The asymptotic representations on the norm of the Fourier operators (Q557008)

The authors obtain new asymptotic formulas for two integrals which are frequently used in Fourier analysis: \[ \int_0^\pi \frac{| \sin nt| }{t}\,dt=\frac{2}{\pi}\ln n+C'+O(n^{-2}),n\to\infty, \] and \[ \frac{2}{\pi}\int_0^\pi\frac{| \sin((2n+1)t/2)| }{2\sin(t/2)}\,dt=\frac{4}{\pi^2}\ln n+C^*+O(n^{-1}), n\to\infty, \] where the constants \(C'\) and \(C^*\) are given explicitly. The second integral represents the norm of the partial sum operator for Fourier series.