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 $b \mod a$, $f(x)=0$.
Simple function help
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sequences-and-series
elementary-number-theory
functions
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0I 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
1 Answers
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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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0It'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