Local properties of functions and approximation by trigonometric polynomials

T. V. Radoslavova

Matematičeskie zametki, Tome 18 (1975) no. 4, pp. 527-539.

Résumé

Suppose $\Phi_{p,E}$ ($p>0$ an integer, $E\subset[0,2\pi]$) is a family of positive nondecreasing functions $\varphi_x(t)$ ($t>0$, $x\in E$) such that $\varphi_x(nt)\leqslant n^p\varphi_x(t)$ ($n=0,1,\dots$), $t_n$ is a trigonometric polynomial of order at most $n$, and $\Delta_h^l(f,x)$ ($l>0$ an integer) is the finite difference of order $l$ with step $h$ of the function $f$. \underline{THEOREM.} Suppose $f(x)$ is a function which is measurable, finite almost everywhere on $[0, 2\pi]$, and integrable in some neighborhood of each point $x\in E$, $\varphi_x\in\Phi_{p,E}$ and $$ \varlimsup_{\delta\to\infty}\left|(2\delta)^{-1}\int_{-\delta}^\delta\Delta_u^l(f,x)\,du\right|\varphi_x^{-1}(\delta)\leqslant C(x)\infty\qquad(x\in E). $$ Then there exists a sequence $\{t_n\}_{n=1}^\infty$, which converges to $f(x)$ almost everywhere, such that for $x\in E$ $$ \varlimsup_{n\to\infty}|f(x)-t_n(x)|\varphi_x^{-1}(1/n)\leqslant AC(x), $$ where $A$ depends on $p$ and $l$.