Functional quantization in information theory creates stimulating questions: How should sharp asymptotes for quantization errors of a large class of Gaussian measures on a Hilbert space be derived? How should high-resolution theory of Gaussian processes be made exact in finite-dimensional theory?

The premise and practice of quantization first emerged in the literature in the late 18th century. In the late 1940s, Oliver, Pierce, and Shannon illuminated the function of quantization in analog-to-digital conversion and modulation. Also in the late 1940s, Bennett provided remarkable insights into the analysis of quantization noise for Gaussian processes, while Shannon shed light on the distortion rate theory in the quantization of analog-to-digital conversion and data compression. The historical development of the theory [1], mathematical underpinnings [2], and information theory and signal processing applications of quantization [3] exist in the literature. However, the concise derivation of asymptotics for quantization errors of Gaussian measures on a Hilbert space did not surface in the literature until 2004 [4].

Regarding the principle of small deviations for Gaussian measures, the major issues deal with the exploration of the asymptotic characterization of the probabilities of domains of subsets of Borel sigma-algebras in separable Banach spaces. Researchers typically double the sums of Gaussian fields to investigate the exact asymptotic representation of large deviation probabilities for Gaussian processes.

Fatalov convincingly examines the problems of small deviations for nonsingular zero-mean Gaussian measures in finite-dimensional Banach and Hilbert spaces. He derives and proves results on the sharp asymptotic representations of deviation probabilities for classes of continuous Gaussian processes in the Höder norm, and presents well-established results for the Wiener--or Slepian--process and the Brownian bridge. Fatalov cleverly outlines methods for computing sharp asymptotics of small deviation probabilities, for a large class on the brink of Gaussian stationary and Markov processes with logarithmic or power law covariance functions, and he presents formulas for calculating sharp asymptotics of small deviation probabilities in the power law norm, in the sup-norm, and for the exact distribution of the supremum. He convincingly relates the asymptotic behavior of quantization errors to the function of a zero-mean Gaussian random variable, with values in a separable finite or infinite-dimensional Banach space, or a unit ball in a Hilbert space.