Approximation of periodic functions in the uniform metric by Jackson type polynomials

V. V. Zhuk

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 23, Tome 357 (2008), pp. 115-142

Résumé

Let $C$ be a space of continuous $2\pi$-periodic functions $f$ with the norm $\|f\|=\max_{x\in\mathbb R}|f(x)|$, $$ J_n(f,x)=\frac1{(n+1)^2}\sum^n_{k=0}f(t_k)\Biggl(\frac{\sin\frac{(n+1)}2(x-t_k)}{\sin\frac{(x-t_k)}2}\Biggr)^2,\quad\text{where}\quad t_k=\frac{2\pi k}{n+1}, $$ be the Jackson polynomials of a function $f$. Let $\omega_r(f,h)$ be the $r$th continuity modulu of $f$, $E_n(f)$ be the best approximation of $f$ in the space $C$ by trigonometric polynomials of order $n$, and let $\widetilde F$ be the function trigonometrically conjugated with the primitive of $f$. The paper establishes results of the following types: \begin{align*} E_n(f)+\|J_{4n-1}(f)-f\|\approx\omega_1\Bigl(f,\frac1{n+1}\Bigr)+(n+1)\omega_2\Bigl(\widetilde F,\frac1{n+1}\Bigr),\\ \sup_{\alpha\in\mathbb R}\|J_n(f,\cdot+\alpha)-f(\cdot+\alpha)\|\approx\omega_1\Bigl(f,\frac1{n+1}\Bigr)+(n+1)\omega_2\Bigl(\widetilde F,\frac1{n+1}\Bigr). \end{align*} Here, the symbol $\approx$ does not depend on $f$ and $n$. Bibl. – 7 titles.