$$ f(t) =\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos(n\pi tL)+\sum_{n=1}^\infty b_n \sin(n\pi tL) $$
In something like this, the coefficients are found for $\cos(1\pi tL)$, $\cos(2\pi tL)$ $\cos(3\pi tL)$ ...
Is there some reason $n$ needs to be a whole number, why can't n be a fraction $\cos(0.5\pi tL)$, $\cos(1\pi tL)$ $\cos(1.5\pi tL)$ ...
If this works, does it lead to faster convergence?
Does this also hold for the complex Fourier series?