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：本博士论文摘要提出了扩展离散傅里叶变换（EDFT）的完整数学表述，该变换直接从傅里叶积分及其正交性推导得出。通过求解连续域与离散域中的加权最小二乘估计量，本文构建了一种自适应频域表示方法，该表示与经典傅里叶框架完全一致。在均匀采样数据且频率网格尺寸与采样点数相同的情况下，EDFT精确退化为经典离散傅里叶变换（DFT）。然而，当分析网格尺寸超过观测样本数量时，EDFT通过优化扩展频率集上的变换基函数，避免了传统补零操作。这使得该方法能够从不完整或非均匀采样数据中实现精确的谱估计。因此，EDFT在强谱分量区域实现了增强的频率分辨率，同时保持全局分辨率平衡，从而遵循不确定性原理。逆EDFT可重构原始信号，并在存在谱信息的区域生成外推或内插样本。EDFT无需显式分离确定性与随机分量，能同时准确捕获宽带、瞬态及正弦特征。仿真研究证实了该方法在非均匀采样、多重奈奎斯特区、数据缺失条件以及包含线谱与连续谱混合结构的信号中的鲁棒性。尽管EDFT的迭代计算相比经典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——但现代算力的发展已基本克服该问题，使得EDFT适用于当代信号分析应用。