Discrete-transform approach to deconvolution problems

If Fourier series are used as the basis for inference in deconvolution problems, the effects of the errors factorise out in a way that is easily exploited empirically. This property is the consequence of elementary addition formulae for sine and cosine functions, and is not readily available when one is using methods based on other orthogonal series or on continuous Fourier transforms. It allows relatively simple estimators to be constructed, founded on the addition of finite series rather than on integration. The performance of these methods can be particularly effective when edge effects are involved, since cosine-series estimators are quite resistant to boundary problems. In this context we point to the advantages of trigonometric-series methods for density deconvolution; they have better mean squared error performance when edge effects are involved, they are particularly easy to code, and they admit a simple approach to empirical choice of smoothing parameter, in which a version of thresholding, familiar in wavelet-based inference, is used in place of conventional smoothing. Applications to other deconvolution problems are briefly discussed.