A spectral representation for the class of band-limited functions

Abstract

In this paper, we give an optimal estimation procedure for the class of band-limited signals. Wegman (1984) suggested a generalized framework for optimal nonparametric function estimation. This framework involves the specification of a class of admissible functions and the specification of a convex objective functional. In this paper, we propose an admissible space of estimators for the class of band-limited broadband signals as a subspace of an appropriate Hilbert space. Specifically, the property of absolute continuity is built into this subspace, which we model as a Sobolev space. We also propose an objective functional which contains a penalty functional for out-of-band energy. It is shown that under this setting, the optimal estimator for the class of broadband signals is a subclass of generalized L-spline functions. Because certain classes of wavelets span the Sobolev space, the optimal solution may be written in terms of a wavelet basis. The signal estimation problem is close analog to the nonparametric regression problem; Eubank (1988) is an excellent source for a general treatment of splines and their use in the statistical nonparametric regression setting. Our treatment here is based on functional analytic framework; Hutson and Pym (1980) is an excellent general reference for the Hilbert space and functional analytic discussions in this paper.