Minimal number of significant directional moduli of smoothness
Let \(\Delta^ k_{he} f(x)\) be the \(k\)th finite difference of the function \(f(x)\), \(x\in D\subset\mathbb{R}^ d\), with respect to the domain \(D\), that is \[ \Delta^ k_{he} f(x)=\begin{cases} 0\quad\text{if Conv-hull } (x,x+khe)\not\subset D,\\ \sum^ k_{m=0} (-1)^{k-m}{k\choose m} f(x+ mhe)\quad\text{otherwise}.\end{cases} \] The directional \(k\)th modulus of smoothness for \(L_ p(D)\) is defined by \[ \omega^ k(f,e,t)_{L_ p(D)}=\sup_{| h|\leq t}\|\Delta^ k_{he} f(x)\|_{L_ p(D)}. \] The vector \(e\) is called the direction of \(\omega^ k(f,e,t)\). Then the modulus of smoothness is defined by \[ \omega^ k(f,t)_{L_ p(D)}= \sup_{| e|=1} \omega^ k(f,e,t)_{L_ p(D)}. \] In this paper the authors show that there exist \(n\) directions \(e_ 1,e_ 2,\dots,e_ n\) such that \[ \omega^ k(f,t)_{L_ p(D)}\leq C(k,p,D,n) \sup_{1\leq m\leq n} \omega^ k(f,e_ m,t)_{L_ p(D)}. \] The obtained results cannot be improved.
scientific article; zbMATH DE number 500208
scientific article