高斯混合范数Fock空间中的采样与插值
Sampling and Interpolation in Gaussian Mixed-Norm Fock Spaces
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中文总结 AI 辅助
本文为高斯混合范数Fock空间建立了采样与插值的尖锐判据，以经典密度α/π为临界值，并给出有界线性右逆，推广了经典Hilbert Fock空间理论。
中文摘要 AI 辅助
我们为径向-角向高斯混合范数Fock空间 $\mathcal F_\alpha^{p,q}$ 建立了尖锐的采样与插值准则，其中 $\alpha>0$ 且 $0<p,q\le\infty$。自然的点赋值尺度导致归一化限制 $R_{\alpha,\Lambda}^{p,q}f = \bigl(f(\lambda)e^{-\frac{\alpha}{2}|\lambda|^2}(1+|\lambda|)^{\frac{1}{q}-\frac{1}{p}}\bigr)_{\lambda\in\Lambda}$。在此归一化下，临界密度在整个Banach和拟Banach范围内均为经典值 $\frac{\alpha}{\pi}$。对于任意局部有限集 $\Lambda$，采样等价于 $D_{\mathrm{sep}}^{-}(\Lambda)>\frac{\alpha}{\pi}$，且当 $p<\infty$ 时还需满足相对分离条件；当 $p=\infty$ 时无需相对分离条件。这里 $D_{\mathrm{sep}}^{-}$ 是 $\Lambda$ 的分离子集上的下Beurling密度的上确界。插值等价于分离性和 $D^{+}(\Lambda)<\frac{\alpha}{\pi}$。同样的准则刻画了端点空间 $f_\alpha^{p,\infty}$ 的插值，且在两种情形下归一化限制映射均存在有界线性右逆。充分性论证基于快速局部化的Hilbert对偶和Lagrange原子，以及全混合范数尺度上的加权局部化综合定理。对于必要性，我们证明了快速局部化矩阵在加权环形混合序列空间上的下稳定性蕴含其在 $\ell^2$ 上的下稳定性。证明首先通过平移多项式截断和交换子估计过渡到 $\ell^\infty$，然后利用Sjöstrand类的 $p$-无关定理。通过固定的Hilbert采样格应用此结果，将严格密度条件归结为经典Hilbert Fock采样与插值定理。
英文摘要
We establish sharp sampling and interpolation criteria for the radial--angular Gaussian mixed-norm Fock spaces $\mathcal F_α^{p,q}$, where $α>0$ and $0<p,q\le\infty$. The natural point-evaluation scale leads to the normalized restriction $R_{α,Λ}^{p,q}f = \bigl(f(λ)e^{-\fracα{2}|λ|^2}(1+|λ|)^{\frac{1}{q}-\frac{1}{p}}\bigr)_{λ\inΛ}$. With this normalization, the critical density is the classical value $\fracαπ$ throughout the Banach and quasi-Banach ranges. For an arbitrary locally finite set $Λ$, sampling is equivalent to $D_{\mathrm{sep}}^{-}(Λ)>\fracαπ$, together with relative separation when $p<\infty$; no relative-separation condition is required when $p=\infty$. Here $D_{\mathrm{sep}}^{-}$ is the supremum of the lower Beurling densities over separated subsets of $Λ$. Interpolation is equivalent to separation and $D^{+}(Λ)<\fracαπ$. The same criterion characterizes interpolation for the little endpoint $f_α^{p,\infty}$, and in both settings the normalized restriction map admits a bounded linear right inverse. The sufficiency arguments are based on rapidly localized Hilbert dual and Lagrange atoms together with a weighted localized synthesis theorem on the full mixed-norm scale. For necessity, we prove that lower stability of a rapidly localized matrix on a weighted annular mixed sequence space implies lower stability on $\ell^2$. The proof first passes to $\ell^\infty$ by translated polynomial cutoffs and a commutator estimate, and then uses the $p$-independence theorem for the Sjöstrand class. Applied through a fixed Hilbert sampling lattice, this reduces the strict density conditions to the classical Hilbert Fock sampling and interpolation theorems.
发表机构
- National Yang Ming Chiao Tung University（国立阳明交通大学）
- VNU University of Science Vietnam National University, Hanoi（越南国立大学河内科学技术大学）
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