Consistency of trigonometric and polynomial regression estimators

The problem of nonparametric regression function estimation is considered using the complete orthonormal system of trigonometric functions or Legendre polynomials ${\displaystyle e_k}$, k=0,1,..., for the observation model ${\displaystyle y_i=f\left(x_i\right)+{\eta }_i}$, i=1,...,n, where the ${\displaystyle {\eta }_i}$ are independent random variables with zero mean value and finite variance, and the observation points ${\displaystyle x_i\in [a,b]}$, i=1,...,n, form a random sample from a distribution with density ${\displaystyle \varrho \in L^1[a,b]}$. Sufficient and necessary conditions are obtained for consistency in the sense of the errors ${\displaystyle Vertf-w{\hat{f}}_{N}Vert,vertf\left(x\right)-w{\hat{f}}_N\left(x\right)vert}$, ${\displaystyle x\in [a,b]}$, and ${\displaystyle EVertf-w{\hat{f}}_{N}Ver{t}^2}$ of the projection estimator ${\displaystyle w{\hat{f}}_N\left(x\right)=\sum _{k=0}^{N}w{\hat{c}}_k{e}_k\left(x\right)}$ for ${\displaystyle w{\hat{c}}_0,w{\hat{c}}_1,...,w{\hat{c}}_N}$ determined by the least squares method and ${\displaystyle f\in L^2[a,b]}$.