半直线伸缩调制框架集

本文研究右半直线平方可积函数空间L^2(R+)中的一类伸缩调制系.实际问题中时间变量不可取负值,L^2(R+)可模拟因果信号空间.但因R+按加法不能作成一个群,它不容许小波与Gabor系.我们研究L^2(R+)中由特征函数生成的伸缩调制系(MD-系)框架,引入了R+中MD-框架集的概念,利用"伸缩等价"与"基数函数"方法刻画了L^2(R+)中MD-Bessel集与完备集;得到了关于MD-Riesz基集的两个充分条件,并证明了通过对MD-Riesz基集进行有限可测分解可得到MD-框架集.

The Dilation-and-Modulation Frame Sets on the Half Real Line

This paper addresses a class of dilation-and-modulation (MD) systems in the space L²(R₊) of square integrable functions defined on the right half real line R₊. In practice, the time variable cannot be negative. L²(R₊) models the causal signal space, but it admits no wavelet and Gabor systems due to R₊ being not a group under addition. We study the dilation-and-modulation systems in L²(R₊) generated by characteristic functions. We introduce the notion of MD-frame sets in R₊. Using "dilation-equivalence" and "cardinality function" methods we characterize MD-Bessel and complete sets; obtain two sufficient conditions for MD-Riesz basis sets; and prove that an arbitrary finite and measurable decomposition of an MD-Riesz basis set leads to an MD-frame set.