On Banaś modulus of smoothness and Gao Pythagorean constant of Lp(μ)

Abstract

In this paper, we first compute the Banaś modulus of smoothness of $L_p\left(\mu \right)$, which gives a solution to the problem posed by Banaś in 1986 (see Problem 4 of Banas (1986) [1]). Then, we introduce and calculate Gao Pythagorean constant of $L_p\left(\mu \right)$, which extends and improves some main results of Gao (2006) [3].