Summation methods for Fourier series yielding best order of approximation (Q1337866)

The author examines the sequence of linear operators \(L_n (f)(x)={1\over\pi}\int^\pi_{- \pi} U_n (t - x) f(t)dt\), where \(U_n (s) = 1/2 + \sum^n_{k = 1} \lambda_{kn} \cos ks\), \(n \geq 1\), on the Banach space \(C_{2 \pi}\) of continuous \(2 \pi\)-periodic functions. By the method of decomposition of meromorphic functions a sequence \(U_n\), \(n \geq 1\), is constructed such that the order of approximation is exact (it is equal to \(1/n^{2m + 2})\) for each \(2(m + 2)\)-differentiable function \(f\) with bounded derivate \(f^{(2m + 2)}\).