I need to calculate the Fourier transform of $e^{i\log(x)}g(x)$, where $g(x)$ is a smooth function compactly supported on $[2^{j-1},2^{j+1}]$. I tried to use Taylor expansion to approximate $\log(x)$ by a linear function around $2^j$ and hence approximate the Fourier transform of $e^{i\log(x)}g(x)$ by a nice function, but the error doesn't seem to be able to be well bounded. Does anyone encountered this kind of problem or know if there is any good way?
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For $\Re(s) \in (0,1)$ and $a > 0$, $\int_0^\infty x^{s-1} e^{-a x}dx =\int_0^\infty (y/a)^{s-1} e^{-y}d(y/a)= a^{-s} \Gamma(s)$. By analytic continuation it stays true for $a = i\omega$, then it stays true in the sense of distributions for every $s$. Next use the convolution theorem. – reuns Oct 17 '17 at 22:01
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Thank you. But it is not clear to me how to play with the analytic continuation for a in this case. Could you suggest a reference with details? – Sam Wang Oct 18 '17 at 02:55