A new method of summation of Fourier series converging in \(C\)-norm (Q818965)

It is known that, if \(f\) is a periodic continuous function on \(\mathbb{R}^1\), the sequence of the usual Fourier sums does not always converge uniformly to the function. In this paper, some modification of the usual Fourier sum is considered. This deformation converges uniformly to the given function. More precisely, the main result states: If \(f\) is continuous \(2\pi\)-periodic function and \[ \sigma_mf(x)=a_0\frac {1}{2 \pi}+\frac{1}{\pi}\sum^m_{k=1}\bigl[ a_k(s)\sin (kx)+a_k(c)\cos (kx) \bigr]k^{-\alpha(m)} \] for \(\alpha(m)=\ln\ln(m+1) [\ln(m+ 1)]^{-1} \), where \[ a_0=\int^{2\pi}_0f(y)dy,\;a_k(s)=\int^{2\pi}_0f(y)\sin (ky) dy,\;a_k(c)=\int^{2\pi}_0f(y)\cos(ky)dy\quad\text{for}\quad k=1, 2, \dots, \] then \[ \lim_{m\to+\infty}\max_{0\leq x\leq 2\pi}\bigl| \sigma_m f(x)-f(x)\bigr|=0. \] The proof is based on the theory of positive operators in Banach spaces developed by M. Krasnoselski.