Square of the Error Octonionic Theorem and hypercomplex Fourier Serier

Square of the Error Octonionic Theorem and hypercomplex Fourier Serier

C.A.P. Martinez^I; A.L.M. Martinez^I; M.F. Borges^II; E.V. Castelani^III,*

^ICoordenação de Matemática, COMAT, UTFPR - Universidade Tecnológica Federal do Paraná, Campus Cornélio Procópio, Av. Alberto Carazzai, 1640, 86300-000 Cornélio Procópio, PR, Brasil. E-mails: [e-mail]; [e-mail]

^IIDepartment of Computing, UNESP - State University of São Paulo at São José do Rio Preto, 15054-000 São José do Rio Preto, SP, Brazil. E-mail: [e-mail]

^IIIDepartamento de Matemática, DMA, UEM - Universidade Estadual de Maringá, Av. Colombo, 5790, 87020-900 Maringá, PR, Brasil. E-mail: [e-mail]

ABSTRACT

The focus of this paper is to address some classical results for a class of hypercomplex numbers. More specifically we present an extension of the Square of the Error Theorem and a Bessel inequality for octonions.

Keywords: hypercomplex functions, Fourier series, octonions.

RESUMO

O foco deste trabalho é abordar alguns resultados clássicos para uma certa classe de números hipercomplexos. Mais específicamente, apresentamos uma extensão do Teorema do Erro Quadrático e da Desigualdade de Bessel para octônios.

Palavras-chave: Funções hipercomplexas, Séries de Fourier, octônios.

1 INTRODUCTION AND MOTIVATION

Octonions are hypercomplex numbers and in many ways may be regarded as non-associative quaternions. Nowadays they have been used in appropriated approaches of higher dimensional physics [5, 8], such as M-theory, Strings and alternative gravity theories. In this paper, with the main purpose in the nearest future of making a larger application of hypercomplex in unifiedfield theories, and motivated both by earlier works by Eilenberg, Niven [7], Deavours [6], Sinegre [16], and on recent results obtained by some of the authors [2, 3, 4, 12, 13, 14], Boek, Gurlebeck [1], and Lam [11], we have concentrated our efforts in constructing an extension of the Algebra of hypercomplex. Hypercomplex numbers are mainly concerned to quaternions and octonions. Their algebras may regarded as extensions of the ordinary bi-dimensional complex algebra [10, 15], either being non-commutative and associative for quaternions or non-commutative and non-associative in the octonionic case.

Quaternions were discovered in 1843 by Willian R. Hamilton (op. cit. [16]), and in that same year John T. Graves, a Hamilton's friend, found an 8-dimensional Algebra whose property on non-associativity in a multiplication table holds. Further on, two years later, in 1845, after some contributions on this subject by Arthur Cayley octonions have also been named "Cayley numbers".

In the context of the present work, and based in a previous result (op. cit. [13]), we extend for octonions the concept of a 2L-periodical function, that we will call Octonionic Fourier Series. We define an octonionic exponential function, and show both a non-associative expansion of the Moivre Theorem and generalizations of the Euler formula. Finally, we obtain octonionic versions of the Square of the Error Theorem and the Bessel inequality.

1.1 Hypercomplex Fourier Series

The octonions are a somewhat nonassociativite extension of the quaternions. They form the 8-dimensional normed division algebra on .

The octonionic algebra, also called octaves denoted for , is an alternative division algebra.

The octonions set is denoted by

= {a, b, c, d, e, f, g, h} ∈,

where,

The octonions do not form a ring due the non-commutativity of the multiplication. Also do not form a group due a nonassociativide of multiplication. They form a Moufang Loop, a Loop with identity element (Jacobson, N., [9]).

Let us consider an octonionic number given by

o = a + bi + cj + dk + el + fli + glj + hlk

The octonion unities 1, i, j, k, l, li, lj, lk form an orthonormal base of the 8-dimensional algebra.

Similarly to [13] we start our contribution in considering f a function defined on interval [-L, L], L > 0, and outside of this interval set as f(x) = f(x + 2L), that is, f(x) is 2L-periodical. If f and f ' are piecewise continuous then the series of function given below,

is convergent and the limit is

The coefficients of Fourier of f, a₀, a_n and b_n given by:

and

The trigonometric series presented in (1.1) with this choice of coefficients is the Fourierseries of f.

Now, we will detail some properties considering octonions o ∈ given by

o = u₁ + u₂i + u₃j + u₄k + u₅l + u₆li + u₇lj + u₈lk = u₁ + .

