How do you find the Fourier coefficients of $f(3x)$ with given Fourier-series of $f(x)$!

Question

Given that $f(x)$ has period $2\pi$ and is represented by a Fourier-series $$f(x) = \sum^{\infty}_{n = 0}a_n\cos(nx) + b_n\sin(nx)$$

what are then the Fourier coefficients, if using the same period $2\pi$, for the function $f(3x)$?

Answer 1

$$ g(x) = f(3x) = \sum_{n=0}^{+\infty}a_n\cos(3nx)+b_n\sin(3nx) = \sum_{n=0}^{+\infty}c_n\cos(nx)+d_n\sin(nx)$$

where $$ c_n = \left\{ \begin{array}{ll} a_n & \mbox{if } n \equiv 0[3] \\ 0 & \text{otherwise} \end{array} \right. $$

and $$ d_n = \left\{ \begin{array}{ll} b_n & \mbox{if } n \equiv 0[3] \\ 0 & \text{otherwise} \end{array} \right. $$

Answer 2

Given that $f(x)$ has period $2\pi$ and is represented by a Fourier-series $$f(x) = \sum^{\infty}_{n = 0}a_n\cos(nx) + b_n\sin(nx)$$ $g(x)=f(3x)$ has period $T= \frac{2\pi}{3}$ and is represented by a Fourier-series

$$f(x) =a_0(g)+ \sum_{n = 1}^{\infty} \left ( a_n (g) \cdot \cos \left ( nx\frac{2\pi}{T} \right ) + b_n(g) \cdot \sin \left ( nx\frac{2\pi}{T} \right )\right )\\ =a_0(g)+ \sum_{n = 1}^{\infty} \left ( a_n (g) \cdot \cos \left ( 3nx \right ) + b_n(g) \cdot \sin \left ( 3 nx \right )\right ).$$ but, $$f(3x) = \sum^{\infty}_{n = 0}a_n\cos(3nx) + b_n\sin(3nx)$$ hence buy identification, $a_n = a_n(g) $ and $b_n = b_n(g)$ if $n=3p$ and $a_n = b_n = 0$ otherwise.