No abstract

This paper studies uncertainty principles for doubly periodic functions. Firstly, the means and the variances of time and of frequency for doubly periodic signal functions are provided. Also, the phase and the amplitude derivatives of doubly periodic signal functions are properly defined. Based on these definitions, we establish two versions of uncertainty principles. An example is presented to illustrate these results.

No abstract

<abstract><p>For a Jordan curve $ \Gamma $ in the complex plane, its constant distance boundary $ \Gamma_ \lambda $ is an inflated version of $ \Gamma $. A flatness condition, $ (1/2, r_0) $-chordal property, guarantees that $ \Gamma_ \lambda $ is a Jordan curve when $ \lambda $ is not too large. We prove that $ \Gamma_ \lambda $ converges to $ \Gamma $, as $ \lambda $ approaching to $ 0 $, in the sense of Hausdorff distance if $ \Gamma $ has the $ (1/2, r_0) $-chordal property.</p></abstract>

The classical uncertainty principle works for smooth signal functions. In our work, we apply the Fourier transform derivatives for the study of uncertainty principle, so that the smoothness condition for the signal functions is not required. At first, the amplitude and phase derivatives of vector-valued signal functions based on the Fourier transform are defined. Then we obtain a strong form of the uncertainty principle.