Poisson summation formula

So, imagine you have a big jar of marbles. Each marble has a number on it, like a ticket number. Now, let's say you want to add up all the numbers on the marbles, but you're lazy and don't want to actually count them all.

Well, lucky for you, there's a formula called the Poisson Summation Formula that can help you out!

First, you take all the numbers on the marbles and list them out in order. Then, you add them all up. This is called the "sum" of the marbles.

Next, you take this sum and do a little math magic to it. You take a certain function and "transform" it using this sum. The function is kind of like a rule that tells you what to do with a number to get a new number.

Once you've transformed the function with the sum, you get a new function that looks different but means the same thing. This new function is called the "dual" function.

Finally, you take this dual function and do another little math magic to it. You transform it back into a sum, but now it's a different kind of sum. It's a sum over a different set of numbers.

And here's the cool part - this new sum is actually equal to the original sum you started with! So now, you can add up all the numbers in the original sum without actually having to count them all.

Basically, the Poisson Summation Formula takes a big list of numbers and turns it into a different kind of sum that's easier to work with. It's like turning a pile of marbles into a pile of candy - they're both still treats, but the candy is easier to eat!