Continuous wavelet design is the endeavor to construct mother wavelets with desirable properties for the continuous wavelet transform (CWT). One class of methods for choosing a mother wavelet involves minimizing a functional, called the wavelet uncertainty functional. Recently, two new wavelet uncertainty functionals were derived from theoretical foundations. In both approaches, the uncertainty of a mother wavelet describes its concentration, or accuracy, as a time-scale probe. While an uncertainty minimizing mother wavelet can be proven to have desirable localization properties, the existence of such a minimizer was never studied. In this paper, we prove the existence of minimizers for the two uncertainty functionals.
翻译:连续波子设计是努力为连续波子变换(CWT)构建具有理想特性的母波子。 选择母波子的一种方法是将功能最小化,称为波子不确定性功能。 最近,两个新的波子不确定性功能来自理论基础。 在这两种方法中,母波子的不确定性都将其集中性或准确性描述为时间尺度的探测。 虽然将母波子最小化的不确定性可以证明具有理想的本地化特性,但从未研究过这种最小化装置的存在。 在本文中,我们证明存在两种不确定性功能的最小化装置。