Quadratic-phase Hilbert transform in the realm of Quaternion domains

Musadiq Shaheen; Firdous A. Shah; Aftab Hussain; Hamed Alsulami; Musadiq Shaheen; Firdous A. Shah; Aftab Hussain; Hamed Alsulami

doi:10.3934/math.20261202

AIMS Mathematics

2026, Volume 11, Issue 9: 30333-30354. doi: 10.3934/math.20261202

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Research article

Quadratic-phase Hilbert transform in the realm of Quaternion domains

1.
Department of Mathematics, University of Kashmir South Campus, Anantnag, J & K, India
2.
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia

Received: 17 September 2025 Revised: 26 November 2025 Accepted: 09 December 2025 Published: 17 September 2026
MSC : 30G35, 42A38, 42B10, 44A15, 46S10, 47G10

In the realm of signal processing, the Hilbert transform and quadratic phase Fourier transform (QPFT) are well-established tools for analyzing complex-valued signals. However, their efficiency to analyze the quaternionic-valued signals is almost negligible due to the higher-dimensional nature of such signals. In this study, we explored this area by introducing a novel integral transform, namely, the quadratic-phase Hilbert transform in the realm of quaternion domains (briefly referred to as the quaternion quadratic-phase Hilbert transform (Q-QPHT)), by combining the merits of the Hilbert transform and QPFT. The study began with a comprehensive investigation of the fundamental properties of the proposed transform including the derivation of Parseval's formula and a detailed characterization of its range. Subsequently, general relationships, illustrative examples, and Hausdorff-Young inequalities associated with the proposed Q-QPHT were established. These findings shed light on the broader applications of the Q-QPHT in signal processing; particularly in handling complex quaternionic-valued signals that require simultaneous control over multiple aspects. Finally, the validation of the obtained results was established by well-constructed examples and numerical simulations.
- quadratic phase Fourier transform,
- Hilbert transform,
- Hausdorff-Young inequality,
- quaternion Fourier transform
Citation: Musadiq Shaheen, Firdous A. Shah, Aftab Hussain, Hamed Alsulami. Quadratic-phase Hilbert transform in the realm of Quaternion domains[J]. AIMS Mathematics, 2026, 11(9): 30333-30354. doi: 10.3934/math.20261202

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Abstract

In the realm of signal processing, the Hilbert transform and quadratic phase Fourier transform (QPFT) are well-established tools for analyzing complex-valued signals. However, their efficiency to analyze the quaternionic-valued signals is almost negligible due to the higher-dimensional nature of such signals. In this study, we explored this area by introducing a novel integral transform, namely, the quadratic-phase Hilbert transform in the realm of quaternion domains (briefly referred to as the quaternion quadratic-phase Hilbert transform (Q-QPHT)), by combining the merits of the Hilbert transform and QPFT. The study began with a comprehensive investigation of the fundamental properties of the proposed transform including the derivation of Parseval's formula and a detailed characterization of its range. Subsequently, general relationships, illustrative examples, and Hausdorff-Young inequalities associated with the proposed Q-QPHT were established. These findings shed light on the broader applications of the Q-QPHT in signal processing; particularly in handling complex quaternionic-valued signals that require simultaneous control over multiple aspects. Finally, the validation of the obtained results was established by well-constructed examples and numerical simulations.

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