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

We give tight bounds on the degree ` homogenous parts f` of a bounded function f on the cube. We show that if f : {±1} → [−1, 1] has degree d, then ‖f`‖∞ is bounded by d/`!, and ‖f̂`‖1 is bounded by d`e( `+1 2 )n `−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.

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