论文标题
傅立叶变换倒置在Alexiewicz Norm中
Fourier transform inversion in the Alexiewicz norm
论文作者
论文摘要
如果$ f \ in l^1（{\ Mathbb r}）$，则证明$ \ lim_ {s \ to \ infty} \ lvert f-f-f \ ast d_s d_s \ rvert = 0 $，其中$ d_s（x）= \ sin（sx）/（sx）/（πx）$是dirichelet ffterffft \ sup_ {α<β} | \int_α^βf（x）\，dx | $是Alexiewicz Norm。这给出了对真实线上傅立叶变换的对称反转。也证明了不对称的反转。结果也适用于$ df $给出的度量，其中$ f $是有限变化的连续函数。相对于Lebesgue度量，此类措施不必绝对连续。一个示例说明l^1（{\ Mathbb r}）$中有$ f \，以便$ \ lim_ {s \ to \ infty} \ rvert f-f-f \ ast d_s d_s \ lvert_1 \ neq 0 $。
If $f\in L^1({\mathbb R})$ it is proved that $\lim_{S\to\infty}\lVert f-f\ast D_S\rVert=0$, where $D_S(x)=\sin(Sx)/(πx)$ is the Dirichlet kernel and $\lVert f\rVert = \sup_{α<β}|\int_α^βf(x)\,dx|$ is the Alexiewicz norm. This gives a symmetric inversion of the Fourier transform on the real line. An asymmetric inversion is also proved. The results also hold for a measure given by $dF$ where $F$ is a continuous function of bounded variation. Such measures need not be absolutely continuous with respect to Lebesgue measure. An example shows there is $f\in L^1({\mathbb R})$ such that $\lim_{S\to\infty} \rVert f-f\ast D_S\lVert_1\neq 0$.
