The τq-Fourier transform: Covariance and uniqueness

Abstract

We propose an alternative definition for a Tsallis entropy composition-inspired Fourier transform, which we call "τ q -Fourier transform". We comment about the underlying "covariance" on the set of algebraic fields that motivates its introduction. We see that the definition of the τ q -Fourier transform is automatically invertible in the proper context. Based on recent results in Fourier analysis, it turns that the τ q -Fourier transform is essentially unique under the assumption of the exchange of the point-wise product of functions with their convolution.

Power-law entropies for continuous systems and generalized operations

“…The success of eq. ( 6) is corroborated by that it is possible, within its context, to define a generalized Fourier transform [31,32] which is free of the problem of invertibility plaguing the other proposed definitions [28,33]. The uniqueness of the generalized Fourier transform is guaranteed in the context of the generalized operations giving rise to eq.…”

“…A generalized multiplication ⊗ q which is distributive with respect to ⊕ q was proposed in [26], and in a different but conjecturally equivalent form in [32], to be…”

“…As a result, a deformation of the reals R q was proposed in [26,32], which is realized through the field isomorphism τ q : R → R q defined as…”

Analiticidade da função exponencial generalizada para argumentos complexos e suas implicações

“…[15]. Com este operador, apresentamos uma estrutura de anel generalizado, que garante a validade de muitas propriedades e teoremas próprios para esse tipo de estrutura algébrica [15][16][17][18]. Nessas discussões acerca de generalizações convenientes, destacamos três princípios básicos que devem ser preservados:…”

“…Uma alternativa a essa generalização do produto é apresentada por Lobão [15] e utilizada, mais tarde, por Kalogeropoulos [17,18]. Nestas referências, o produto é definido por extinção de espécies em dinâmica de populações [4].…”