An extension of Kramer’s sampling theorem for not necessarily “bandlimited” signals – the aliasing error. (English) Zbl 0835.94004

Butzer, P. L.; Stens, R. L.

Acta Sci. Math. 60, No. 1-2, 59-69 (1995).

Summary: Kramer’s sampling theorem, a generalization of Shannon’s sampling theorem, states that a function \(f\) which is representable as a finite integral transform can be reconstructed from sample values \(f(t_k)\) in terms of a series expansion with respect to a complete orthogonal set. The aim of this paper is to investigate the error occurring when this expansion is used for a function \(f\) which is representable as an infinite rather than as a finite integral transform. In particular, it is shown that in many applications this error tends to zero when the distance between the sampling points \(t_k\) tends to zero.

MSC:

94A12	Signal theory (characterization, reconstruction, filtering, etc.)
41A25	Rate of convergence, degree of approximation
41A58	Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
34B24	Sturm-Liouville theory

Keywords:

Kramer’s sampling theorem; infinite integral transform; generalized Shannon sampling theorem; expansion