Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Generalization of the Octonion Fourier Transform to 3-D
  Octonion-Valued Signals -- Properties and Possible Applications to 3-D LTI
  Partial Differential Systems   The paper is devoted to the development of the octonion Fourier transform
(OFT) theory initiated in 2011 in articles by Hahn and Snopek. It is also a
continuation and generalization of earlier work by Blaszczyk and Snopek, where
they proved few essential properties of the OFT of real-valued functions, e.g.
symmetry properties. The results of this article focus on proving that the OFT
is well-defined for octonion-valued functions and almost all well-known
properties of classical (complex) Fourier transform (e.g. argument scaling,
modulation and shift theorems) have their direct equivalents in octonion setup.
Those theorems, illustrated with some examples, lead to the generalization of
another result presented in earlier work, i.e. Parseval and Plancherel
Theorems, important from the signal and system processing point of view.
Moreover, results presented in this paper associate the OFT with 3-D LTI
systems of linear PDEs with constant coefficients. Properties of the OFT in
context of signal-domain operations such as derivation and convolution of
$\mathbb{R}$-valued functions will be stated. There are known results for QFT,
but they use the notion of other hypercomplex algebra, i.e. double-complex
numbers. Considerations presented here require defining other higher-order
hypercomplex structure, i.e. quadruple-complex numbers. This hypercomplex
generalization of the Fourier transformation provides an excellent tool for the
analysis of 3-D LTI systems.

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The categories of this paper are: patt-sol
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The categories of this paper are: gr-qc
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The categories of this paper are: ao-sci
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The categories of this paper are: quant-ph
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The categories of this paper are: adap-org
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The categories of this paper are: hep-ph
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The categories of this paper are: econ
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: chem-ph
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The categories of this paper are: q-bio
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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-th
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The categories of this paper are: dg-ga