Is there any close-form solution for a function $f(t)$ satisfied the below equation:
$f(t)=g(t)+\frac{1}{t^2}(h(t)*f(t))$. Operator $*$ is convolution integral, and $g(t)$ and $h(t)$ are known functions.

Please provide more detail, for example what class of functions are you considering? Measurable? Smooth? C^1? Schwarz? etc
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David RobertsJul 8 '13 at 10:56

1

On that set $f$ is defined?
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AndrewJul 8 '13 at 11:11

$t>0$ and is a real value. At the first, we have no condition on $f$.
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Venous007Jul 8 '13 at 12:10

And what is a "closed form"? A formula with Fourier transforms and convolutions is a "closed form" or not?
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Alexandre EremenkoJul 8 '13 at 22:13

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I think you should give explicitly $g$ and $h$. For $t\ne 0$, a Fourier transform of this equation gives $f''(\omega)+h(\omega)f(\omega)=g''(\omega)$ that has known solutions just for some special cases of h.
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JonJul 9 '13 at 12:41