## Advanced topics in Shannon sampling and interpolation theory.(English)Zbl 0905.94002

This volume is intended as a companion to the text “Introduction to Shannon sampling and interpolation theory” (Springer Verlag 1991; Zbl 0729.94001). The contributions included here are written by well-known specialists in various areas where sampling and interpolation are widely used. The contents is as follows: (1) Gabor’s signal expansion and its relation to sampling of the sliding-window spectrum (M. J. Baastians); (2) Sampling in optics (F. Gori); (3) A multidimensional extension of Papoulis’ generalized sampling expansion with the application in minimal density sampling (K. F. Cheung); (4) Nonuniform sampling (F. Marvasti); (5) Linear prediction by samples from the past (P. L. Butzer and R. L. Stens); (6) Polar, spiral, and generalized sampling and interpolation (H. Stark); (7) Error analysis in application of the generalizations of the sampling theorem (A. J. Jerri); Bibliography; Index.
The contributions are devoted to problems of different interests and practical aspects. So, the contribution (1) provides an interesting analysis pertaining to Gabor expansion which constitutes a cornerstone of modern signal theory. Some results on the usage of sampling theory in optics are given in (2). It is worth pointing out that in optics some processing operations are more easily performed than in (electrical) signal theory. Papoulis’ extension of the Shannon theorem is discussed in (3). As usual, any discussion on sampling theory must include an analysis of nonuniform sampling; this is done in (4). In (5) a kind of realizability condition for the sampling process is presented. It is not at all a trivial problem: the sampling theorem insures the reconstruction of the signal at the time $$t_0$$ by the use of all samples (in the past as well as in the future), but in reality only past samples are available. The 2-D sampling which is nowadays used in computer tomography is discussed in (6). An excellent analysis with many examples related to errors occurring in sampling processes is given in the final paper (7). The (exhaustive) bibliography including 1003 items, represents the collective work of all contributors.
The book provides an excellent presentation of the modern problems arising in sampling theory and we think that it could stimulate creative ideas to researchers. It is the real merit of R. J. Marks II for bringing together such an excellent group of well-known workers in signal theory.

### MSC:

 94-02 Research exposition (monographs, survey articles) pertaining to information and communication theory 94A12 Signal theory (characterization, reconstruction, filtering, etc.) 41A05 Interpolation in approximation theory 94-06 Proceedings, conferences, collections, etc. pertaining to information and communication theory 00B15 Collections of articles of miscellaneous specific interest

Zbl 0729.94001