Best Mean-Square Approximations by Entire Functions of Exponential Type and Mean $\nu$-Widths of Classes of Functions on the Line

S. B. Vakarchuk

Matematičeskie zametki, Tome 96 (2014) no. 6, pp. 827-848

Résumé

For the classes $L^r_2(\mathbb{R})$, $r\in \mathbb{Z}_{+}$, we establish the upper and lower bounds for the quantities $$ \chi_{\sigma,k,r,\mu,p}(\psi,t):=\sup\biggl\{\mathcal{A}_{\sigma} (f^{(r-\mu)})\Bigm/\biggl(\int_0^t \omega^p_k(f^{(r)},\tau) \psi(\tau)\,d\tau\biggr)^{1/p}:f \in L^r_2(\mathbb{R})\biggr\}, $$ where $\mu, r \in \mathbb{Z}_{+}$, $\mu \le r$, $k \in \mathbb{N}$, $0 p \le 2$, $0 \sigma \infty$, $0$, and $\psi$ is a nonnegative, measurable function summable on the closed interval $[0,t]$ and not equivalent to zero. In the cases $\chi_{\sigma,k,r,\mu,p}(1,t)$, where $\mu\in \mathbb{N}$, $1/\mu\le p \le 2$, and $\chi_{\sigma,k,r,\mu,2/k}(1,t)$, where $0$, we obtain the exact values of these quantities. We also obtain the exact values of the average $\nu$-widths of classes of functions defined in terms of the modulus of continuity $\omega^{*}$ and the majorant $\Psi$.

Détail

Keywords: entire function of exponential type, best mean-square approximation, average $\nu$-width, modulus of continuity, Jackson-type inequality, Plancherel's theorem, Paley–Wiener theorem, Hölder's inequality, Kolmogorov width, Bernstein width, Bernstein's inequality.
Mots-clés : Fourier transform, majorant