An Exact Upper Bound on the $L^p$ Lebesgue Constant and The $\infty$-R\'enyi Entropy Power Inequality for Integer Valued Random Variables

TL;DR

This paper establishes a precise asymptotic upper bound for the $L^p$ Lebesgue constant and demonstrates its application to a new $ abla$-Rényi entropy power inequality for integer-valued variables.

Contribution

It provides the first exact asymptotic upper bound for the $L^p$ Lebesgue constant and applies it to derive a novel entropy power inequality for integer-valued random variables.

Findings

Exact asymptotic upper bound for $L^p$ Lebesgue constant for $p \\ge 2$

Verification of a new $ abla$-Rényi entropy power inequality for integer-valued variables

Implications for analysis of Dirichlet kernels and entropy inequalities

Abstract

In this paper, we proved an exact asymptotically sharp upper bound of the $L^p$ Lebesgue Constant (i.e. the $L^p$ norm of Dirichlet kernel) for $p\ge 2$. As an application, we also verified the implication of a new $\infty $-R\'enyi entropy power inequality for integer valued random variables.