Reconstruction of signals in L p (ℝ)-space by generalized sampling series based on linear combinations of B-splines†

Abstract

In her plenary lecture [Daubechies, I., 2005, Redundancy and robust A/D and D/A conversion (see ref. 28). SampTA 05, Sampling Theory and Applications. Proceedings of International Workshop, Samsun, Turkey, 2005 (Samsun: Ondokuz Mayis University, Department of Mathematics), p. 18.] at the workshop SampTa 05, Samsun, Turkey, July 2005, Daubechies, Princeton, presented three major reasons as to why the classical sampling theorem was fully impractical for real-life signal processing. One needs infinitely many samples extended over the whole real axis, the sinc-kernel decaying much too slowly, and band limitation is a too restrictive assumption. This paper presents an approach to overcome these difficulties, which actually began to develop at Aachen 1977. The sinc-function is replaced by certain simple linear combinations of B-splines, only a finite number of samples need be available. This approach can be used to process arbitrarily continuous and even discontinuous signals. Best possible error estimates in terms of the L p -average modulus of smoothness are presented. Three typical examples exhibiting the various problems involved are worked out in detail. †This paper is based in part on a plenary lecture held by the first-named author at the International Conference ‘Analytical Methods of Analysis and Differential Equations (AMADE—2006)’, 13—19 September 2006, Minsk, Belarus.