If I have $A(x)=B(x) C(x)$ (sine periodic from 0 to 1) rewritten as 


$\sum\_n A\_n \sin(n \pi x)=\sum\_m B\_m \sin(m \pi x)\sum\_p C\_p \sin(p \pi x)$


is there any easier way to compute $A\_n$ from $B\_m,C\_p$ other than 


$A\_n=\sum\_m \sum\_p B\_m C\_p\int\_0^1 \sin(n \pi x) \sin(m \pi x) \sin(p \pi x) dx$


?


Thank you.



Multiplying two Fourier series gives one Fourier series, but what are the new coefficients?
As @Gerald suggests, write things in terms of complex exponentials, and then the magic words are "convolution" and "Fast Fourier Transform".


