多重傅里叶级数球面部分和的广义局部化
Generalized localization for spherical partial sums of multiple Fourier series
摘要：
本文证明了多重傅里叶级数的球面部分和在$L_2$类中的广义局部化原理，即如果$f\in L_2(T^N)$和$f=0$在开集$Ω\subset T^N$上，则证明了该函数的球面部分和在$Ω$上几乎处处收敛于零.以前已经知道，当$L_p(T^N)$。因此，球面部分和的广义局部化问题在$L_p(T^N)$，$p\geq 1$中得到了完全解决：如果$p\geq2$，那么我们有广义局部化，如果$p<2$，那么广义局部化失败。
英文摘要：
In this paper the generalized localization principle for the spherical partial sums of the multiple Fourier series in the $L_2$ - class is proved, that is, if $f\in L_2(T^N)$ and $f=0$ on an open set $Ω\subset T^N$, then it is shown that the spherical partial sums of this function converge to zero almost - everywhere on $Ω$. It has been previously known that the generalized localization is not valid in $L_p(T^N)$ when $1\leq p<2$. Thus the problem of generalized localization for the spherical partial sums is completely solved in $L_p(T^N)$, $p\geq 1$: if $p\geq2$ then we have the generalized localization and if $p<2$, then the generalized localization fails.
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