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PreprintCC BY 4.0arXiv

Characterization of the group Fourier transform of Hilbert space Valued functions

  • Satyajyoti Achar
  • Md Hasan Ali Biswas
  • Ramesh Manna
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Abstract

A remarkable theorem of S. Alesker, S. Artstein-Avidan, and V. Milman characterizes the Fourier transform on ℝn as, in essence, the unique transformation on the space of tempered distributions that exchanges convolution with pointwise multiplication. In this paper, we extend this perspective to the setting of the Heisenberg group He, focusing on functions taking values in a separable Hilbert space Hi. We establish a characterization of the group Fourier transform on He in terms of convolution. More precisely, we characterize the group Fourier transform on L1(He,Hi) and L2(He,Hi). In addition, we obtain a similar characterization for the group Fourier transform of Hi-valued Schwartz class functions on the Heisenberg group.

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Cite

BibTeXRIS
BibTeX
@misc{achar2026characterization,
  title = {{Characterization of the group Fourier transform of Hilbert space Valued functions}},
  author = {Satyajyoti Achar and Md Hasan Ali Biswas and Ramesh Manna},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.03267},
  eprint = {2611.03267},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.03267},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Characterization of the group Fourier transform of Hilbert space Valued functions
AU  - Satyajyoti Achar
AU  - Md Hasan Ali Biswas
AU  - Ramesh Manna
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.03267
UR  - https://arxiv.org/abs/2611.03267
AB  - A remarkable theorem of S. Alesker, S. Artstein-Avidan, and V. Milman characterizes the Fourier transform on ℝ^n as, in essence, the unique transformation on the space of tempered distributions that exchanges convolution with pointwise multiplication. In this paper, we extend this perspective to the setting of the Heisenberg group He, focusing on functions taking values in a separable Hilbert space Hi. We establish a characterization of the group Fourier transform on He in terms of convolution. More precisely, we characterize the group Fourier transform on L^1(He,Hi) and L^2(He,Hi). In addition, we obtain a similar characterization for the group Fourier transform of Hi-valued Schwartz class functions on the Heisenberg group.
ER  -