Sequential Convergence in the Testing Function Space of the Two-Sided FKF Transform

Abstract

The Fourier–Kontorovich–Lebedev (FKF) transform is a hybrid integral transform whose kernel is the product of a Fourier exponential factor and the Macdonald function of purely imaginary order. Its rigorous extension to generalized functions rests on a carefully constructed testing-function space equipped with a countable family of seminorms. This paper presents a detailed mathematical account of convergence in that testing space. We define the multinorm topology, state the precise notion of sequential convergence, contrast it with weaker pointwise notions, and explain why the topological formulation is indispensable for justifying differentiation under the transform sign, passage of limits, continuity of the distributional pairing, and spectral monitoring applications in multi-scale supply-chain systems. Illustrative figures depict seminorm decay, the underlying domain Ω, the spectral plane, and the distinction between pointwise and topological residuals.

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BibTeX RIS

BibTeX

@article{gulhane2026sequential,
  title = {{Sequential Convergence in the Testing Function Space of the Two-Sided FKF Transform}},
  author = {Akarshan Gulhane},
  journal = {International Journal of Creative and Open Research in Engineering and Management},
  year = {2026},
  month = aug,
  doi = {10.55041/ijcope.v2i8.138},
  url = {https://doi.org/10.55041/ijcope.v2i8.138},
  note = {CC BY-NC},
}

RIS

TY  - JOUR
TI  - Sequential Convergence in the Testing Function Space of the Two-Sided FKF Transform
AU  - Akarshan Gulhane
PY  - 2026
DA  - 2026/08/17
JO  - International Journal of Creative and Open Research in Engineering and Management
DO  - 10.55041/ijcope.v2i8.138
UR  - https://doi.org/10.55041/ijcope.v2i8.138
AB  - The Fourier–Kontorovich–Lebedev (FKF) transform is a hybrid integral transform whose kernel is the product of a Fourier exponential factor and the Macdonald function of purely imaginary order. Its rigorous extension to generalized functions rests on a carefully constructed testing-function space equipped with a countable family of seminorms. This paper presents a detailed mathematical account of convergence in that testing space. We define the multinorm topology, state the precise notion of sequential convergence, contrast it with weaker pointwise notions, and explain why the topological formulation is indispensable for justifying differentiation under the transform sign, passage of limits, continuity of the distributional pairing, and spectral monitoring applications in multi-scale supply-chain systems. Illustrative figures depict seminorm decay, the underlying domain Ω, the spectral plane, and the distinction between pointwise and topological residuals.
SN  - 3108-1754
ER  -

Abstract

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