An approximation of \(| \sin \,x|\) by rational Fourier series (Q921400)

For the real knots \(\alpha_ 0=0\), \(| \alpha_ k| <1\), \(\alpha_{2k}=-\alpha_{2k+1}\) the system of functions \[ \phi_ 0(z)=\frac{1}{\sqrt{2\pi}}\cdot \phi_ k(z)=\frac{\sqrt{1-\alpha_ k^ 2}}{\sqrt{2\pi}(1-\alpha_ kz)}\prod^{k-1}_{j=0}\frac{z-\alpha_ j}{1-\alpha_ jz} \] is orthogonal on \(| z| =1\). The author investigates the Fourier expansion of \(| \sin x|\) with respect to the system \(\{\phi_ k(e^{ix})\), \(k\in N\cup \{0\}\), \(\overline{\phi_ k(e^{ix})}\), \(k\in N\}\) for both fixed and variable knots \(\alpha_ j\).