtained using $S_{N}^{(1)}[p]$ and $S_{N}^{(1.1)}[p]$ for $N=50$. Remarkably, the approximation provided by the expansion obtained for $q=1.1$ does not fluctuate in the interval $[-\nicefrac{{T}}{{4}},\nicefrac{{T}}{{4}}]$, where $p$ is different from zero and the Gibbs phenomenon appears to be suppressed, even close to the discontinuity points, despite the fact that both series have been truncated at the same value of $N$. The possible application of the series in Eq. (34), and the effects of the $q$-deformation on the Gibbs phenomenon, still deserve further investigation.

Conclusions

In the present paper, we have discussed a new representation of the Dirac delta function in $d$ dimensions, given in terms of $q$-exponential functions. Using this representation, we have proved the invertibility of the $q$-FT in $d$ dimensions. We have finally numerically illustrated the effects of the $q$-deformation on the Gibbs phenomenon in a Fourier-like series expansion.

The new tools introduced in the present paper are expected to be useful in further investigations on the mathematical foundations of generalized thermostatistics. In particular, the $q$-FT in $d$ dimensions and the $q$-generalized representation of the Dirac delta function can be useful in the study of generalized versions of the Central Limit Theorem. A more detailed and rigorous investigation of the applicability of the previous tools is also of great mathematical interest.

Acknowledgments

The authors thank Max Jáuregui for useful discussions. They also acknowledge partial financial support by CNPq and Faperj (Brazilian agencies) and by the John Templeton Foundation (USA).