The exact $L_1$-norm supremum of Walsh--Kaczmarz--Fejér kernels

Abstract

We study the $L_1(G)$-norms of Fejér kernels for the Walsh--Kaczmarz system. In the Walsh--Paley case the identity \[ \|K^w_{2^n}\|_1=1 \qquad (n\in\mathbb N) \] follows directly from the binary structure. Toledo's sharp result gives the global supremum \[ \sup_{n\in\mathbb P}\|K^w_n\|_1=\frac{17}{15}. \] The Walsh--Kaczmarz kernels behave differently. We first prove that \[ \sup_{n\in\mathbb N}\|K^κ_{2^n}\|_1=\lim_{n\to\infty}\|K^κ_{2^n}\|_1=\frac43, \] and we prove that the sequence $(\Vert K^{\hat{I}}{º}_{2^n}{\Vert }_1:n\in \mathbb{N})$ is non-decreasing and is strictly increasing from $n=2$ on. Then we use this result, an exact splitting formula from Skvortsov's decomposition, and the sharp Walsh--Paley estimates. This gives the exact global supremum \[ \sup_{n\in\mathbb P}\|K^κ_n\|_1=\frac{71}{50}. \]

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