# On the Octonion Cross Wigner Distribution of 3-D Signals
## 2 Basic Definitions
### Octonion Fourier Transform  
Definition of the octonion Fourier transform (OFT) of the real-valued function of three variables was introduced in [53] and used in later publications, as indicated in Section 1. In [49], it was proved that the OFT of real-valued function is well-defined and has some interesting properties. These results were further generalized to the case of octonion-valued functions (see [46,50]). In the next part of this section, we recall the most important results on this topic. Consider the octonion-valued function of three variables \(u\colon\mathbb{R}^{3}\to\mathbb{O}\), i.e.,  
\[u(\mathbf{x})=u_{0}(\mathbf{x})+u_{1}(\mathbf{x})\mathbf{e}_{1}+\ldots+u_{7}( \mathbf{x})\mathbf{e}_{7},\]  
where \(u_{i}\colon\mathbb{R}^{3}\to\mathbb{R}\), \(i=0,\ldots,7\), and \(\mathbf{x}=(x_{1},x_{2},x_{3})\). The octonion Fourier transform of the integrable (in Lebesgue sense) function \(u\) is given by the formula  
\[U_{\text{OFT}}(\mathbf{f})=\int_{\mathbb{R}^{3}}u(\mathbf{x})e^{-\mathbf{e}_{ 1}2\pi f_{1}x_{1}}e^{-\mathbf{e}_{2}2\pi f_{2}x_{2}}e^{-\mathbf{e}_{4}2\pi f_{ 3}x_{3}}\,\text{d}\mathbf{x}.\]  
Since the octonion algebra is non-associative, we assume that the multiplication in the above integrals is performed from left to right. The conditions for the existence of the OFT and its advantages over the classical approach have been discussed in detail in previous works [49,50]. **Theorem 1**. : _Let \(u\in\mathcal{S}(\mathbb{R}^{3};\mathbb{O})\). Then for all \(\mathbf{x}\in\mathbb{R}^{3}\) we have_  
\[u(\mathbf{x})=\int_{\mathbb{R}^{3}}U_{\text{OFT}}(\mathbf{f})e^{\mathbf{e}_{ 4}2\pi f_{3}x_{3}}e^{\mathbf{e}_{2}2\pi f_{2}x_{2}}e^{\mathbf{e}_{1}2\pi f_{1 }x_{1}}\,\text{d}\mathbf{f}\]  
_(where multiplication is performed from left to right)._  
Let us recall that the symbol \(\mathcal{S}(\mathbb{R}^{3};\mathbb{O})\) denotes the Schwartz class of rapidly decreasing functions (generalized to the case of octonion-valued functions). The above claim was proved in one of the previous articles [50] and we omit the details here. It is worth adding, however, that the above theorem is also true under other assumptions, which were also discussed. It should be noted that for the function \(u\in L^{1}(\mathbb{R}^{3};\mathbb{O})\), its octonion Fourier transform is uniformly continuous, and the claim of the Riemann-Lebesgue lemma holds:  
\[\lim_{|\mathbf{f}|\to\infty}U_{\text{OFT}}(\mathbf{f})=0.\]  
In previous studies, we also proved a number of OFT properties [46,47,49,50]. The most important of them (and those that we need in further discussion on the Wigner distribution) are collected in Table 1. In the further part of the paper, we focus our attention on the most critical signal theory theorems, i.e., the [PERSON]'s theorem and the [PERSON]'s theorem proved in our earlier work [49]. **Theorem 2**. : _Let \(u,v\in L^{2}(\mathbb{R}^{3};\mathbb{R})\). Then_  
\[(u,v)_{L^{2}(\mathbb{R}^{3};\mathcal{O})}=(U_{\mathrm{OFF}},V_{\mathrm{OFF}})_{L^ {2}(\mathbb{R}^{3};\mathcal{O})},\]  
_where \((\cdot,\cdot)_{L^{2}(\mathbb{R}^{3};\mathcal{O})}\) denotes the classical scalar product of functions (real- or octonion-valued)._  
The assumption that the functions \(u\) and \(v\) are real-valued is relevant [46]. In the case of real-valued functions, [PERSON]'s theorem is a direct corollary of the above-mentioned result, but it is also valid in the general case of octonion-valued functions [44; 46]. **Theorem 3**.