Let $\tau : [0,1]\rightarrow [0,1] $ be a continuous invertible map. Then the 'extension of $\tau$ to the space of square integrable real valued functions on $[0,1]$ is defined by the linear operator $T$ which maps $f$ to $g=Tf$ by $g(x):=f(\tau^{-1}(x))$.
Let $\{e_i\}_{i>1}$ denote an orthogonal basis on $L^2[0 1]$, and let $$C_{ik}=<Te_i,e_k>$$ for $i,k=1,2,..$ be a matrix representation of $T$.
Question: Is there a relation between the Fourier coefficients $<e_i,\tau>$ of the function $\tau$ and the matrix $C$?