Least-squares trigonometric regression estimation

The problem of nonparametric function fitting using the complete orthogonal system of trigonometric functions ${\displaystyle e_k}$, k=0,1,2,..., for the observation model ${\displaystyle y_i=f\left(x_{\in }\right)+{\eta }_i}$, i=1,...,n, is considered, where ${\displaystyle {\eta }_i}$ are uncorrelated random variables with zero mean value and finite variance, and the observation points ${\displaystyle x_{\in }\in [0,2\pi ]}$, i=1,...,n, are equidistant. Conditions for convergence of the mean-square prediction error ${\displaystyle \left(\frac{1}{n}\right)\sum _{i=1}^{n}E{(f\left(x_{\in }\right)-w{\hat{f}}_{N\left(n\right)}\left(x_{\in }\right))}^2}$, the integrated mean-square error ${\displaystyle E\Vert f-w{\hat{f}}_{N\left(n\right)}{\Vert }^2}$ and the pointwise mean-square error ${\displaystyle E{(f\left(x\right)-w{\hat{f}}_{N\left(n\right)}\left(x\right))}^2}$ of the estimator ${\displaystyle w{\hat{f}}_{N\left(n\right)}\left(x\right)=\sum _{k=0}^{N\left(n\right)}w{\hat{c}}_k{e}_k\left(x\right)}$ for f ∈ C[0,2π] and ${\displaystyle w{\hat{c}}_0,w{\hat{c}}_1,...,w{\hat{c}}_{N\left(n\right)}}$ obtained by the least squares method are studied.