Suppose I have function $f $ on $\mathbb R^2$ and I want to compute the fourier transform of $f $ relative to first variable,
Is it true that for every $ t\in\mathbb R $
$\hat f ( \xi,t)=\int f (x,t)e^{-2\pi i x\xi} dx $
?
Is my definition correct?
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Btw, in that other problem I was not calculating an FT with respect to one variable, as you claimed - I was calculating $\hat f(0,t)$. – David C. Ullrich Dec 22 '15 at 19:05
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As a definition, of course. For a fixed $t$, define $g_t(x) := f(x,t)$. Whether or not the integral is defined for every fixed $t$ is a separate concern... – Chester Dec 22 '15 at 20:40