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<p>This paper introduces a novel Laguerre Ridgelet Transform (LRT) that integrates the radial localization properties of Laguerre polynomials with the directional sensitivity of ridgelet transforms. The construction of the LRT is presented with a new kernel formulation involving Laguerre-based translation and convolution. Fundamental properties—including Parseval’s identity, an inversion formula, and a Heisenberg-type uncertainty principle are established. An illustrative example demonstrates the transform’s effectiveness in multidimensional function representation. Furthermore, the LRT is applied to obtain an analytical solution to the Poisson equation, showcasing its potential in solving partial differential equations and its applicability across diverse domains such as signal processing, quantum mechanics, and medical imaging.</p>

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A novel Laguerre ridgelet transform and its application to the Poisson equation

  • 期刊论文

  • 发表日期：

    2026年1月3日

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  • Deepika Patel,
  • Sag Ram Verma

摘要

This paper introduces a novel Laguerre Ridgelet Transform (LRT) that integrates the radial localization properties of Laguerre polynomials with the directional sensitivity of ridgelet transforms. The construction of the LRT is presented with a new kernel formulation involving Laguerre-based translation and convolution. Fundamental properties—including Parseval’s identity, an inversion formula, and a Heisenberg-type uncertainty principle are established. An illustrative example demonstrates the transform’s effectiveness in multidimensional function representation. Furthermore, the LRT is applied to obtain an analytical solution to the Poisson equation, showcasing its potential in solving partial differential equations and its applicability across diverse domains such as signal processing, quantum mechanics, and medical imaging.