Almost Orthogonality in Fourier Analysis: From Singular Integrals to Function Spaces to Leibniz Rules for Fractional Derivatives

Explore a 50-minute conference talk by Rodolfo Torres of the University of California, Riverside, presented at IPAM's LatMath 2025 Workshop on March 8, 2025. Delve into the powerful world of Fourier analysis and its applications across diverse scientific fields including digital image processing, forensics, option pricing, cryptography, and protein structure analysis. Discover how Fourier analysis transforms signals into mathematical spectra of wave components, revealing hidden properties in data much like a prism decomposes light into colors. Learn about decomposition techniques such as atomic, molecular, wavelet, and wave-packet expansions that provide multi-scale refinement by exploiting the concept that "waves with different frequencies are almost invisible to each other." Follow Torres as he presents his contributions to the study of multilinear singular integrals and function spaces, and their applications to developing equivalents of the calculus Leibniz rule for fractional derivatives. This talk bridges harmonic analysis, complex analysis, and partial differential equations, showcasing how decomposition techniques capture subtle cancelations and quantify properties of operators through norm estimates in function spaces.