On determination of jumps in terms of Abel–Poisson mean of Fourier series

In this paper, we generalize some well-known results (Theorems A, C, and D) by establishing two general results (Theorems 1 and 3). As special applications, we find that the (generalized) jumps of f can be determined by the higher order partial derivatives of its Abel–Poisson means. This is different from the determination of jumps by higher order derivatives of the partial sums. We also give some estimates of the higher order partial derivatives of the Abel–Poisson mean of an integrable function F at those points at which F is smooth.