Iterated Integral (Fourier Integral)

[MHD93]

What conditions are there that allow to change the order of iterated integrals (improper ones).
For example, the following doesn't seem to work:
$$\int_{0}^\infty\left(\int_{0}^\infty f(t) \cos(\alpha t) dt\right)\cos(\alpha x)d\alpha = \int_{0}^\infty f(t)\left(\int_{0}^\infty \cos(\alpha t) \cos(\alpha x) d\alpha\right)dt$$
The integral in parentheses on the RHS obviously fails to converge.

 

[lurflurf]

See theorem II on wikipedia http://en.wikipedia.org/wiki/Order_of_integration_(calculus). That is the commonly referenced theorem. The trouble with it is it is very weak in the sense that many integrals that can be reversed do not meet the conditions. In those cases other methods are used, but they are not general.