A note about approximation by trigonometric polynomials

N. A. Shirokov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 46, Tome 467 (2018), pp. 238-243

Résumé

Let $E=\bigcup^n_{k=1}[a_k,b_k]\subset\mathbb R$; if $n>1$ then we assume that the segments $[a_k,b_k]$ are pairwise disjoint. Suppose that the following property holds: \begin{equation} E\cap(E+2\pi\nu)=\varnothing,\qquad\nu\in\mathbb Z,\quad\nu\ne0. \end{equation} We denote by $H^{\omega+r}(E)$ the space of functions $f$ defined on $E$ such that $|f^{(r)}(x_2)-f^{(r)}(x_1)|\leq c_f\omega (|x_2-x_1|)$, $x_1,x_2\in E$, $f^{(0)}\equiv f$. We assume that a modulus of continuity $\omega$ satisfies the condition \begin{equation} \int^x_0\frac{\omega(t)}t\,dt+x\int^\infty_x\frac{\omega(t)}{t^2}\,dt\leq c\omega(x). \end{equation} We find a constructive description of the space $H^{\omega+r}(E)$ in terms of the rate of nonuniform approximation of $f\in H^{\omega+r}(E)$ by means of trigonometric polynomials if $E$ satisfies (1) and $\omega$ satisfies (2).