# On the κ-Deformed Cyclic Functions and the Generalized Fourier Series in the Framework of the κ-Algebra

## Abstract

**:**

## 1. Introduction

_{n}(x) is a complete set of functions on the interval [a, b] fulfilling the orthogonality condition:

_{n}(x) are related to a Sturm-Liouville problem for a second order differential equation:

_{i}and b

_{i}are constants.

## 2. κ-Mathematics Formalism

#### 2.1. κ-Algebra

^{κ}= {x

^{{}

^{κ}

^{}}: −∞ < x < ∞} as follows:

^{{}

^{κ}

^{}}→ x (and x → x

_{{}

_{κ}

_{}}), we get:

^{κ}. In addition, there exist the null element for the sum: x ⊕ Ø = Ø ⊕ x = x; and the identity for the product: x ⊗ I = I ⊗ x = x; as well as the opposite: x ⊕ (−x) = (−x) ⊕ x = Ø; and the inverse: x ⊗ (1/x) = (1/x) ⊗ x = I, for any x ∊ ℜ

^{κ}. Therefore, the algebraic structure (ℜ

^{κ}, ⊕, ⊗) forms an Abelian field isomorph to the field of the ordinary real numbers (ℜ, +, ·).

_{κ}≡ {x

_{{}

_{κ}

_{}}: −∞ < x < ∞} isomorphic to (ℜ, +, ·), endowed by a generalized sum and product different from the ones given in Equations (11) and (12). We remand the interested reader to [34] for the details.

^{−1}sinh κ, (−x) ≡ −x and (1/x) ≡ κ

^{−1}sinh (κ

^{2}/arcsinh κx). In this way, the difference x ⊖ y = x ⊕ (−y) and the quotient x ⊘ y = x ⊗(1/y) arise from Equations (15) and (16) as:

^{κ}, ⊕, ⊗) reduces to (ℜ, +, ·).

_{0}x = exp x and ln

_{0}x = ln x. Useful relations concerning these functions are the symmetry under reflection of the deformation parameter: exp

_{κ}x = exp

_{−}

_{κ}x, ln

_{−}

_{κ}x = ln

_{−}

_{κ}x; the scaling relations: exp

_{κ}(ax) = (exp

_{κ}, x)

^{a}, ln

_{κ}x

^{a}= a ln

_{κ}

_{′}x, with κ′ = aκ; and the algebraic properties:

_{{}

_{κ}

_{}}= f (x) and x

^{{}

^{κ}

^{}}= f

^{−1}(x), with f(x) = κ

^{−1}arcsinh(κx), the κ-sum and the κ-product can be redefined as:

#### 2.2. κ-Calculus

_{κ}x according to:

_{κ}(a ⊕ x) = d

_{κ}x, d

_{κ}(x ⊕ y) = d

_{κ}x + d

_{κ}y, as well as d

_{κ}(a ⊗ x) = a

_{{}

_{κ}

_{}}· d

_{κ}x. In this sense, the κ-differential is κ-linear:

_{κ}(x) d

_{κ}x coincides with the exact differential d exp

_{κ}x. Many relations of the standard calculus still hold if opportunely reformulated in the κ-formalism. For example, we have:

## 3. Euler Formula and κ-Cyclic Trigonometric Functions

#### 3.1. First Case

_{k}x)

^{i}describes the unitary circle in the complex plane like the function exp(ix) does. Therefore, Function (54) has a unitary modulus for any x ∊ ℜ. However, noting that |x| > |x

_{{}

_{κ}

_{}}|, as well as the difference |x − x

_{{}

_{κ}

_{}}| increases as |x| → ∞, it follows that the circle is revolving around slowly as |x| grows. This implies that the function (exp

_{κ}x)

^{i}maps the unitary circle with a period that increases as |x| increases.

_{{}

_{κ}

_{}}. As a consequence, the κ-cyclic functions are periodic like the standard functions sin x and cos x, although their period is not constant, but increases for |x| → ∞, in agreement with our previous considerations.

