Certain properties of sine series with monotone coefficients (Q1261265)

The author studies sine series (1) \(\sum^ \infty_{k=1}a_ k\sin kx\) with coefficients \(a_ k\) tending to zero in a decreasing way. Denote by \(g(x)\) the sum of series (1) and let \(g^*(x):=\sup\left\{\left|\sum^ n_{k=1}a_ k\sin kx\right|:n\geq 1\right\}\). He proves various estimates of the integrals \(\int| g(x)| dx\) and \(\int g^*(x)dx\) over some subintervals of \((0,\pi]\). These estimates are uniform with respect to the coefficients and the endpoints of the subintervals. As samples we present two corollaries here. Corollary 1. \(\int^ \pi_{\alpha/m}| g(x)| dx=\sum^ m_{k=1}a_ k/k+O(a_ 1)\), where \(\alpha\in[1/2,\pi]\). Corollary 5. \(\int^ \pi_ \varepsilon g^*(x)dx=2\int^ \pi_ \varepsilon| g(x)| dx+O(a_ 1)\), where \(0<\varepsilon<\pi\). The author also gives estimates of the integral modulus of continuity of \(g\) of order \(s=1,2,\dots\;\).