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.
Fourier series of the fractional part
-1
$\begingroup$
fourier-series
-
1I'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? – 2012-12-03
-
0I get a complicated double series, which im not sure how to simplify. – 2012-12-04
-
0What 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? – 2012-12-04
1 Answers
0
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 $c$.
-
0But I need it to match up at the discontinuites of both a, and b – 2012-12-03
-
0I think $\{a\}$ and $\{b\}$ are meant as real functions in the variables $a$ and $b$ respectively, not fixed numbers. – 2012-12-03
-
0@WimC Just define $\{c\} = \{a\}\{b\}$, which is the fractional part of some real number $c$. – 2012-12-03