AI 中文总结
该文为一维时频局部化算子特征值计数建立显式上界，提出残差能量原理及多种多项式与截断方法，并处理软、硬及无界窗口，给出严格且测度敏感的界。
AI 中文摘要
我们证明了一维时频局部化算子在开放过渡窗口中的特征值数量的显式上界。一个抽象的残差能量原理结合了秩-N近似与其余项的Hilbert-Schmidt能量，并保留了由开放阈值所强制的严格整数修正。对于Fourier核，中心最小二乘多项式、优化矩和Chebyshev-Bessel截断改进或匹配了直径-Taylor证书；在区间块上，矩包络获得了精确的因子1/(2N+1)。有限可测划分产生了最佳多项式矩阵和残差能量界，以及利用空间或Fourier侧正交性的两级阈值分配，并在大的空间隙下保持稳定。一个互补的缺陷通道给出了有限区间并集的分量wise变分Schatten包络。对于任意有限测度硬窗口，精确的跨边界和对称差公式给出了严格的过渡计数和一个可认证的近一簇，该簇可从直接计数中减去。我们还处理了可积软掩模、根平方和核心-尾部分解，以及由矩控制的无界窗口。一个显式的无界有限测度集合具有所有多项式矩和对于每个阶数在(0,1]中的无限分数平移周长，然而承认一个O(log(1/epsilon)/log log(1/epsilon))证书。最后，具有有限多个有界分离相位交互和有限分离振幅秩的核承认阶乘、最小二乘、Chebyshev-Bessel、跨度压缩和各向异性度界。对于有界硬Fourier窗口，最终的混合不大于此处证明的相应早期证书；不主张与所有正则域估计的普遍比较。
英文摘要
We prove explicit upper bounds for the number of eigenvalues of a one-dimensional time-frequency localization operator in an open transition window. An abstract residual-energy principle combines a rank-N approximant with the Hilbert-Schmidt energy of its remainder and retains the strict integer correction imposed by the open threshold. For the Fourier kernel, centered least-squares polynomials, optimized moments, and Chebyshev-Bessel truncations improve or match the diameter-Taylor certificate; on interval blocks the moment envelope gains the exact factor 1/(2N+1). Finite measurable partitions yield best-polynomial matrix and residual-energy bounds, together with two-level threshold allocations that exploit spatial or Fourier-side orthogonality and remain stable under large empty gaps. A complementary defect channel gives a componentwise variational Schatten envelope for finite interval unions. For arbitrary finite-measure hard windows, exact cross-boundary and symmetric-difference formulas give strict transition counting and a certified near-one cluster that can be subtracted from direct counts. We also treat integrable soft masks, a root-sum-square core-tail decomposition, and unbounded windows controlled by moments. An explicit unbounded finite-measure set has all polynomial moments and infinite fractional translation perimeter for every order in (0,1], yet admits an O(log(1/epsilon)/log log(1/epsilon)) certificate. Finally, kernels with finitely many bounded separated phase interactions and finite separated amplitude rank admit factorial, least-squares, Chebyshev-Bessel, span-compressed, and anisotropic-degree bounds. For bounded hard Fourier windows, the final hybrid is no larger than the corresponding earlier certificates proved here; no universal comparison with all regular-domain estimates is asserted.
Comments80 pages; no figures
