Fuzzy Henstock-Kurzweil Fourier and Laplace transforms and their applications in fuzzy signals

Dengxiang Zhang; Yabin Shao; Yong Shi; Dengxiang Zhang; Yabin Shao; Yong Shi

doi:10.3934/era.2026276

Electronic Research Archive

2026, Volume 34, Issue 9: 6315-6348. doi: 10.3934/era.2026276

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Fuzzy Henstock-Kurzweil Fourier and Laplace transforms and their applications in fuzzy signals

1.
School of Electronic Science and Engineering, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
2.
School of Mathematics and Statistics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

Received: 07 April 2026 Revised: 26 June 2026 Accepted: 08 July 2026 Published: 17 July 2026

Most existing fuzzy integral transforms rely on non-additive measures and absolute integrability, making them invalid for discontinuous fuzzy signals. In this paper, we developed a unified frequency-domain analysis framework via fuzzy Henstock-Kurzweil (FHK) integrals without measure restrictions. We formulated the bilateral FHK Laplace transform (bi-FHKLT) and elaborated its inherent linkage with the fuzzy Henstock-Kurzweil Fourier transform (FHKFT) over complex domains. Under the FHK integral framework, the fuzzy Dirac delta distribution was constructed to derive the inversion formula of the FHKFT. Rigorous time/frequency domain convolution theorems are established for general discontinuous fuzzy functions. Numerical comparisons on rationally defined and countably discontinuous fuzzy signals demonstrated that the proposed transforms produce well-defined bounded magnitude outputs, whereas classical fuzzy Riemann transforms collapse completely. The constructed system offers an effective frequency analysis tool for fuzzy signals beyond the scope of traditional fuzzy Riemann integral methods.
- fuzzy number,
- fuzzy Henstock-Kurzweil integral,
- bilateral fuzzy Laplace transform,
- fuzzy Fourier transform,
- fuzzy convolution,
- discontinuous fuzzy signal
Citation: Dengxiang Zhang, Yabin Shao, Yong Shi. Fuzzy Henstock-Kurzweil Fourier and Laplace transforms and their applications in fuzzy signals[J]. Electronic Research Archive, 2026, 34(9): 6315-6348. doi: 10.3934/era.2026276

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Abstract

Most existing fuzzy integral transforms rely on non-additive measures and absolute integrability, making them invalid for discontinuous fuzzy signals. In this paper, we developed a unified frequency-domain analysis framework via fuzzy Henstock-Kurzweil (FHK) integrals without measure restrictions. We formulated the bilateral FHK Laplace transform (bi-FHKLT) and elaborated its inherent linkage with the fuzzy Henstock-Kurzweil Fourier transform (FHKFT) over complex domains. Under the FHK integral framework, the fuzzy Dirac delta distribution was constructed to derive the inversion formula of the FHKFT. Rigorous time/frequency domain convolution theorems are established for general discontinuous fuzzy functions. Numerical comparisons on rationally defined and countably discontinuous fuzzy signals demonstrated that the proposed transforms produce well-defined bounded magnitude outputs, whereas classical fuzzy Riemann transforms collapse completely. The constructed system offers an effective frequency analysis tool for fuzzy signals beyond the scope of traditional fuzzy Riemann integral methods.

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