Convergence rates for trigonometric and polynomial-trigonometric regression estimators

Upper bounds are derived for the rates of convergence for trigonometric series regression estimators of an unknown, smooth regression function. The resulting rates depend on the regression function satisfying certain periodic boundary conditions that may not hold in practice. To overcome such difficulties alternative estimators are proposed which are obtained by regression on trigonometric functions and low-order polynomials. These estimators are shown to always be capable of obtaining the optimal rates of convergence over a particular smoothness class of functions, irregardless of whether or not the regression function is periodic.