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<p>The main aim of this paper is to introduce a novel framework for the construction of multiresolution structures within the realm of the special affine Fourier transform (SAFT), termed as the special-affine multiresolution analysis (SAMRA). The proposed SAMRA offers enhanced flexibility for analyzing non-stationary and chirp-like signals. Moreover, it unifies and generalizes existing frameworks such as the classical, fractional, and linear canonical MRAs. Unlike existing methods based on the discretization of continuous affine wavelets, our approach builds directly on a SAFT-domain sampling theorem, leading to the systematic construction of orthonormal special-affine wavelets in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^2(\mathbb {R})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, we demonstrate the function approximation capabilities of the proposed SAMRA, illustrating how signals with localized frequency variations can be effectively represented and reconstructed within this framework.</p>

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SPECIAL-AFFINE WAVELETS: MULTI-RESOLUTION ANALYSIS AND FUNCTION APPROXIMATION IN \(\varvec{L^2(\mathbb {R})}\)

  • 期刊论文

  • 发表日期：

    2026年1月6日

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  • Vikash K. Sahu,
  • Waseem Z. Lone,
  • Amit K. Verma

摘要

The main aim of this paper is to introduce a novel framework for the construction of multiresolution structures within the realm of the special affine Fourier transform (SAFT), termed as the special-affine multiresolution analysis (SAMRA). The proposed SAMRA offers enhanced flexibility for analyzing non-stationary and chirp-like signals. Moreover, it unifies and generalizes existing frameworks such as the classical, fractional, and linear canonical MRAs. Unlike existing methods based on the discretization of continuous affine wavelets, our approach builds directly on a SAFT-domain sampling theorem, leading to the systematic construction of orthonormal special-affine wavelets in \(L^2(\mathbb {R})\) L 2 ( R ) . Furthermore, we demonstrate the function approximation capabilities of the proposed SAMRA, illustrating how signals with localized frequency variations can be effectively represented and reconstructed within this framework.