On the rate of convergence of Fourier series of functions of bounded variation (Q1810268)

The author strengthens a result of Bojanic. The new theorem reads as follows. If the function is \(2\pi\)-periodic and of bounded variation, the sequence \(\{n_j\}\) satisfies the condition \[ \sum^\infty_{j=m} {1\over n_j}\leq {A\over n_m}, \] then for all \(n\) and \(x\) the following estimate is valid: \[ \begin{aligned} |f(x)- s_n(f, x)|&\leq \Biggl|\sum^{n_i-1}_{k=n+1} (a_k\cos kx+ b_k\sin kx)\Biggr|+ \sum^\infty_{j=i} \Biggl|\sum^{n_{j+1}- 1}_{k= n_j} (a_k\cos kx+ b_k\sin kx)\Biggr|\\ &\leq O\Biggl({A\over n+1}\Biggr) \sum^{n+1}_{k=1} V\Biggl(g_x,\Biggl[0,{\pi\over k}\Biggr]\Biggr),\end{aligned} \] where \(n_{i=1}\leq n< n_i\), \(g_x(f):= f(x+ t)+ f(x- t)- 2f(x)\), and \(V(f,[\alpha, \beta])\) is the variation of \(f\) on \([\alpha,\beta]\).