Please, give me the cue: does exist analytical representation of Fourier Transform of $sin(\frac{1}{x})$ for$ x>0$ (or $x>1$). Maybe exist an approximation of $FT(sin(\frac{1}{x}))$ by Bessel functions? Thank you.
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$\begingroup$ Is this function even in $L^1$ or $L^2$? $\endgroup$– Cameron WilliamsCommented Dec 13, 2014 at 18:45
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1$\begingroup$ Why do you need $L^1$ or $L^2$? It is a tempered distribution. In fact it is in $L^2$, it is like $1/x$ at infinity. $\endgroup$– Alexandre EremenkoCommented Dec 13, 2014 at 18:56
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1$\begingroup$ What is an "analytical representation"? Is not Fourier integral itself an analytical representation? $\endgroup$– Alexandre EremenkoCommented Dec 13, 2014 at 18:58
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$\begingroup$ @AlexandreEremenko True that but based on the lack of detail I wasn't sure what the level of understanding OP has. I would approach this via distributions. $\endgroup$– Cameron WilliamsCommented Dec 13, 2014 at 19:05
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2$\begingroup$ Since $\int_0^\infty \exp(-ax-b/x) \, dx$ is basically a Bessel function I'd expect that there's a formula for $\int_0^\infty \sin(1/x) \, e^{ixy} \, dx$ in terms of Bessel functions too. But it looks like this question will be closed before I can find this formula... Try Gradshteyn & Ryzhik. $\endgroup$– Noam D. ElkiesCommented Dec 13, 2014 at 19:48
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1 Answer
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The explicit answer is the formula 2.5.24.1 on page 433 from Brychkov, Marichev, Prudnikov Integral and Series, vol. 1. Note that for the odd function the FT reduces to sinT, and take $\alpha=1, \delta=1$. And yes, the answer is via Bessel and Macdonald functions.
