On some contact points with the multiple Fourier series and lattice point problems with weights

对于d维欧几里德空间上径向函数周期化的多重傅里叶级数，研究了球面部分和的性质。我们展示了吉布斯-威尔伯拉罕现象，平斯基现象以及傅里叶级数的第三种现象。第三种现象与格点问题密切相关，这是解析数论的经典主题。我们还证明了傅里叶级数的收敛问题在某种意义上等价于格点问题。

For the multiple Fourier series of periodization of radial functions on d-dimensional Euclidean space, we studied the behavior of the spherical partial sum. We showed the Gibbs-Wilbraham phenomenon, the Pinsky phenomenon and the third phenomenon for the multiple Fourier series. The third phenomenon is closely related to the lattice point problems, which is classical theme of analytic number theory. We also proved that the convergence problems on the Fourier series is equivalent to the lattice point problems in a sense.