CCM Seminar: Fourier multipliers, Hausdorff measure and classification with scattering transform

Next Thursday December 9, Javier Minguillón Sánchez, ERC-DyCon Internship student at UAM will talk on “Fourier multipliers, Hausdorff measure and classification with scattering transform” as part of our DeustoCCM’s seminars.

Thursday December 9, 2021 at 11:00H

Abstract. We begin by defining what a Fourier multiplier operator is and state some important properties. After that we give several examples of operators, out of which the most relevant are the disk multiplier and the spherical multiplier.

Then we proceed to discuss the boundedness of the disk operator and the spherical maximal operator. The first one is related to a counterexample due to Fefferman. For the second one, we use a result of Rubio de Francia.

We move on to define the Hausdorff measure and Hausdorff dimension on \mathbb{R}^d. We state several results:
Frostman’s Lemma, one result about constructing sets of arbitrary dimension, and another one about the projection of Borel sets onto lines.

After that we talk a bit about funcional data statistics and discuss in detail the Principal Components Analysis method. We give its theoretical basis (Karhunen-Loève’s Theorem).

As a closure, we define the scattering transform of a function with domain in \mathbb {R}^2 . We explain how to classify images with the above and the help of principal components analysis.