FOURIER TRANSFORMS AND SPECTRAL APPROXIMATION IN MODERN MACHINE LEARNING

Summary

Spectral representations play a central role in modern machine learning, providing efficient tools for data representation, approximation, and computation. Fourier series and Fourier transforms provide a rigorous mathematical framework for decomposing functions and signals into their frequency components, thereby linking smoothness, convergence, and approximation accuracy to learning performance (Titchmarsh, 1948). In practical learning systems, data are finite and discretely sampled, making the Discrete Fourier Transform (DFT) and the Fast Fourier Transform (FFT) essential for scalable computation of spectral features, convolution operations, and numerical reconstructions (Cooley & Tukey, 1965). These methods underpin many signal-processing and neural network architectures used in contemporary applications. Approximation theory further clarifies the expressive power of Fourier-based models, demonstrating their efficiency for smooth functions and motivating hybrid spectral–data-driven approaches (Trefethen, 2013). Recent machine-learning developments, such as random Fourier features, explicitly incorporate frequency-domain representations to scale kernel methods and improve generalization (Rahimi & Recht, 2007). This article presents a unified perspective on Fourier-based data processing, connecting classical Fourier series, discrete transforms, approximation theory, and modern machine-learning methods within a common spectral framework.