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I was wondering about one particular example of this interchange.

In Mallat's book, at the proof of Poisson Formula

it's visible that the equation at the beginning of the 42nd page features the limit outside of the integral. It is my understanding that this limit had to be in, together with the sum it puts limits on, and then Mallat interchanged the limit and the integral.

I was wondering how would I justifty it, and I wasn't sure. I can't use the dominated convergence theorem, since this sum doesn't converge to anything, (unless we allow Dirac impulses, then it converges in weak sense to this distribution; if I am wrong and someone could correct me on this, I'd be much obliged).

So, could anyone help me with this? I should say I am not that good at math, I am just a simple engineer student.

Many thanks!

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# Interchange of limits and integrals

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