Periodic Splines and Spectral Estimation

The theory of periodic smoothing splines is presented, with application to the estimation of periodic functions. Several theorems relating the order of the differential operator defining the spline to the saturation (order of bias) of the estimator are proven. The linear operator which maps a function to its periodic continuous smoothing spline approximation is represented as a convolution operator with a given convolution kernel. This operator is shown to be the limit of a sequence of operators which map a function into the periodic version of the usual lattice smoothing spline. The convolution kernel above appears as the kernel in a kernel type estimate of the spectral density. Thus, it is shown that, a smoothing spline spectral density estimate, is also asymptotically a kernel type spectral density estimate. Some numerical results are presented.