Convexity, Moduli of Smoothness and a Jackson-Type Inequality

Abstract

For a Banach space $B$ of functions which satisfies for some $m>0$ $$\max (\Vert F+G{\Vert }_B,\Vert F-G{\Vert }_B)\ge (\Vert F{\Vert }_B^s+m\Vert G{\Vert }_B^s{)}^{1/s},\forall F,G\in B(\ast )$$ 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 $(\ast )$ where $s=\max (p,2),$ and many Orlicz spaces are shown to satisfy $(\ast )$ with appropriate $s.$