_{κ}x for several values of κ. The same picture is reproduced in Figure 2 in a linear-log scale, where it is clear as the periods become constant for large |x|, whilst it grows as the parameter κ increases, in agreement with Equation (61).

_{κ}x and cos

_{κ}x can be derived starting from the following κ-differential equation:

_{{}

_{κ}

_{}}a constant, which can be rewritten in the form of a Sturm-Liouville equation:

_{n}≡ (nπ/h

_{{}

_{κ}

_{}})

^{{κ}}.

#### 3.2. Second Case

_{[}

_{κ}

_{]}and their dual x

^{[}

^{κ}

^{]}are given by:

^{[}

^{κ}

^{]})

_{[}

_{κ}

_{]}= (x

_{[}

_{κ}

_{]})

^{[}

^{κ}

^{]}= x. They are related to the numbers x

_{{}

_{κ}

_{}}, that is x

_{[}

_{κ}

_{]}= x

_{{}

_{κ}

_{′}}, according to the parameter transformation:

_{κ}x, Equation (70) is related to Equation (54) as:

_{κ}(i x)| increases monotonically:

_{κ}take real values for |x| ≤ 1/|κ|, and consequently, like exp

_{κ}(ix), the functions in ${\mathrm{C}}^{\left(2\right)}$ have a real image and modulus unitary only in the interval |x| ≤ 1/|κ|. In additions, the wavelength of Sin

_{κ}x and Cos

_{κ}x reduces for |x| → 1/|κ| as well as it increases as |κ| grows. For |κ| > 1/4, the periods of these functions become grater than their real domain, and the functions cease to be periodic.

_{κ}x iven in Equation (77), in its real domain, for several values of the deformation parameter κ.

_{{}

_{κ}

_{}}= ix

_{[}

_{κ}

_{]}introduces a different definition for the κ-sum and the κ-product, which follows from Equation (33) with f (x) = κ

^{−1}arcsin(κx). They are given by:

_{[}

_{κ}

_{]}x = dx

_{[}

_{κ}

_{]}.

_{κ}x and Cos

_{κ}x from a Sturm-Liouville problem. In fact, they are related to the following differential equation:

_{n}= (nπ/h

_{[}

_{κ}

_{]})

^{[}

^{κ}

^{]}.

## 4. Orthogonality and Completeness Relations

^{2}[−h, h] the scalar product:

^{(1)}be the set of square-integrable functions L

^{2}[−h, h] given by:

_{n}are given constant, with n = 1, 2, … and m = 0, 1,

^{2}[−h, h] when:

_{{}

_{κ}

_{}}and ${a}_{n}^{\prime}={\left({a}_{n}\right)}_{\left\{\kappa \right\}}$, we obtain:

_{n}(x) are eigenfunctions of the given problem. In order to prove the completeness relation for the system ${\mathrm{C}}^{\left(1\right)}$ we start by the well-known relation:

_{{}

_{κ}

_{}}, the sine function transforms as:

_{n}given in Equation (90).

^{2}(−h, h) ⊆ l

^{2}(−1/|κ|, 1/|k|).

## 5. Generalized Fourier Series

_{n}∊ Φ

^{(1)}according to:

_{n}are unique constants given by:

_{n}∊ Φ

^{(1)}, according to:

_{0}and c

_{n}are unique constants given by:

^{1)}, according to:

_{−}

_{n}= (c

_{n}− i s

_{n})/2 and Y

_{n}= (c

_{n}+ i s

_{n})/2.

_{n}and s

_{n}:

^{2}(−h, h) and than expandable in the complex Fourier series with coefficients γ

_{n}and δ

_{n}, we have:

^{{}

^{κ}

^{}}, it is straightforward to change Equation (107) in:

^{{}

^{κ}

^{}}).

^{(2)}as:

^{[}

^{κ}

^{]}).

