Characterization of Fourier transform of Hilbert space Valued functions
Abstract
The aim of this paper is to obtain the characterization of Fourier transform for functions taking values in a separable Hilbert space H, defined on the groups G=Rn, G=Z/nZ. In the first part of the paper, we consider the Fourier transform as a bijective mapping on SH, the space of all H-valued smooth rapidly decaying functions on Rn, which naturally extends bijectively to the dual space SH'. Within this framework, we characterize the Fourier transform in terms of the convolution theorem, supplemented by additional structural properties of the Fourier transform. In the second part of the paper, we consider the discrete setting, where the Fourier transform is an operator on L1(Z/nZ, H), and we provide corresponding characterization based on its essential properties. In contrast to the classical setting, a key assumption in our framework involves the interplay between two distinct convolution operations. As a consequence, the structure of our proof diverges notably from the classical case and introduces additional complexities.
Related
- Characterization of the group Fourier transform of Hilbert space Valued functions
- Characterization of Integral Transforms for Hilbert Space-Valued Functions
- Sparse domination and the bi-parameter Hilbert transform
- A curved three-dimensional simplex Hilbert transform
- Optimal distributional estimates of the multiple Hilbert transform
BibTeX
@misc{achar2026characterization,
title = {{Characterization of Fourier transform of Hilbert space Valued functions}},
author = {Satyajyoti Achar and Md Hasan Ali Biswas and Ramesh Manna},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36957},
eprint = {2610.36957},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36957},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Characterization of Fourier transform of Hilbert space Valued functions
AU - Satyajyoti Achar
AU - Md Hasan Ali Biswas
AU - Ramesh Manna
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36957
UR - https://arxiv.org/abs/2610.36957
AB - The aim of this paper is to obtain the characterization of Fourier transform for functions taking values in a separable Hilbert space H, defined on the groups G=R^n, G=Z/nZ. In the first part of the paper, we consider the Fourier transform as a bijective mapping on S_H, the space of all H-valued smooth rapidly decaying functions on R^n, which naturally extends bijectively to the dual space S_H'. Within this framework, we characterize the Fourier transform in terms of the convolution theorem, supplemented by additional structural properties of the Fourier transform. In the second part of the paper, we consider the discrete setting, where the Fourier transform is an operator on L^1(Z/nZ, H), and we provide corresponding characterization based on its essential properties. In contrast to the classical setting, a key assumption in our framework involves the interplay between two distinct convolution operations. As a consequence, the structure of our proof diverges notably from the classical case and introduces additional complexities.
ER -