From Signal Processing to AGI: A Mathematical Foundation

From Signal Processing to AGI: A Mathematical Foundation develops the central claim that artificial intelligence is best understood not as a break from classical signal processing, but as its high-dimensional, adaptive, and learned continuation. The book begins with the mathematics of signals, representation spaces, uncertainty, Fourier and wavelet analysis, optimization, statistical learning, kernels, and nonlinear operators, showing that the essential problems of AI-perception, estimation, compression, prediction, and decision-already live inside the deeper structure of signal-processing theory. From that foundation, it builds a unified language in which observations become structured signals, learned models become operators on representation spaces, and intelligence itself becomes the transformation of uncertain measurements into useful internal state, inference, and action. The result is a mathematically rigorous bridge from classical analysis to modern machine learning, grounded in Hilbert spaces, stochastic processes, spectral methods, and variational principles.As the book progresses, it extends this framework into the core architectures and frontier problems of contemporary AI: convolutional networks, recurrent and state-space models, transformers, self-supervised learning, multimodal fusion, generative modeling, diffusion, causal representation learning, world models, agentic planning, safety, and the search for a unified theory of intelligent systems. Rather than treating these as disconnected technologies, the manuscript argues that they are all instances of a common mathematical pattern: structured observation, representation, latent dynamics, operator adaptation, and decision under uncertainty. In that sense, the book is both a graduate-level theoretical synthesis and a research program. It offers a coherent view of how signal processing, probability, geometry, optimization, and dynamical systems can be brought together to explain modern AI and to frame the path toward more general, robust, and scientifically grounded intelligence.