I have two one dimensional functions which are represented by some polynomial expansions (let's assume Chebyshev polynomials)
$f(x)=\sum_{k=1}^{N} f_k T_k(x)$ and $g(x)=\sum_{k=1}^{N} g_k T_k(x)$.
I need to multiply these two expansions. It is known that the result will be some new $2N$ degree polynomial
$s(x)=f(x) g(x)=\sum_{k=0}^{2N}s_kT_k(x)$,
with coefficients $s_k$ which can be calculated by known algorithms.
My questions is whether it is possible to somehow reduce the degree of final expansion? Namely, I am interested to obtain resulting expansion of the same $N$ degree rather $2N$. So I am looking for some algorithm which is able to transform original $\{s_k\}_{k=0}^{2N}$ coefficients to some new set $\{\tilde{s}_k\}_{k=0}^{N}$ with reduced dimension and still good enough to represent function of interest with reasonable precision.