The Kontorovich–Lebedev Transform: Inversion Theory, Distributional Extensions and Applications to Multi-Echelon Supply-Chain Dynamics

Sharma, V. A.

doi:https://doi.org/10.55041/ijcope.v2i8.179

Volume 02, Issue 8

Published on: August 2026

THE KONTOROVICH–LEBEDEV TRANSFORM: INVERSION THEORY, DISTRIBUTIONAL EXTENSIONS AND APPLICATIONS TO MULTI-ECHELON SUPPLY-CHAIN DYNAMICS

V. A. Sharma

Department of Mathematics, Arts, Commerce and Science College, Amravati

DOI:https://doi.org/10.55041/ijcope.v2i8.179

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Abstract

The Kontorovich–Lebedev (KL) transform, defined by integration against the Macdonald function of purely imaginary order, is a classical continuous analogue of the Fourier–Bessel expansion. This paper develops a self-contained account of the operator, its formal adjoint and a detailed proof of the inversion theorem under the natural weighted integrability condition . The complex-contour form of the inversion is shown to furnish the analytic foundation for the distributional extensions that appear in the recent FKF literature. The classical spectral decay estimates are then mapped onto a multi-scale framework for resilient multi echelon supply chains: spectral filtering of high-frequency shocks, exact time-shift and modulation of lead time profiles, residue-based resonance extraction in hybrid vehicle logistics, and sequential convergence treatments of discontinuous disruptions. All related work ranging from the classical treatise of Zemanian to the contemporary FKF, digital-twin, quantum annealing and AI-risk papers and related to concrete analytic ingredients. Five figures and a flowchart illustrate the kernel behavior, the large-order decay, the logistics architecture, the shift property, the distributional hierarchy and the inversion-proof pipeline.

Keywords: Kontorovich–Lebedev transform, Macdonald function, inversion theorem, FKF transform, distributional extension, supply-chain resilience, multi-echelon lead times, spectral truncation, residue calculus, sequential convergence.

How to Cite this Paper

Sharma, V. A. (2026). The Kontorovich–Lebedev Transform: Inversion Theory, Distributional Extensions and Applications to Multi-Echelon Supply-Chain Dynamics. International Journal of Creative and Open Research in Engineering and Management, <i>02</i>(8), 1-9. https://doi.org/10.55041/ijcope.v2i8.179

Sharma, V.. "The Kontorovich–Lebedev Transform: Inversion Theory, Distributional Extensions and Applications to Multi-Echelon Supply-Chain Dynamics." International Journal of Creative and Open Research in Engineering and Management, vol. 02, no. 8, 2026, pp. 1-9. doi:https://doi.org/10.55041/ijcope.v2i8.179.

Sharma, V.. "The Kontorovich–Lebedev Transform: Inversion Theory, Distributional Extensions and Applications to Multi-Echelon Supply-Chain Dynamics." International Journal of Creative and Open Research in Engineering and Management 02, no. 8 (2026): 1-9. https://doi.org/https://doi.org/10.55041/ijcope.v2i8.179.

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References

Akarshan Gulhane, “Shift-Invariant Spectral Analysis of Multi-Echelon Supply-Chain Lead Times via the FKF Transform,” International Journal of Creative and Open Research in Engineering and Management, Vol. 02, Issue 08, 2026, DOI: 10.55041/ijcope.v2i8.136.

Akarshan Gulhane, “Spectral Framework Based on the FKF Transform for Supply-Chain Dynamics and Resilience,” ibid., DOI: 10.55041/ijcope.v2i8.140.

Akarshan Gulhane, “Spectral Differential Properties of the Distributional FKF Transform with Applications to Resilient Supply-Chain Systems,” ibid., DOI: 10.55041/ijcope.v2i8.139.

Akarshan Gulhane, “FKL Transform Key Properties and Applications to Supply-Chain Analysis,” Int. J. Eng. Sci. & Adv. Tech., Vol. 25 No. 12, 2025, pp. 596–605.

Akarshan Gulhane, “Expanded Treatment of the Time-Shift Property of FKF Transform with Supply-Chain Applications,” ibid., pp. 606–616.

Akarshan Gulhane, “The FKF Transform: Exponential Modulation Properties and Applications to Secure Digital Supply-Chain Networks,” ibid., pp. 617–623.

Gudadhe, “On LS Spaces of Gelfand–Shilov Technique,” Bull. Pure & Appl. Sci., Vol. 25, Issue 1, pp. 351–354, 2006.

Gudadhe, “Distributional LK Transform, Distributional Impulsive Signals and System Response,” Proc. 7th WSEAS Int. Conf. Applied Mathematics, 2005.

Akarshan Gulhane, “Digital Twin Technology for Building Resilient and Adaptive Global Supply Chains: A Review,” Int. Res. J. Modernization in Eng. Tech. & Sci., Vol. 6, Issue 12, 2024.

Akarshan Gulhane, “Industry 5.0 and Intelligent Logistics,” Int. J. Eng. Sci. & Adv. Tech., Vol. 24, Issue 12, 2024.

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