Tight bounds on the Fourier growth of bounded functions on the hypercube

Abstract: We give tight bounds on the degree $\ell$ homogenous parts $f_\ell$ of a bounded function $f$ on the cube. We show that if $f: {\pm 1}^n \rightarrow [-1,1]$ has degree $d$, then $| f_\ell |\infty$ is bounded by $d^\ell/\ell!$, and $| \hat{f}\ell |_1$ is bounded by $d^\ell e^{\binom{\ell+1}{2}} n^{\frac{\ell-1}{2}}$. We describe applications to pseudorandomness and learning theory. We use similar methods to generalize the classical Pisier's inequality from convex analysis. Our analysis involves properties of real-rooted polynomials that may be useful elsewhere.