| Résume | Every function $f$ on the $n$-dimensional discrete cube $\{-1,1\}^n$ admits a unique representation as a multilinear polynomial of total degree at most $n$, called the Walsh expansion of $f$. We will review the basics of Fourier analysis on the discrete cube and explain a duality argument (inspired by classical work of Figiel) which leads to approximation theoretic estimates for functions whose Walsh coefficients are supported on frequencies bounded above or below. These include Bernstein--Markov type inequalities and their reverses, moment comparison for vector-valued Rademacher chaos of low degree and estimates on the $\ell_p$ sum of influences of bounded functions. The talk is based on joint work with Paata Ivanisvili. |
