On least squares estimation of Fourier coefficients and of the regression function

The problem of nonparametric function fitting with the observation model ${\displaystyle y_i=f\left(x_i\right)+{\eta }_i}$, i=1,...,n, is considered, where ${\displaystyle {\eta }_i}$ are independent random variables with zero mean value and finite variance, and ${\displaystyle x_i\in [a,b]\subset R^1}$, i=1,...,n, form a random sample from a distribution with density ${\displaystyle \varrho \in L^1[a,b]}$ and are independent of the errors ${\displaystyle {\eta }_i}$, i=1,...,n. The asymptotic properties of the estimator ${\displaystyle w{\hat{f}}_{N\left(n\right)}\left(x\right)=\sum _{k=1}^{N\left(n\right)}w{\hat{c}}_k{e}_k\left(x\right)}$ for ${\displaystyle f\in L^2[a,b]}$ and ${\displaystyle w{\hat{c}}^{N\left(n\right)}={(w{\hat{c}}_1,...,w{\hat{c}}_{N\left(n\right)})}^T}$ obtained by the least squares method as well as the limits in probability of the estimators ${\displaystyle w{\hat{c}}_k}$, k=1,...,N, for fixed N, are studied in the case when the functions ${\displaystyle e_k}$, k=1,2,..., forming a complete orthonormal system in ${\displaystyle L^2[a,b]}$$$a,b$$!$! are analytic.