Time-frequency Localization in the Fourier Symmetric Sobolev Space
Abstract
Abstract We study concentration operators acting on the Fourier symmetric Sobolev space $$\mathcal {H}$$ H consisting of functions f such that $$\int _{\mathbb {R}} |f(x)|^2(1+x^2) dx + \int _{\mathbb {R}} |\hat{f}(\xi )|^2(1+\xi ^2) d\xi < \infty $$ ∫ R | f ( x ) | 2 ( 1 + x 2 ) d x + ∫ R | f ^ ( ξ ) | 2 ( 1 + ξ 2 ) d ξ < ∞ . We find that the Bargmann transform is a unitary operator from $$\mathcal {H}$$ H to a weighted Fock space. After identifying the reproducing kernel of $$\mathcal {H}$$ H , we discover an unexpected phenomenon about the decay of the eigenvalues of a two-sided concentration operator, namely that the plunge region is of the same order of magnitude as the region where the eigenvalues are close to 1, contrasting the classical case of Paley–Wiener spaces.
Fields of Study & Subjects
Mathematical Analysis and Transform Methods
Spectral Theory in Mathematical Physics
Holomorphic and Operator Theory
Publication details
- Journal / source
- Journal of Fourier Analysis and Applications
- Publisher
- Springer Science and Business Media LLC
- Volume
- 32
- Issue
- 2
- Publication date
- 16 March 2026
- Language
- EN