Ferenc Lukacs type theorems in terms of the Abel-Poisson mean of conjugate series

摘要

A theorem of Ferenc Lukacs determines the generalized jumps of a periodic, Lebesgue integrable function f in terms of the partial sum of the conjugate series to the Fourier series of f. The main aim of this paper is to prove an analogous theorem in terms of the Abel-Poisson mean. We also prove an estimate of the partial derivative (with respect to the angle) of the Abel-Poisson mean of an integrable function F at those points at which F is smooth. Finally, we reveal the intimate relation between these two results. [References: 4]