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### Revision(s):

Revision #1 to TR21-102 | 19th July 2021 19:41

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

Revision #1
Authors: Siddharth Iyer, Anup Rao, Victor Reis, Thomas Rothvoss, Amir Yehudayoff
Accepted on: 19th July 2021 19:41
Keywords:

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 $d^\ell e^{{\ell+1 \choose 2}} n^{\frac{\ell-1}{2}}$. 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.

### Paper:

TR21-102 | 13th July 2021 20:55

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

TR21-102
Authors: Siddharth Iyer, Anup Rao, Victor Reis, Thomas Rothvoss, Amir Yehudayoff
Publication: 13th July 2021 20:55
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 $d^\ell e^{{\ell+1 \choose 2}} n^{\frac{\ell-1}{2}}$. 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.