Consistency of a least squares orthonormal series estimator for a regression function

This paper establishes the almost. sure consistency of least. squares regression series estimators, in the L2-norm and the sup-norm, under very large assumptions on the underlying model. Three examples are considered in order to illustrate the general results: trigonometric series, Legendre polynomials and wavelet. series estimators. Then optimal choices for the number of functions in the series are discussed and convergence rates are derived. It is shown that. for the wavelet. case, the best. possible convergence rate is attained.