On Fourier coefficient estimators consistent in the mean-square sense

The properties of two recursive estimators of the Fourier coefficients of a regression function ${\displaystyle f\in L^2[a,b]}$ with respect to a complete orthonormal system of bounded functions (e_k) , k=1,2,..., are considered in the case of the observation model ${\displaystyle y_i=f\left(x_i\right)+{\eta }_i}$, i=1,...,n , where ${\displaystyle {\eta }_i}$ are independent random variables with zero mean and finite variance, ${\displaystyle x_i\in [a,b]\subset {\left\{symR\right\}}^1}$, i=1,...,n, form a random sample from a distribution with density ϱ =1/(b-a) (uniform distribution) and are independent of the errors ${\displaystyle {\eta }_i}$, i=1,...,n . Unbiasedness and mean-square consistency of the examined estimators are proved and their mean-square errors are compared.