The Jackson inequality and widths of function classes in \(L^2([0,1],x^{2v+1})\) (Q454826)

Let \(J_{\nu}(x)\) be the Bessel function of order \(\nu>-1/2\) and let \(\{\mu_l^{(\nu)}\}\) be the set of all positive roots of \(J_{\nu}(x)\), which are numbered such that \(0< \mu_l^{(\nu)} < \mu_{l+1}^{(\nu)}\) for all \(l\in {\mathbb N}\). Then for \(f\in L^2([0,1], x^{2\nu +1})\) we have the expansion NEWLINE\[NEWLINE f(x)=\sum_{l=1}^{\infty}a_l\varphi_l(x), \quad a_l=\frac{(f,\varphi_l)}{\varphi_l,\varphi_l}, \eqno (1) NEWLINE\]NEWLINE where \(\varphi_l(x)=j_{\nu}(\mu_l^{(\nu)}x)\) with \(j_{\nu}(x)=2^{\nu}\Gamma(\nu + 1)J_{\nu}(x)/x^{\nu}\) for \(x\neq 0\) and \(j_{\nu}(0)=1\). Using (1), the authors define the (generalized) derivative \(\widetilde D^{(\lambda)}f\) and the continuous modulus \(\omega_{\alpha}(f,\tau)\), where \(\lambda\in {\mathbb R}_+\), \(\alpha, \, \tau > 0\). We write \(f\in L^{2,\lambda}([0,1], x^{2\nu +1})\) if the derivative \(\widetilde D^{(\lambda)}f\) exists and belongs to the space \(L^2([0,1], x^{2\nu +1})\). Further, let \(E_n(f,{\mathcal L}_n)\) denote the best approximation of \(f\) by the subspace \({\mathcal L}_n=\text{span}\,\{\varphi_1, \dots, \varphi_n\}\) in the metric \(L^2([0,1], x^{2\nu +1})\). For small enough \(t\), the problem on the sharp Jackson constant \(K_{n,\alpha,\lambda}(t)\) in the inequality NEWLINE\[NEWLINE E_n(f,{\mathcal L}_n)\leq K_{n,\alpha,\lambda}(t)(\mu_{n+1}^{(\nu)})^{-\lambda} \omega_{\alpha}(\widetilde D^{(\lambda)}f,t), \quad f\in L^{2,\lambda}([0,1], x^{2\nu +1}), NEWLINE\]NEWLINE is solved. Moreover, the exact values of the \(n\)-widths of the classes \(S(\alpha, \lambda,\Psi)\) and \(\mathfrak{S}(\alpha, \lambda)\) in the space \(L^2([0,1], x^{2\nu +1})\) are found.