Certain summation methods for Fourier series giving the best order of approximation (Q1325250)

Let \(L_ n(f; x)= {1\over \pi} \int^ \pi_{-\pi} f(t) U_ n(t- x) dt\), \(f\in C_{2\pi}\), \(n\in \mathbb{N}\), be a sequence of convolution operators with \(U_ n\) a \(2\pi\) periodic even function. For a fixed \(n\) the operator \(L_ n\) is called of class \(S_{2m}\) if \(U_ n\) changes its sign on \((-\pi,\pi)\) at most \(2m\) times. The operators \(L_ n\) are called polynomial operators if for each \(n\)\ \(U_ n(t)\) and \(L_ n(f; x)\) are trigonometric polynomials of order \(\leq n\). Assume that \(L_ n(1; x)\equiv 1\). P. P. Korovkin has proved that the order of approximation of a continuous \(2\pi\) periodic function by a sequence of polynomial operators of class \(S_{2m}\) cannot be better as \(n^{-(2m+ 2)}\). The author gives a general method of construction of operators of class \(S_{2m}\) giving the best order of approximation. Then he gives various examples of extremal operators of the classes \(S_ 2\) and \(S_ 4\).