A Fejér type theorem to determine jumps in terms of the Abel-Poisson mean of double Fourier series

Abstract:

We extend from single to double Fourier series a theorem of Zygmund to determine the generalized jumps of a periodic integrable function at a simple discontinuity point. As a by-product of the proof, we obtain an estimate of the fourth mixed partial derivative of the Abel-Poisson mean of any integrable function $F(x,y)$ at such a point where $F$ is smooth. We also consider the extension of the Zygmund classes $\lambda _{*}$ and $\Lambda _{*}$ to the two-dimensional torus $\mathcal {T} ^{2}$.