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<p>This article investigates the octonionic Fourier transform (OFT) in the octonionic Schwartz space. The results related to the multi-index derivative of the OFT for octonion-valued functions are presented. Expanding upon these foundations, the octonionic Moritoh wavelet transform (OMT) is introduced using the convolution of octonion-valued functions and the continuity of OMT in the octonionic Sobolev space is proved. The fundamental properties, including an inner-product relation and inverse transform for the OMT are provided. A necessary and sufficient condition for the Moritoh transform of quaternion-valued functions is also presented. Furthermore, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {O}\)</EquationSource> </InlineEquation>-para-linearity of OFT and OMT are discussed and an uncertainty principle for OMT is derived. A modified <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\beta , \gamma \)</EquationSource> </InlineEquation>-indexed uncertainty principle for OMT is also obtained. An application of OMT to the octonionic wave equation is discussed. This article establishes a mathematical foundation to expand the application scopes of the quaternion wavelet transform.</p>

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The octonionic wavelet transform: a function space perspective

  • 期刊论文

  • 发表日期：

    2026年4月23日

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  • Awniya Kumar,
  • Sunil Kumar Singh

摘要

This article investigates the octonionic Fourier transform (OFT) in the octonionic Schwartz space. The results related to the multi-index derivative of the OFT for octonion-valued functions are presented. Expanding upon these foundations, the octonionic Moritoh wavelet transform (OMT) is introduced using the convolution of octonion-valued functions and the continuity of OMT in the octonionic Sobolev space is proved. The fundamental properties, including an inner-product relation and inverse transform for the OMT are provided. A necessary and sufficient condition for the Moritoh transform of quaternion-valued functions is also presented. Furthermore, \(\mathbb {O}\) -para-linearity of OFT and OMT are discussed and an uncertainty principle for OMT is derived. A modified \(\beta , \gamma \) -indexed uncertainty principle for OMT is also obtained. An application of OMT to the octonionic wave equation is discussed. This article establishes a mathematical foundation to expand the application scopes of the quaternion wavelet transform.