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What is the Fourier series for $\{a\}\{b\}$, i.e. the product of the fractional parts of $a$ and $b$. I know what the Fourier series looks like for a single value of either $a$ or $b$, but I want to know what it is when the two are multiplied together.

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I'm not quite sure what a Fourier series for a function of two variables looks like, but what happens if you just multiply the two individual Fourier series together? –  Gerry Myerson Dec 3 '12 at 6:00
    
I get a complicated double series, which im not sure how to simplify. –  Ethan Dec 4 '12 at 0:57
    
What kind of simplification are you looking for? and do you have any reason to think the kind of simplification you are looking for actually exists? –  Gerry Myerson Dec 4 '12 at 1:58

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If $0 \leq \{a\} < 1$ and $0 \leq \{b\} < 1$, then $0 \leq \{a\}\{b\} < 1$. So the Fourier series of the product would look just like the Fourier series of $\{c\}$ of some real number $c$.

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But I need it to match up at the discontinuites of both a, and b –  Ethan Dec 3 '12 at 5:21
    
I think $\{a\}$ and $\{b\}$ are meant as real functions in the variables $a$ and $b$ respectively, not fixed numbers. –  WimC Dec 3 '12 at 6:02
    
@WimC Just define $\{c\} = \{a\}\{b\}$, which is the fractional part of some real number $c$. –  glebovg Dec 3 '12 at 17:47

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