Widths of Classes of Periodic Differentiable Functions in the Space~$L_{2}[0,2\pi]$

M. Sh. Shabozov

Matematičeskie zametki, Tome 87 (2010) no. 4, pp. 616-623

Résumé

We obtain exact values of different $n$-widths for classes of differentiable periodic functions in the space $L_{2}[0,2\pi]$ satisfying the constraint $$ \biggl(\int_{0}^{h}\omega_{m}^{p}(f^{(r)};t)\,dt\biggr)^{1/p}\le\Phi(h), $$ where $0$, $1/r$, $r\in\mathbb{N}$, and $\omega_{m}(f^{(r)};t)$ is the modulus of continuity of $m$th order of the derivative $f^{(r)}(x)\in L_{2}[0,2\pi]$.

Détail

Keywords: differentiable periodic function, width in the sense of Bernstein, Kolmogorov, Gelfand, the space $L_{2}[0,2\pi]$, trigonometric polynomial, Fourier series, modulus of continuity, linear operator.