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Bohnenblust--Hille inequality for cyclic groups

Joseph Slote, Alexander Volberg, Haonan Zhang

Abstract

For any $K>2$ and the multiplicative cyclic group $Ω_K$ of order $K$, consider any function $f:Ω_K^n\to\mathbf{C}$ and its Fourier expansion $f(z)=\sum_{α\in\{0,1,\ldots,K-1\}^n}a_αz^α$, with $d:=\text{deg}(f)$ denoting its degree as a multivariate polynomial. We prove a Bohnenblust--Hille (BH) inequality in this setting: the $\ell_{2d/(d+1)}$ norm of the Fourier coefficients of $f$ is bounded by $C(d,K)\|f\|_\infty$ with $C(d,K)$ independent of $n$. This is the interpolating case between the now well-understood BH inequalities for functions on the poly-torus ($K =\infty$) and the hypercube ($K=2$) but those extreme cases of $K$ have special properties whose absence for intermediate $K$ prevent a proof by the standard BH framework. New techniques are developed exploiting the group structure of $Ω_K^n$. By known reductions, the cyclic group BH inequality also entails a noncommutative BH inequality for tensor products of the $K \times K$ complex matrix algebra (or in the language of quantum mechanics, systems of $K$-level qudits). These new BH inequalities generalize several applications in harmonic analysis and statistical learning theory to broader classes of functions and operators.

Bohnenblust--Hille inequality for cyclic groups

Abstract

For any and the multiplicative cyclic group of order , consider any function and its Fourier expansion , with denoting its degree as a multivariate polynomial. We prove a Bohnenblust--Hille (BH) inequality in this setting: the norm of the Fourier coefficients of is bounded by with independent of . This is the interpolating case between the now well-understood BH inequalities for functions on the poly-torus () and the hypercube () but those extreme cases of have special properties whose absence for intermediate prevent a proof by the standard BH framework. New techniques are developed exploiting the group structure of . By known reductions, the cyclic group BH inequality also entails a noncommutative BH inequality for tensor products of the complex matrix algebra (or in the language of quantum mechanics, systems of -level qudits). These new BH inequalities generalize several applications in harmonic analysis and statistical learning theory to broader classes of functions and operators.
Paper Structure (13 sections, 11 theorems, 187 equations)

This paper contains 13 sections, 11 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. The obstacle: a generalized maximum modulus principle
  3. Proof of Theorem \ref{['thm:bh cyclic']}
  4. Applications
  5. Sidon constants
  6. Bohr's radius
  7. Juntas and learning bounded low-degree polynomials
  8. Noncommutative Bohnenblust--Hille inequality for Heisenberg--Weyl matrix basis
  9. Learning bounded quantum observables of low-degrees
  10. Retracing the proofs of Bohnenblust--Hille inequality
  11. A partial answer to Conjecture \ref{['conjectuer']}: convex hull of rows of the discrete Fourier transform matrices
  12. Constant cannot be $1$
  13. A partial solution

Key Result

Theorem 1

Fix $d\ge 1$ and $K>2$. Then there exists $C(d,K)>0$ depending only on $d$ and $K$ such that for any $n\ge 1$ and for any that is of degree at most $d$, we have If $K$ is prime, we may choose $C(d,K)=C_K^{d^2}$ for some constant $C_K$ depending only on $K$.

Theorems & Definitions (27)

  • Theorem 1: Cyclic BH inequalities
  • Conjecture 1: Generalized Maximum Modulus Principle
  • Remark 1
  • Definition : Support-homogeneous polynomials
  • Definition : Maximal support pseudo-projection
  • Proposition 1
  • Proposition 2: Dimension-free boundedness of $\mathfrak{D}_{\xi}$
  • proof
  • Lemma 2: Support-homogeneous cyclic BH inequalities
  • proof
  • ...and 17 more