A SPECTRAL REPRESENTATION FOR THE CLASS OF BAND-LIMITED FUNCTIONS

In this paper, we give an optimal estimation procedure for the class o f band-limited signals. Wegman (1984) suggested a generalized framewor k for optimal nonparametric function estimation. This framework involv es the specification of a class of admissible functions and the specif ication of a convex objective functional- In this paper, we propose an admissible space of estimators for the class of band-limited broadban d signals as a subspace of an appropriate Hilbert space. Specifically, the property of absolute continuity is built into this subspace, whic h we model as a Sobolev space. We also propose an objective functional which contains a penalty functional for out-of-band energy. It is sho wn that under this setting, the optimal estimator for the class of bro adband signals is a subclass of generalized L-spline functions. Becaus e certain classes of wavelets span the Sobolev space, the optimal solu tion may be written in terms of a wavelet basis. The signal estimation problem is close analog to the nonparametric regression problem; Euba nk (1988) is an excellent source for a general treatment of splines an d their use in the statistical nonparametric regression setting. Our t reatment here is based on functional analytic framework; Hutson and Py m (1980) is an excellent general reference for the Hilbert space and f unctional analytic discussions in this paper.