# Iterated Integral (Fourier Integral)

## Main Question or Discussion Point

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.

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