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Approximation of polynomials from Walsh tail spaces

Alexandros Eskenazis, Haonan Zhang

Abstract

We derive various bounds for the $L_p$ distance of polynomials on the hypercube from Walsh tail spaces, extending some of Oleszkiewicz's results (2017) for Rademacher sums.

Approximation of polynomials from Walsh tail spaces

Abstract

We derive various bounds for the distance of polynomials on the hypercube from Walsh tail spaces, extending some of Oleszkiewicz's results (2017) for Rademacher sums.
Paper Structure (2 sections, 7 theorems, 54 equations)

This paper contains 2 sections, 7 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Proofs

Key Result

Proposition 1

For every $1\leq p\leq q< \infty$ and $k\in\mathbb{N}$, the constant $\mathsf{M}_{p,q}(k)$ in inequality eq:mom is also the least constant for which every function $f:\{-1,1\}^n\to \mathbb{R}$, where $n\geq k$, satisfies where the conjugate exponent $r^\ast$ of $r\in[1,\infty]$ satisfies $\frac{1}{r^\ast}+\frac{1}{r} = 1$.

Theorems & Definitions (17)

  • Proposition 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 5
  • proof : Proof of Proposition \ref{['prop:dual-mom']}
  • proof : Proof of Theorem \ref{['thm:dual-hit']}
  • Proposition 6
  • Lemma 7
  • proof
  • ...and 7 more