## 6. Final Remarks

_{{}

_{κ}

_{}}and x

_{[}

_{κ}

_{]}, which are endowed by a generalized sum and product, defined as (in the text, for the sake of notations, the generalized operations associated with the κ-numbers x

_{[}

_{κ}

_{]}have been denoted by $\overline{\oplus}$ and $\overline{\otimes}$, respectively):

^{(}

^{m}

^{)}related to two different Sturm-Liouville problems. Orthogonality and completeness relations for the systems Φ

^{(}

^{m}

^{)}are then obtained in the framework of the Sturm-Liouville theory and have been used to introduce two different κ-deformed Fourier series in the space of the square-integrable functions in L

^{2}(−h, h) and L

^{2}(−1/|κ|, 1/|κ|) respectively. In both cases, the corresponding κ-Fourier series can be recast in an ordinary Fourier series of a suitably κ-deformed function.

_{n}and c

_{n}of Function (124), with m = 50, evaluated in the period [e, e

^{2}] (a) for κ = 0.0, corresponding to the standard case and (b) for κ = 0.1. Clearly, in both cases, a large dispersion of the coefficient values around the m = 50 harmonic is observed, although in (b), this dispersion is sensibly narrower than in (a). Thus, in this situation, it seems that there is not a significative advantage in the use of the κ-Fourier analysis with respect to the standard one. This occurs since the onset of the log-periodic behavior of the κ-cyclic functions arises in the asymptotic region and then is not yet established in the windows [e, e

^{2}]. However, taking advantage of the periodicity of f (x), we can analyze its spectrum equivalently in the highest region, namely [e

^{19}, e

^{20}], as shown in (c). Here, the log-periodic behavior of the κ-sine and κ-cosine functions is almost exact, and as expected, the spectrum corresponds to that of an exact monochromatic oscillation with only the s

_{50}and c

_{50}coefficients different from zero. Actually, one should expect the sole coefficient c

_{50}to be different from zero, since, after all, we are decomposing a monochromatic cos - log function. The presence of both of the coefficients, is caused by the initial region where the functions φ

_{n}and ϕ

_{n}do not have a log-periodic behavior. This causes a phase shift between the analyzed function and the harmonic waves used in the expansions. This phase shift may be tuned by acting on the deformation parameter, as shown in (d), where the only harmonic, corresponding to c

_{50}= 1, exists.

## Conflicts of Interest

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**Figure 1.**Linear-linear plot of sin

_{κ}x given in Equation (58) for several values of the deformation parameter κ.

**Figure 2.**Linear-log plot of sin

_{κ}x given in Equation (58) for several values of the deformation parameter κ.

**Figure 3.**Linear-linear plot of Sin

_{κ}x given in Equation (77), in its real domain |x| ≤ 1/|κ| for several values of the deformation parameter κ.

**Figure 4.**First n = 100 Fourier coefficients of the log-periodic function cos(2 πm ln(x)), with m = 50, evaluated for:

**(a)**the standard Fourier series (k = 0.0) in the period [e, e

^{2}];

**(b)**the κ-Fourier series in the period [e, e

^{2}] with κ = 0.1;

**(c)**the κ-Fourier series in the period [e

^{19}, e

^{20}] with κ = 0.1; and

**(d)**the κ-Fourier series in the period [e

^{19}, e

^{20}] with κ = 0.5.

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**MDPI and ACS Style**

Scarfone, A.M.
On the *κ*-Deformed Cyclic Functions and the Generalized Fourier Series in the Framework of the *κ*-Algebra. *Entropy* **2015**, *17*, 2812-2833.
https://doi.org/10.3390/e17052812

**AMA Style**

Scarfone AM.
On the *κ*-Deformed Cyclic Functions and the Generalized Fourier Series in the Framework of the *κ*-Algebra. *Entropy*. 2015; 17(5):2812-2833.
https://doi.org/10.3390/e17052812

**Chicago/Turabian Style**

Scarfone, Antonio Maria.
2015. "On the *κ*-Deformed Cyclic Functions and the Generalized Fourier Series in the Framework of the *κ*-Algebra" *Entropy* 17, no. 5: 2812-2833.
https://doi.org/10.3390/e17052812