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I am not familiar with Fourier series, (I'm guessing that has something to do with what I want), and I want to know if someone could construct a convergent series for a function $f(x)$ with the property that if $x\equiv b\mod{a}$, $f(x)=1$, and if it's not congruent to $b \mod a$, $f(x)=0$.

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    Eerr....I think you just did construct such a function! What did you expect?2012-11-22
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    I need it in terms of continuous functions, like how one can represent the floor function with an infinite sum of sine functions.2012-11-22

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$$f(x)={1\over a}\sum_{t=0}^{a-1}e^{2\pi it(x-b)/a}$$

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    Gerry how did you come up with this? Also thanks2012-11-23
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    It's a standard trick in parts of Number Theory, where it is used for turning the number of solutions of a congruence into an exponential sum, all the better to estimate it.2012-11-23