## Sampling theory for functions with fractal spectrum.(English)Zbl 1014.94006

The authors investigate in greater detail a sampling formula given by the second author [J. Anal. Math. 81, 209-238 (2000; Zbl 0976.42020)] for functions whose spectrum lies in a Cantor set $$K$$ of a special type introduced by P. Jorgensen and S. Pedersen [ibid. 75, 185-228 (1998)], where the sampling set is extremely thin and the sampling function is quite different from the usual sinc function. The authors obtain new properties of the sampling function and give approximate descriptions of both local and global behavior of functions with spectrum in $$K$$. Some experimental results are described, and more can be found at http://www.mathlab.cornell.edu/$$\sim$$tillman.

### MSC:

 94C10 Switching theory, application of Boolean algebra; Boolean functions (MSC2010) 28A80 Fractals 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems

Zbl 0976.42020

MATHLAB
Full Text:

### References:

 [1] DOI: 10.1007/BF02475985 · Zbl 0908.42003 [2] DOI: 10.1006/jath.1993.1087 · Zbl 0787.41019 [3] Gulisashvili A., ”On the -dimension and -entropy of classes of functions with fractal spectral sets” · Zbl 0787.41019 [4] DOI: 10.1080/10586458.1992.10504561 · Zbl 0788.65129 [5] DOI: 10.1007/BF02788699 · Zbl 0959.28008 [6] Kuratowski K., Topology (1968) [7] DOI: 10.1512/iumj.1990.39.39038 · Zbl 0695.28003 [8] DOI: 10.2307/2154350 · Zbl 0765.28007 [9] DOI: 10.1512/iumj.1993.42.42018 · Zbl 0790.28003 [10] DOI: 10.1007/BF02788700 · Zbl 0959.28009 [11] DOI: 10.1007/BF02788990 · Zbl 0976.42020 [12] Strichartz R., Math. Res. Lett. 7 pp 317– (2000) · Zbl 0977.42021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.