The local property of several operators on sampling

Abstract

The main purpose of this article is to investigate the local property of several operators on sampling problems. We find that if the generator of the shift-invariant space decays rapidly, especially if the generator is compactly supported, then the behavior of these operators is well localized, and any signal in the shift-invariant space can be recovered from its discrete samples locally. The results are useful for designing local reconstruction algorithms.

Keywords:

• Sampling
• Local property
• Reproducing kernel
• Local bound estimation
• Oscillation operator
• Local quasi-interpolant
• AMS Subject Classifications: 42A16
• 42A38
• 42C40
• 42C99
• 65G99
• 41A45
• 46E15

Acknowledgments

The author would like to thank professor Wei Lin for his warm-hearted help in correcting the mistakes in this article and the convenience he presented.