Nonparametric methods for deconvolving multiperiodic functions

Multiperiodic functions, or functions that can be represented as finite additive mixtures of periodic functions, arise in problems related to stellar radiation. There they represent the overall variation in radiation intensity with time. The individual periodic components generally correspond to different sources of radiation and have intrinsic physical meaning provided that they can be ‘deconvolved’ from the mixture. We suggest a combination of kernel and orthogonal series methods for performing the deconvolution, and we show how to estimate both the sequence of periods and the periodic functions themselves. We pay particular attention to the issue of identifiability, in a nonparametric sense, of the components. This aspect of the problem is shown to exhibit particularly unusual features, and to have connections to number theory. The matter of rates of convergence of estimators also has links there, although we show that the rate-of-convergence problem can be treated from a relatively conventional viewpoint by considering an appropriate prior distribution for the periods.