Time frequency localization in the Fourier Symmetric Sobolev space
Denis Zelent
TL;DR
We study time-frequency localization in the Fourier symmetric Sobolev space $\mathcal{H}$, whose symmetric norm $\|f\|^2_{\mathcal{H}}=\|f\|^2_{\mathcal{H}_1}+\|\hat f\|^2_{\mathcal{H}_1}$ reveals a Fourier-invariant setting; we show the Bargmann transform is unitary from $\mathcal{H}$ to a weighted Fock space and obtain an explicit reproducing kernel via a scaled Hermite basis. Concentration operators $T_{I,J}$ are analyzed to reveal eigenvalue distributions: in the two-sided problem with $I=[-R,R]$, $J=[-T,T]$ there are at least $4RT+O(\log R)+O(T)$ eigenvalues near $1$, with a long plunge region, unlike the Paley–Wiener case; in the Paley–Wiener with weight, the trace scales as $4RT+O(\log R)$, indicating similar eigenvalue growth. For one-sided concentration, eigenvalues cluster around $1/2$ with a precise moment structure: $\sum_n \lambda_n \sim \pi R^2$, $\sum_n \lambda_n^2 \sim (\pi/2)R^2$, and the bulk mass lies near $1/2$, implying $o(R^2)$ eigenvalues exceed $1/2+\varepsilon$ while about $2\pi R^2$ lie in $[1/2-\varepsilon,1/2+\varepsilon]$. The results advance understanding of time-frequency localization under a symmetric Sobolev norm and have implications for sampling/interpolation in time-frequency analysis.
Abstract
We study concentration operators acting on the Fourier symmetric Sobolev space $H$ consisting of functions $f$ such that $\int_{\mathbb{R}} |f(x)|^2(1+x^2) dx + \int_{\mathbb{R}} |\hat{f}(ξ)|^2(1+ξ^2) dξ< \infty $. We find that the Bargmann transform is a unitary operator from $H$ to a weighted Fock space. After identifying the reproducing kernel of $H$, 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.