(C,1) summability of the differentiated Fourier series (Q1804785)

This paper is devoted to the study of \((C,1)\) summability of the differentiated Fourier series. A typical result of the authors is as follows. Theorem: Let \(f\) be differentiable on the closed interval \(I\). For each \(x\in I\) define \(g_ x\) to be the \(2\pi\)-periodic function such that on \([0, 2\pi)\), \[ g_ x(t)= \begin{cases} {f(x+ t)- f(x)\over t},\quad & t\neq 0,\\ f'(x),\quad & t= 0.\end{cases} \] If \(V_ H(g_ x, [-\delta, \delta])\to 0\) uniformly in \(x\in I\) as \(\delta\to 0^ +\), where \(V_ H\) denotes the harmonic variation, then the differentiated Fourier series of \(f\) at \(x\) is summable \((C,1)\) to \(f'(x)\) uniformly on \(I\).