Almost orthogonality in Fourier analysis: From singular integrals, to function spaces, to the structural coloration of biological tissues

School of Mathematics Colloquium
Thursday, March 10, 2016 - 16:05
1 hour (actually 50 minutes)
Skiles 005
University of Kansas
Decomposition techniques such as atomic, molecular, wavelet and wave-packet expansions provide a multi-scale refinement of Fourier analysis and exploit a rather simple concept: “waves with very different frequencies are almost invisible to each other”. Starting with the classical Calderon-Zygmund and Littlewood-Paley decompositions, many of these useful techniques have been developed around the study of singular integral operators. By breaking an operator or splitting the functions on which it acts into non-interacting almost orthogonal pieces, these tools capture subtle cancelations and quantify properties of an operator in terms of norm estimates in function spaces. This type of analysis has been used to study linear operators with tremendous success. More recently, similar decomposition techniques have been pushed to the analysis of new multilinear operators that arise in the study of (para) product-like operations, commutators, null-forms and other nonlinear functional expressions. In this talk we will present some of our contributions in the study of multilinear singular integrals, function spaces, and the analysis of nanostructure in biological tissues, not all immediately connected topics, yet all centered on some notion of almost orthogonality.