I was looking into a problem and I arrived to something in which I want to expand some function $\varphi(x)$ in series of hyperbolic sines, something like:
$\varphi(x)=\sum\limits_{n=1}^{\infty}a_n\sinh(q_nx)$ where $q_n,a_n\in\mathbb{R}$.
Is there any way to do it? Would it be too weird?
Fourier series expansion works because $\sin(nx),\cos(nx)$ are an orthonormal basis (after normalizing). You will need to massage things a bit, in order to turn $\sinh(nx),\cosh(nx)$ into an orthonormal basis. For instance, if you are trying to approximate a function on $[0,1]$, you will need your expansion to satisfy $$ \int_0^1 \sinh(nx)\sinh(mx)=0 $$ when $n\not=m$. This is not literally true, so you could use a procedure like Gram-Schmidt to find a linear combination of these functions that form an orthonormal set.