This summary of the doctoral thesis provides a comprehensive formulation of the Extended Discrete Fourier Transform (EDFT), derived directly from the Fourier integral and its orthogonality properties. The method is obtained by solving weighted least-squares estimators in both continuous and discrete domains, yielding an adaptive frequency-domain representation that remains fully consistent with the classical Fourier framework. In the special case of uniformly sampled data on a uniform frequency grid of the same size, the EDFT reduces exactly to the classical Discrete Fourier Transform (DFT). However, when the analysis grid exceeds the number of observed samples, EDFT circumvents conventional zero-padding by optimizing the transformation basis over the extended frequency set. This enables accurate spectral estimation from incomplete or nonuniformly sampled data. Consequently, the EDFT achieves enhanced frequency resolution in regions of strong spectral content while maintaining global resolution balance, thereby remaining consistent with the uncertainty principle. The inverse EDFT reconstructs the original signal and produces extrapolated or interpolated samples wherever spectral information is available. The EDFT requires no explicit separation of deterministic and stochastic components and accurately captures broadband, transient, and sinusoidal features simultaneously. Simulation studies confirm its robustness under nonuniform sampling, multiple Nyquist zones, missing-data conditions, and signals with mixed spectra comprising both line and continuous components. Although iterative computation of the EDFT entails higher numerical cost compared to the classical DFT, this limitation - significant in the 1990s - has been largely mitigated by modern computational resources, rendering the EDFT practical for contemporary signal analysis applications.

翻译：本博士论文摘要全面阐述了扩展离散傅里叶变换（EDFT）的构建方法，该变换直接源于傅里叶积分及其正交性特性。该方法通过求解连续域和离散域中的加权最小二乘估计量而获得，从而产生一种自适应频域表示，该表示与经典傅里叶框架保持完全一致。在均匀采样数据与相同尺寸的均匀频率网格这一特殊情况下，EDFT精确退化为经典离散傅里叶变换（DFT）。然而，当分析网格点数超过观测样本数时，EDFT通过在扩展频率集上优化变换基，规避了传统的零填充方法。这使得从非完整或非均匀采样数据中进行精确的频谱估计成为可能。因此，EDFT在强频谱分量区域实现了增强的频率分辨率，同时保持了全局分辨率平衡，从而与不确定性原理保持一致。逆EDFT能够重建原始信号，并在频谱信息可用的任何位置生成外推或内插样本。EDFT无需显式分离确定性和随机分量，即可同时精确捕获宽带、瞬态和正弦特征。仿真研究证实了其在非均匀采样、多个奈奎斯特区域、数据缺失条件以及包含线谱和连续谱分量的混合频谱信号下的鲁棒性。尽管EDFT的迭代计算相较于经典DFT具有更高的数值成本——这一限制在20世纪90年代较为显著——但现代计算资源已很大程度上缓解了此问题，使得EDFT在当代信号分析应用中具有实用性。