AI 中文总结
研究在规定勒贝格测度的可测集里，使Toeplitz算子前K个特征值之和最大化的集合的存在性，通过证明得出相关结论，还将证明扩展到局部化算子，可应用于Donoho - Stark集中问题及推广。
AI 中文摘要
我们证明，在所有具有规定勒贝格测度的可测集Ω⊂C中，存在一个集合，它能使福克空间上相关Toeplitz算子的前K个特征值（K≥1）之和最大化。在福克情形下，K = 1的情况是已知的，最优集是具有规定测度的球，而对于K>1，最优集的存在似乎是新的（极值化子未明确知晓，球的最优性仍是猜想）。此外，在温和假设下，我们的证明扩展到与抽象小波变换相关的局部化算子。在这个更广泛的情形下，即使对于K = 1，结果也是新的。作为应用，我们证明了Donoho - Stark集中问题及其对正交系统推广的最优集的存在性。
英文摘要
We prove that, among all measurable sets $Ω\subset\mathbb{C}$ of prescribed Lebesgue measure, there exists a set maximizing the sum of the first $K$ eigenvalues ($K\geq 1$) of the associated Toeplitz operator on the Fock space. In the Fock setting, the case $K=1$ is well known, the optimal sets being balls of prescribed measure, whereas for $K>1$ the existence of optimal sets appears to be new (maximizers are not known explicitly, and the optimality of balls remains conjectural). Moreover, under mild assumptions, our proof extends to localization operators associated with abstract wavelet transforms. In this broader setting, the result is new even for $K=1$. As an application, we prove the existence of optimal sets for the Donoho--Stark concentration problem and its generalization to orthonormal systems.
CommentsPreliminary version
