Approximation of integrable functions by linear methods almost everywhere

T. V. Radoslavova

Matematičeskie zametki, Tome 18 (1975) no. 1, pp. 77-90

Résumé

It is shown that $2\pi$ periodic functions whose $(r-1)$-th derivatives have bounded variation $(r>0)$ can be approximated by de La Vallée-Poussin $\sigma_{n,m}(an\le m=m(n)\le An, 0<a<A<1)$ at almost all points with a rate $o(n^{--r})$. For functions belonging to the class $\operatorname{Lip}(\alpha,L)(0<\alpha<1)$, any natural $N$, and a positive $\varepsilon>0$, we have almost everywhere $$ |f(x)-\sigma_{n,m}(f;x)|\le c(f,x)n^{-\alpha}\ln n\dots\ln_N^{1+\varepsilon}n, $$ where $\ln_kx=\underbrace{\ln\dots\ln x}_k(k=1,2,\dots)$. For any triangular method of summation $T$ with bounded coefficients we construct functions belonging to $\operatorname{Lip}(\alpha,L)(0<\alpha<1)$ and such that almost everywhere, $$ \varlimsup_{n\to\infty}|f(x)-\tau_n(f;x)|n^\alpha(\ln n\dots\ln_Nn)^{-\alpha}=\infty, $$ where the $\tau_n(f;x)$ are the means of the method $T$.