Convexity, Moduli of Smoothness and a Jackson-Type Inequality

Abstract: For a Banach space $B$ of functions which satisfies for some $m>0$ $$ \max(|F+G|_B,|F-G|_B) \ge (|F|^s_B + m|G|^s_B)^{1/s}, \forall F,G\in B \ () $$ a significant improvement for lower estimates of the moduli of smoothness $\omega^r(f,t)_B$ is achieved. As a result of these estimates, sharp Jackson inequalities which are superior to the classical Jackson type inequality are derived. Our investigation covers Banach spaces of functions on $R^d$ or $T^d$ for which translations are isometries or on $S^{d-1}$ for which rotations are isometries. Results for $C_0$ semigroups of contractions are derived. As applications of the technique used in this paper, many new theorems are deduced. An $L_p$ space with $1<p<\infty $ satisfies $()$ where $s=\max(p,2),$ and many Orlicz spaces are shown to satisfy $(*)$ with appropriate $s.$