### A note on orthogonal series regression function estimators

The problem of nonparametric estimation of the regression function f(x) = E(Y | X=x) using the orthonormal system of trigonometric functions or Legendre polynomials ${e}_{k}$, k=0,1,2,..., is considered in the case where a sample of i.i.d. copies $({X}_{i},{Y}_{i})$, i=1,...,n, of the random variable (X,Y) is available and the marginal distribution of X has density ϱ ∈ ${L}^{1}$[a,b]. The constructed estimators are of the form ${\widehat{f}}_{n}\left(x\right)={\sum}_{k=0}^{N\left(n\right)}{\widehat{c}}_{k}{e}_{k}\left(x\right)$, where the coefficients ${\widehat{c}}_{0},{\widehat{c}}_{1},...,{\widehat{c}}_{N}$ are determined by minimizing the empirical risk ${n}^{-1}{\sum}_{i=1}^{n}{({Y}_{i}-{\sum}_{k=0}^{N}{c}_{k}{e}_{k}\left({X}_{i}\right))}^{2}$. Sufficient conditions for...