The opposite is possible, throught simple fourier analysis. For this question, suppose we have the periodic functions square wave functions:
$f(x) = \begin{cases} 1 & 0\leq x < 1 \\-1 & 1\leq x < 2 \\\end{cases}$
$g(x) = \begin{cases} 1 & 0\leq x < 1/2 \\-1 & 1/2\leq x < 3/2 \\1 & 3/2\leq x < 2 \\\end{cases}$
with period 2. Notice $g(x) = f(x + 1/2)$. Can they be used for a generalized fourier anlysis, so that
$sin( \pi x)= e + \sum \limits_{n=1}^\infty a_n f(nx) + \sum \limits_{n=1}^\infty b_n g(nx)$
I've checked that they are orthogonal (biorthogonal actually), but I'm not sure they are complete. I've tried applying the generalized Fourier formulas to derive the coefficients $a_n$ and $b_n$, but I get nonsense.
The questions: Can the set of square wave functions $ \{f(nx), g(nx)\} _{n=0}^\infty $ be used as basis for generalized Fourier Series. If yes, what am I missing, if no, why.
As you are using sums, not integrals, you need to make sure the periods are commensurable. You should be trying to express $\sin (\pi x)$. Yes, they are complete and you can express $\sin (\pi x)$ this way. You are basically inverting the expansion of a square wave into sines and cosines. The discontinuity says the coefficients will only fall off as $1/n$