From Signal Processing to AGI: A Mathematical Foundation: Volume I: Mathematical and Signal Foundations

Volume I lays the mathematical groundwork for the entire trilogy by showing that the conceptual roots of artificial intelligence lie in the deep structure of signal processing, linear operators, probability, and information. It develops a unified language built from linear algebra, Hilbert spaces, random processes, statistical estimation, Fourier and harmonic analysis, sampling theory, quantization, optimization, learning theory, kernels, and multiresolution methods. Rather than presenting these subjects as isolated prerequisites, the volume frames them as the native mathematical environment in which intelligent systems operate. Signals become data, operators become transformations of representation, estimation becomes inference, and compression becomes the first step toward abstraction. In this way, Volume I argues that intelligence begins not with modern neural architectures, but with the disciplined study of how structure, uncertainty, and information can be encoded, transformed, and recovered. At the same time, the volume is forward-looking in its treatment of classical theory. Each chapter is oriented toward AI, showing how foundational results in signal processing anticipate core problems in modern learning: generalization, latent representation, robustness, approximation, similarity, sparsity, and multiscale structure. The reader is gradually prepared to see learning systems as mathematically organized signal-processing pipelines rather than opaque empirical machines. By the end of the volume, the essential conceptual bridge has been built: artificial intelligence is revealed as an extension of the mathematics of signals, estimation, and operators into adaptive and representation-driven regimes. Volume I therefore serves both as a rigorous foundation and as a philosophical thesis for the trilogy, establishing that any serious theory of advanced intelligence must begin with first principles.