Convergence rates of orthogonal series regression estimators

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 (Y_i,X_i), i=1,...,n, where ${\displaystyle X_i\in A\subset {ℝ}^d}$ have marginal distribution with density ${\displaystyle \varrho \in L^1\left(A\right)}$ and Var( Y | X = x) is bounded on A. Convergence rates of the errors ${\displaystyle E_X{(f\left(X\right)-w{\hat{f}}_N\left(X\right))}^2}$ and ${\displaystyle Vertf-w{\hat{f}}_{N}Ver{t}_{\infty }}$ for the estimator ${\displaystyle w{\hat{f}}_N\left(x\right)=\sum _{k=1}^{N}w{\hat{c}}_k{e}_k\left(x\right)}$, constructed using an orthonormal system ${\displaystyle e_k}$, k=1,2,..., in ${\displaystyle L^2\left(A\right)}$ are obtained.