Fourier transform inversion in the Alexiewicz norm

E. Talvila
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Abstract

Abstract. If f P LpRq it is proved that limSÑ8‖f ́ f ̊ DS‖ “ 0, where DSpxq “ sinpSxq{pπxq is the Dirichlet kernel and ‖f‖ “ supαăβ | şβ α fpxq 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 P LpRq such that limSÑ8‖f ́ f ̊ DS‖1‰ 0.
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Alexewicz范数中的傅立叶变换反演
摘要如果f P LpRq,则证明了limSñ8‖f́fõDS‖“0,其中DSpxq”sinpSxq{Pπxq是Dirichlet核“supαăβ|şβαfpxq dx |是Alexewicz范数。这给出了实线上傅立叶变换的对称反演。还证明了非对称反演。结果也适用于dF给出的测度,其中F是有界变差的连续函数。这样的测度不必相对于Lebesgue测度是绝对连续的。一个例子表明其极限为Sñ8½f́f́DS½1‰0。
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