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From the Wikipedia page on integral transforms, it states that:

...an integral transform is any transform $T$ of the following form: $ (Tf)(u)=\int^{t_2}_{t_1}K(t,u)f(t)dt $ ...There are numerous useful integral transforms. Each is specified by a choice of the function $K$ of two variables, the kernel function or nucleus of the transform.

I was wondering if the kernel function has to have any special properties in order for the integral transform to work. On the bottom of the Wikipedia page, it shows some of the common kernel functions such as $\frac{e^{-iut}}{\sqrt{2\pi}}$ and $e^{-ut}$ for the Fourier Transform and the Laplace Transform. However, as an example, with the kernel function $ K(t,u)=\ln|t+u|,\space t_1=0,\space t_2=\infty $ the integral transform would diverge for almost all $f(t)$. Or, for another example, the kernel function $ K(t,u)=ut,\space t_1=-\infty,\space t_2=0 $ also diverges for almost all $f(t)$. My thinking is that there has to be some certain property of the Fourier Transform and the Laplace transform that make it so they don't diverge for almost all $f(t)$. So, my basic question is whether a certain property has to hold in order for the integral transform to not diverge to infinity given any function $f(t)$.

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    A more typical question http://math.stackexchange.com/questions/362011.2013-04-15

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