论文标题
关于贝塞尔的$ l^p $融合的评论 - 碟片上
Remarks on the $L^p$ convergence of Bessel--Fourier series on the disc
论文作者
论文摘要
$ l^p $在平面域上针对拉普拉斯人的特征功能扩展的融合在很大程度上是$ p \ neq 2 $的不知名。在讨论了2-torus上的经典傅立叶系列之后，我们转到椎间盘上，其本征函数可显式计算为三角和贝塞尔函数的产物。我们总结了Balodis andCórdoba（1999）的结果，内容涉及混合空间中Bessel-fourier系列的$ l^p $融合，$ l^p _ {\ p _ {\ mathrm {rad}}（l^2 _ {\ 2 _ {\ mathrm {ang {ang}}} $ for the disk for the disk for the for the for $ $ $ \ tffr p。然后，我们描述如何修改它们的结果以获得$ l^p（\ mathbb {d}，r \，\ mathrm {d} r \，\ mathrm {d} t）$ $ l^p _ {\ mathrm {rad}}（l^q _ {\ mathrm {ang}}）$（$ \ tfrac {1} {1} {p}+\ tfrac {1} {1} {1} {q} = 1 $）用于约束范围$ {2 \ leq p <4} $。
The $L^p$ convergence of eigenfunction expansions for the Laplacian on planar domains is largely unknown for $p\neq 2$. After discussing the classical Fourier series on the 2-torus, we move onto the disc, whose eigenfunctions are explicitly computable as products of trigonometric and Bessel functions. We summarise a result of Balodis and Córdoba (1999) regarding the $L^p$ convergence of the Bessel--Fourier series in the mixed norm space $L^p_{\mathrm{rad}}(L^2_{\mathrm{ang}})$ on the disk for the range $\tfrac{4}{3}<p<4$. We then describe how to modify their result to obtain $L^p(\mathbb{D}, r\,\mathrm{d}r\,\mathrm{d}t)$ norm convergence in the subspace $L^p_{\mathrm{rad}}(L^q_{\mathrm{ang}})$($\tfrac{1}{p}+\tfrac{1}{q}=1$) for the restricted range ${2\leq p < 4}$.
