Local Singularities of Functions: Cusps, Chirps, and Oscillating Singularities - Part II

Explore the second part of a comprehensive lecture on local singularities of functions, their types, detection, and characterization. Delve into the harmonic analysis methods developed for identifying and analyzing pointwise singularities in various fields such as turbulence, gravitational waves, and physiological data. Examine the oscillatory behavior of signals at singular locations and understand how this information relates to the underlying physical processes. Learn about different types of singularities, their common features, and differences, leading to a classification system based on the p-exponent of Calderón and Zygmund and fractional integration. Discover how wavelet coefficients can be used to characterize classification parameters. Gain insights into multifractal analysis and its application in estimating fractional dimensions of point sets with specific regularity exponents. Investigate the extension of these techniques to broader classifications of pointwise singularities. Consider the relevance of this classification system for non-pointwise singularities, such as chirps in gravitational waves, and explore alternative analysis methods including time-frequency analysis using Wilson or Malvar bases and redundant decomposition systems.