I'm approximating $f(t)$ by $P_n(t)$, a real polynomial of arbitrary degree $n$, by minimizing the $L_2$ norm of its difference in $[0,1]$
$$R_n=|| f(t)-P_n(t)||_2^2=\int_0^1(f(t)-P_n(t))^2\,dt$$
considering the inner product $\displaystyle <f,g>=\int_0^1f(t)g(t)dt\quad$ and $\quad||f||_2=\sqrt{<f,f>}$.
It's well known that, if $\{\phi_n(t)\}$ is an orthonormal basis, this polynomial (that minimize $R_n$) can be calculate as
$$P_n(t)=\sum_{k=0}^n<f,\phi_k>\,\phi_k(t)$$
We also have $\displaystyle\lim_{n\to\infty}R_n=0$.
Question: I wonder if it's possible to find a bound $M_n\quad $of $\displaystyle \int_0^1(f(t)-P_n(t))^2\,dt\leq M_n$ with $\displaystyle \lim_{n\to\infty}M_n=0$
Something like the speed of convergence from $P_n(t)$ to $f(t)$ when $n\to\infty$.
Any bibliografy is also wellcomed.