Approximation in $L_p$ by polynomials in the Walsh system

V. I. Ivanov

Sbornik. Mathematics, Tome 62 (1989) no. 2, pp. 385-402

Résumé

For $0$ and $q=1$, $1\le p\infty$ we calculate the quantity $$ \varkappa_{2^n}(L_p,L_q)=\sup_{f\in L_p}\frac{E_{2^n}(f)_q} {\dot\omega\bigl(\frac1{2^n},f\bigr)_p}\,, $$ where $E_{2^n}(f)_q$ is the best $L_q$-approximation of the function $f$ by Walsh polynomials of order $2^n$ and $$ \dot\omega(\delta,f)_p=\sup_{0\delta}\|f(x\dot+t)-f(x)\|_p $$ is the dyadic modulus of continuity of $f$ in $L_p$ determined by the operation $\dot+$ of addition of numbers from the interval $[0,1]$ in the dyadic system. Bibliography: 21 titles.