论文标题
多个傅立叶系列和晶格点问题
Multiple Fourier series and lattice point problems
论文作者
论文摘要
对于在$ \ mathbb {r}^d $上的某些径向函数的周期化的多个傅立叶序列，我们研究了球形部分总和的行为。我们显示了Gibbs-Wilbraham现象，Pinsky现象和多个傅立叶系列的第三个现象，涉及它们的收敛性。第三个现象与晶格点问题密切相关，这是分析数理论的经典主题。我们还证明，对于两个或三个维度的情况，傅立叶系列的收敛问题在某种意义上等同于晶格点问题。特别是，在二维中，原点的收敛问题等同于Hardy对高斯圆问题的猜想。
For the multiple Fourier series of the periodization of some radial functions on $\mathbb{R}^d$, we investigate the behavior of the spherical partial sum. We show the Gibbs-Wilbraham phenomenon, the Pinsky phenomenon and the third phenomenon for the multiple Fourier series, involving the convergence properties of them. The third phenomenon is closely related to the lattice point problems, which is a classical theme of the analytic number theory. We also prove that, for the case of two or three dimension, the convergence problem on the Fourier series is equivalent to the lattice point problems in a sense. In particular, the convergence problem at the origin in two dimension is equivalent to Hardy's conjecture on Gauss's circle problem.
