Smoothing Periodic Curves by a Method of Regularization

Abstract

The present work develops an extension of the theory of polynomial smoothing splines for the statistical problem of estimating a periodic curve based on discrete and noisy observations of the curve itself or of a convolution functional of the curve. These estimates, called periodic $\alpha $-splines, are still of the method of regularization type, with a penalty involving Fourier coefficients. The questions of asymptotic rate of convergence of the integrated mean square error and the relationships with kernel estimates in the case of equispaced design points are addressed.