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Abstract:
In practice, the time variable cannot be negative. The space L-2(R+) of square integrable functions defined on the right half real line R+ models the causal signal space. This paper focuses on a class of dilationand-modulation systems in L-2(R+). The density theorem for Gabor systems in L-2 (R) states a necessary and sufficient condition for the existence of complete Gabor systems or Gabor frames in L-2 (R) in terms of the index set alone-independently of window functions. The space L-2(R+) admits no nontrivial Gabor system since R+ is not a group according to the usual addition. In this paper, we introduce a class of dilation-and-modulation systems in L-2 (R+) and the notion of T-transform matrix. Using the T-transform matrix method we obtain the density theorem of the dilation-and-modulation systems in L-2 (R+) under the condition that logb a is a positive rational number, where a and b are the dilation and modulation parameters respectively. Precisely, we prove that a necessary and sufficient condition for the existence of such a complete dilation-and-modulation system or dilation-and-modulation system frame in L-2 (R+) is that log(b) a <= 1. Simultaneously, we obtain a T-transform matrix-based expression of all complete dilation-and-modulation systems and all dilation-and-modulation system frames in L-2(R+).
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Source :
RESULTS IN MATHEMATICS
ISSN: 1422-6383
Year: 2019
Issue: 4
Volume: 74
2 . 2 0 0
JCR@2022
ESI Discipline: MATHEMATICS;
ESI HC Threshold:54
JCR Journal Grade:2
Cited Count:
WoS CC Cited Count: 8
SCOPUS Cited Count: 9
ESI Highly Cited Papers on the List: 0 Unfold All
WanFang Cited Count:
Chinese Cited Count:
30 Days PV: 3
Affiliated Colleges: