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

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.