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<p>The main objective of this paper is to construct orthonormal wavelets and investigate some of their properties in the generalized Sobolev space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( H_{\mu }^{\omega ,A} (\mathbb {R}),\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>H</mi> <mrow> <mi>μ</mi> </mrow> <mrow> <mi>ω</mi> <mo>,</mo> <mi>A</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> within the framework of the Linear Canonical Transform (LCT). We also develop the theory of multiresolution analysis (MRA) in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\( H_{\mu }^{\omega ,A} (\mathbb {R}) ,\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>H</mi> <mrow> <mi>μ</mi> </mrow> <mrow> <mi>ω</mi> <mo>,</mo> <mi>A</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> under the same framework. Theoretical developments are supported by worked examples that illustrate the construction of wavelets specifically adapted to the structure of the generalized Sobolev space. Furthermore, a global regularity theorem is established for applicability of the proposed framework.</p>

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Multiresolution analysis and orthonormal linear canonical wavelets in generalized Sobolev spaces

  • 期刊论文

  • 发表日期：

    2025年11月3日

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  • Akhilesh Prasad,
  • Raj Kumar

摘要

The main objective of this paper is to construct orthonormal wavelets and investigate some of their properties in the generalized Sobolev space \( H_{\mu }^{\omega ,A} (\mathbb {R}),\) H μ ω , A ( R ) , within the framework of the Linear Canonical Transform (LCT). We also develop the theory of multiresolution analysis (MRA) in \( H_{\mu }^{\omega ,A} (\mathbb {R}) ,\) H μ ω , A ( R ) , under the same framework. Theoretical developments are supported by worked examples that illustrate the construction of wavelets specifically adapted to the structure of the generalized Sobolev space. Furthermore, a global regularity theorem is established for applicability of the proposed framework.