窗口Fourier变换反演公式的级数表示

Inversion of the continuous windowed Fourier transform using discrete series

本文研究连续窗口Fourier变换的反演公式.与经典的积分重构公式不同,本文证明当窗函数满足合适的条件时,窗口Fourier变换的反演公式可以表示为一个离散级数.此外,本文还研究这一重构级数的逐点收敛及其在Lebesgue空间的收敛性.对于L2空间,本文给出重构级数收敛的充分必要条件.

We study the inversion formula for the continuous windowed Fourier transform. Different from the classical ones where a single or double integral is involved, we show that for a large class of window functions, a function can be reconstructed from its continuous windowed Fourier transform with a discrete series. Moreover, we show that the series is convergent almost everywhere on R as well as in L^p（R） if the function to be reconstructed is. In particular, for the case of p = 2, we give a necessary and sufficient condition for the series to be convergent to the original function.