I trying to find a reference on the optimal rate of convergence in Sobolev space. Given a regression model on interval $[0,1]$ $$ Y_{i}=f(x_{i})+\epsilon_{i},\ i=1,\ldots,N $$ with fixed design and standard error assumptions $E(\epsilon_{i})=0;\ E(\epsilon_{i}\epsilon_{j})=\delta_{i,j}\sigma^{2}$. The regression function $f$ is from Sobolev space $$ f\in W^{q}\left[0,1\right]=\left[f,\ldots,f^{(q-1)} \text{ are absolutely continuous, } \int_{0}^{1}\left| f^{(q)}(x)\right| ^{2}<\infty\right] $$ Is the optimal convergence rate with respect to $L_{2}$-norm $N^{-q/(2q+1)}$ ? or which minimal additional assumptions do I need?
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