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<p>This paper investigates the wave packet transform and wavelet convolution product associated with the Mehler-Fock-Clifford transform (MFC-transform). MFC-wavelets are constructed via translation and dilation operators, and their boundedness, continuity are established in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^p\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-spaces. A wave packet transform (WP-transform) is introduced in both integral and convolution forms, and its key properties, including linearity, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-boundedness are derived. A Plancherel-Parseval relation is proved, showing that the transform preserves energy in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-spaces. Reconstruction results are obtained through admissibility conditions, leading to Calderón Formula and convergence in the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-norm. Furthermore, a generalized convolution framework and Hilbert space setting are developed, ensuring stability of the associated operators.</p>

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Wavelet analysis for the Mehler-Fock-Clifford transform

  • 期刊论文

  • 发表日期：

    2026年7月7日

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  • Supriya,
  • Praveen Kumar,
  • Jeetendrasingh Maan

摘要

This paper investigates the wave packet transform and wavelet convolution product associated with the Mehler-Fock-Clifford transform (MFC-transform). MFC-wavelets are constructed via translation and dilation operators, and their boundedness, continuity are established in \(L^p\) L p -spaces. A wave packet transform (WP-transform) is introduced in both integral and convolution forms, and its key properties, including linearity, \(L^1\) L 1 -boundedness are derived. A Plancherel-Parseval relation is proved, showing that the transform preserves energy in \(L^2\) L 2 -spaces. Reconstruction results are obtained through admissibility conditions, leading to Calderón Formula and convergence in the \(L^2\) L 2 -norm. Furthermore, a generalized convolution framework and Hilbert space setting are developed, ensuring stability of the associated operators.