just as we can expand any given function $f(x)$ as a sum of sine and cosine functions, i. e., Fourier series
$f(x)=a_0+\sum a_k\cos(kx\omega)+\sum b_k\sin(kx\omega)$
or as power series, i. e., Taylor series
$f(x)=\sum c_k x^k$
Is there a way to systematically construct a series using a random function as the basis? Let us say use $\tan(x)$ as the basis and then have something like
$f(x)=\sum h_k\tan(kx\omega)$ or perhaps $f(x)=\sum p_k\tan(x)^k$.
In particular, I am interested in expand the identity function $f(x)=x$ as a sum of $\tanh(x)$ terms. So far I have tried to construct this backwards from the Taylor series of $\tanh(x)$
Since
$\tanh(x)=x−x^3/3+2x^5/15−17x^7/315+⋯=\sum 2^{2n}(2^{2n}−1)B_{2n}(2n−1)/(2n)!$ with $B_n$ the Bernoulli numbers
I can construct the following terms
$\frac{2^{2n}(2^{2n}−1)B_{2n}(2n−1)\tanh(x)^{2n-1}}{2n!}|_{n=2}=x^3/3+⋯$
$\frac{2^{2n}(2^{2n}−1)B_{2n}(2n−1)\tanh(x)^{2n-1}}{2n!}|_{n=3}=-2x^5/15+⋯$
then by summing the left-hand-side expression with $\tanh(x)$ I get a function that is closer to the identity for longer than just $\tanh(x)$ jet I am not sure how robust my idea is.
Any ideas here?