General conditions for convergence rates of nonparametric orthogonal series estimators of the regression function f(x)=E(Y | X = x) are considered. The estimators are obtained by the least squares method on the basis of a random observation sample (Yi,Xi), i=1,...,n, where $X_i\in A\subset {ℝ}^d$ have marginal distribution with density $\varrho \in L^1\left(A\right)$ and Var( Y | X = x) is bounded on A. Convergence rates of the errors $E_X{(f\left(X\right)-{\widehat{f}}_N\left(X\right))}^2$ and $\parallel f-{\widehat{f}}_N{\parallel }_{\infty }$ for the estimator ${\widehat{f}}_N\left(x\right)={\sum }_{k=1}^N{\widehat{c}}_k{e}_k\left(x\right)$, constructed using an orthonormal system $e_k$, k=1,2,..., in $L^2\left(A\right)$ are obtained.