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

We give tight bounds on the degree $\mathrm{\ell }$ homogenous parts $f_{\mathrm{\ell }}$ of a bounded function $f$ on the cube. We show that if $f: {\pm 1}^n \rightarrow [-1,1]$hasdegree$d$,then$| f_\ell |\infty$isboundedby$d^\ell/\ell!$,and$| \hat{f}\ell |_1$isboundedby$d^\ell e^{\binom{\ell+1}{2}} n^{\frac{\ell-1}{2}}$. 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.