Prof. Javier Parcet (ICMAT Madrid)

Encoding Fourier multipliers in matrix algebras

Fourier multipliers are among the most important operators in analysis. They act on a given function $f$ by pointwise multiplication of its Fourier transform with a fixed function $f (m f pt )^$. This action can be vastly extended using more general notions of Fourier transform via group representation theory. In this talk, we will explain how these fundamental maps can be encoded as Schur multipliers in matrix algebras, which act simply as $$A ( M(j,k) A_j,k ).$$ In particular, remarkable inequalities for Fourier multipliers extend naturally to general Schur multipliers, even in the absence of a Fourier transform connection. If time permits, we will discuss applications to harmonic analysis on Lie groups.