Approximation of continuous functions by trigonometric polynomials almost everywhere

T. V. Radoslavova

Matematičeskie zametki, Tome 19 (1976) no. 1, pp. 49-62.

Résumé

We consider the problem of the rate of approximation of continuous $2\pi$-periodic functions of class $W^rH[\omega]_C$ by trigonometric polynomials of order $n$ on sets of total measure. We prove that when $r\ge0$, $\omega(\delta)\delta^{-1}\to\infty$ ($\delta\to0$) there exists a function $f\in W^rH[\omega]_C$ such that $\widetilde f\in W^rH[\omega]_C$ and for any sequence $\{t_n\}_{n=1}^\infty$ we have almost everywhere on $[0,2\pi]$ \begin{gather*} \varlimsup_{n\to\infty}|f(x)-t_n(x)|n^r\omega^{-1}(1/n)>C_x>0 \\ \varlimsup_{n\to\infty}|\widetilde f(x)-t_n(x)|n^r\omega^{-1}(1/n)>C_x>0 \end{gather*}