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

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