The extended equation of De Moivre for octonions is given by:

Then, using (1.4) we can obtain:

Since cos(-y) = cos(y) and sin(-y) = -sin(y) we have that

Following the same arguments we can get

and

From (1.5) and (1.6), we obtain,

Consequently,

Noting that

and using (1.2) and (1.3) we have:

Again, using the same argument we can get:

Extending the ideas used, we conclude that:

and

since (j)² = -1.

and

since (k)² = -1.

and

since (l)² = -1.

and

since (li)² = -1.

and

since (lj)² = -1.

and

since (lk)² = -1.

From (1.7), (1.8), (1.9), (1.10), (1.11), (1.12), (1.13), (1.14),(1.15), (1.16), (1.17), (1.18), (1.19) and (1.20) we can get

Since f : R → R is periodic, with period 2L, f and f ' are piecewise continuous, we have that the Fourier series of f presented in (1.1) can be written as follows

where considering c₀ = .

The series (1.21) is the octonionic Fourier series of f.

Example 1. Let

We will determine the octonionic Fourier series of f. In fact, calculating the coefficients c₀, we have

thus

Example 2. Let f(t) = t, with t ∈ (-1, 1) and f(t + 2) = f(t).

The octonionic coefficients are given by:

Then, the octonionic Fourier series of f is

2 AN EXTENSION OF THE SQUARE OF THE ERROR THEOREM FOR OCTONIONIC FOURIER SERIES AND THE BESSEL INEQUALITY

In this Section we will present an important property of the Hypercomplex Fourier Series derived from some results from the classical Complex Analysis [10]. We also make an extension of the so called Square of the Error Theorem (Walter Rudin [15]).

For the purposes of this section, we denote ϕ the octonionic function defined by

ϕ : R → ,

ϕ(x) = ϕ₁(x) + ϕ₂(x)i + ϕ₃(x)j + ϕ₄(x)k + ϕ₅(x)l + ϕ₆(x)li + ϕ₇(x)lj + ϕ₈(x)lk

where ϕ₁, ϕ₂, ϕ₃, ϕ₄, ϕ₅, ϕ₆, ϕ₇ and ϕ₈ are real functions.

Thus we can consider

Definition 2.1. Let {ϕ_n}, n = 1, 2, 3, 4, ... be a sequence of octonionic functions on [a, b], such that

Then, we say that {ϕ_n} is an orthogonal system of octonionic functions on [a, b]. In addition, if

for all n, we say that {ϕ_n} is orthonormal. Here are some examples.

Example 3. The sequence of functions

form orthonormal system on [-L, L].

Example 4. The sequence of functions

form orthonormal system on [-L, L].

Motivated by example 4, we define

where {ϕ_n} is an orthonormal sequence in [a, b] and and arereal sequences.

We call c_n the n-th octonionic Fourier coefficient of f (relative to {ϕ_n}). We write

and call this series the octonionic Fourier series of f.

The following theorem extends some classical results (op. cit. [15]), which have already been extended for the quaternionic case (op. cit. [13]). More specifically we show that partial sums of octonionic Fourier Series of a function f, have certain minimum property. Let us assume that f is a real function.

Theorem 2.1. Let {ϕ_n} be orthonormal sequence of octonionic functions on [a, b]. Consider the n-th partial sum of the octonionic Fourier series of f

where

and

are given in (2.3). In addition, define

where

and

are real sequences. Then

and equality holds if and only if

That means, among all functions t_n, s_n gives the best possible mean square approximation to f.

Proof. To simplify the notation, let ∫ be the integral over [a, b] and Σ the sum from 1 to n. From the definition of (2.6) and (2.3), we have

Consequently,

Furthermore,

since {ϕ_n} is orthonormal. Therefore,

which is minimized if and only if

Corollary 2.1. Let {ϕ_n} be orthonormal sequence of octonionic functions on [a, b], and if

then

Proof. In the proof of Theorem (2) we found

putting and in this calculation, we obtain

since ∫|f - t_n|²> 0. Letting n → ∞ in (2.10), we obtain (2.9), this inequality is a generalization for octonions of Bessel.

3 CONCLUDING REMARKS

In this paper we have discussed on an extended version of Fourier Series for octonions. Fourier Series are remarkable in the way through an orthonormal basis they approximate functions associated to physical problems such as conducting heat and vibrations.

Furthermore, as not few models of Theoretical Physics may be analysed through the geometry and algebra of hypercomplex, it will be our concern to concentrate the next steps in making all possible applications of our results in the context of unified physical theories for higher dimensional space-times.

Received on March 21, 2013

Accepted on October 13, 2013

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*

Corresponding author: Emerson Vitor Castelani

Publication Dates

• Publication in this collection
  07 Mar 2014
• Date of issue
  Dec 2013

History

• Received
  21 Mar 2013
• Accepted
  13 Oct 2013