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I am told that the functions $f(x), g(x)$ and $h(x)$ satisfy:

$\hat{f}(k) = \dfrac{\hat{h}(k)}{A+\hat{g}(k)} $, where $\hat{f}(k)$ is the Fourier Transform of $f(x)$ (likewise for $h(x)$ and $g(x)$ ), and $A$ is some positive real constant.

Now I am asked:

i)What conditions must be placed upon $h(x)$ and $g(x)$ for a solution to exist if $A=0$?

ii)Does a solution exist if $g(x) = e^{-a|x|}$ and $h(x) = e^{-b|x|}$, $a$ and $b$ being positive real constants?

For i) I know to invert the transform we have:

$f(x) =\dfrac{1}{2\pi} \int_{-\infty}^{\infty}{\dfrac{\hat{h}(k)}{\hat{g}(k)}e^{-i k x} dk} $ (if $f(x)$ isn't continuous, replace the LHS by half the sum of the limit from each side).

For part i), is it sufficient to say that $h(x) $ and $g(x) $ must both be absolutely integrable and of bounded variation for this inverse to exist? Or is there another condition that comes in, perhaps $g(x) $ must be non-zero?

Mel
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  • i) "for a solution to exist"... but what is a solution? Are you looking for $f$ in $L^1$, in $L^2$, or some other space? – 75064 May 04 '13 at 18:27
  • @user75064 I'm not sure what you mean by $L^k$. This is an introductory Fourier methods course so I presume it would be the more straightforward of the two. I need an expression for $f(x)$ – Mel May 04 '13 at 20:13

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