Hölder-Continuity of Extreme Spectral Values of Pseudodifferential Operators, Gabor Frame Bounds, and Saturation

Abstract

We build on our recent results on the Lipschitz dependence of the extreme spectral values of one-parameter families of pseudodifferential operators with symbols in a weighted Sj\"ostrand class. We prove that larger symbol classes lead to H\"older continuity with respect to the parameter. This result is then used to investigate the behavior of frame bounds of families of Gabor systems $\mathcal{G}(g,\alpha\Lambda)$ with respect to the parameter $\alpha>0$, where $\Lambda$ is a set of non-uniform, relatively separated time-frequency shifts, and $g\in M^1_s(\mathbb{R}^d)$, $0\leq s\leq 2$. In particular, we show that the frame bounds depend continuously on $\alpha$ if $g\in M^1(\mathbb{R}^d)$, and are H\"older continuous if $g\in M^1_s(\mathbb{R}^d)$, $0

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伪微分算子极谱值的荷尔德连续性、Gabor 框架边界和饱和度

我们在最近关于符号在加权 Sj\"ostrand 类中的伪微分算子的单参数族的极值谱值的 Lipschitz 依赖性的结果的基础上进行研究。我们证明，符号类越大，参数的连续性就越大。然后，我们利用这一结果来研究 Gabor 系统$mathcal{G}(g,\alpha\Lambda)$族的帧边界在参数$\alpha>0$方面的行为，其中$Lambda$是一组非均匀的、相对分离的时频偏移，并且$g\in M^1_s(\mathbb{R}^d)$, $0\leq s\leq 2$。我们特别指出，如果$g/in M^1(\mathbb{R}^d)$, $0

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