For $f, g$ in Schwartz function, I have $\widehat{D^\alpha f}=|\xi|^\alpha{}\hat{f}(\xi).$
My question is that if
$$(|\xi|^\alpha\widehat{fg}(\xi))\check{}=(|\xi|^\alpha{})\check{}*(\widehat{fg}(\xi))\check{}$$
holds or not. Does it?
2026-04-07 03:22:52.1775532172
Fourier Inversion and Convolution
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