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<p>Generally speaking, in signal processing, functions are typically represented in either the time domain or the frequency domain. Depending on the transform technique employed (e.g., Fourier, Laplace, etc.), the transformed version of a function in the frequency domain is unique, and these respective representations are interchangeable. A long-standing open problem has been that of obtaining an approximation in the time domain that yields an approximation that is also valid in the frequency domain, and <i>vice versa</i>. Nguyen and Oommen (IEEE Trans Pattern Anal Mach Intell 19:84–91, 1997) proposed a pioneering solution to this problem by leveraging the various <i>moments</i> of the function. This approach was further developed by subsequent authors&#xa0;as reported by Goneid and AbuSeif (Proceedings of the internationl conference on computer, communication and control technologies (CCCT’03), 2003). However, a significant limitation of the existing solution is the involvement of singular matrices when higher-order moments are utilized. This paper addresses the singularity issue by introducing the use of Legendre moments. The proposed method ensures that an approximation in the time domain yields a corresponding approximation in the frequency domain, and <i>vice versa</i>, in a smooth and analytically consistent manner. This work presents the theoretical properties of Legendre moments and their corresponding frequency-domain representations for any given time-domain function. We formally demonstrate the validity of the approximations in <i>both</i> domains. Rigorous theoretical and experimental results confirm the claims that we have made. To the best of our knowledge, this is the first paper to employ Legendre moments for frequency-domain analysis, marking a significant contribution to the field.</p>

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Function approximations valid in both time and frequency domains using legendre moments

  • 期刊论文

  • 发表日期：

    2025年11月22日

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  • Hamid Reza Aghamiri,
  • James R. Green,
  • B. John Oommen

摘要

Generally speaking, in signal processing, functions are typically represented in either the time domain or the frequency domain. Depending on the transform technique employed (e.g., Fourier, Laplace, etc.), the transformed version of a function in the frequency domain is unique, and these respective representations are interchangeable. A long-standing open problem has been that of obtaining an approximation in the time domain that yields an approximation that is also valid in the frequency domain, and vice versa. Nguyen and Oommen (IEEE Trans Pattern Anal Mach Intell 19:84–91, 1997) proposed a pioneering solution to this problem by leveraging the various moments of the function. This approach was further developed by subsequent authors as reported by Goneid and AbuSeif (Proceedings of the internationl conference on computer, communication and control technologies (CCCT’03), 2003). However, a significant limitation of the existing solution is the involvement of singular matrices when higher-order moments are utilized. This paper addresses the singularity issue by introducing the use of Legendre moments. The proposed method ensures that an approximation in the time domain yields a corresponding approximation in the frequency domain, and vice versa, in a smooth and analytically consistent manner. This work presents the theoretical properties of Legendre moments and their corresponding frequency-domain representations for any given time-domain function. We formally demonstrate the validity of the approximations in both domains. Rigorous theoretical and experimental results confirm the claims that we have made. To the best of our knowledge, this is the first paper to employ Legendre moments for frequency-domain analysis, marking a significant contribution to the field.