Pitt's inequality and the uncertainty principle associated with the quaternion Fourier transform

被引:84
作者
Chen, Li-Ping [1 ]
Kou, Kit Ian [2 ]
Liu, Ming-Sheng [1 ]
机构
[1] S China Normal Univ, Sch Math Sci, Guangzhou 510631, Guangdong, Peoples R China
[2] Univ Macau, Fac Sci & Technol, Dept Math, Taipa, Macao, Peoples R China
关键词
Quaternion Fourier transform; Pitt's inequality; Logarithmic uncertainty estimate; Uncertainty principle; PHASE-SPACE; DOMAINS;
D O I
10.1016/j.jmaa.2014.10.003
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
The quaternion Fourier transform - a generalized form of the classical Fourier transform has been shown to be a powerful analyzing tool in image and signal processing. This paper investigates Pitt's inequality and uncertainty principle associated with the two-sided quaternion Fourier transform. It is shown that by applying the symmetric form f = f(1)+if(2)+f(3)j+if(4)j of quaternion from Hitzer and the novel module or LP-norm of the quaternion Fourier transform f, then any nonzero quaternion signal and its quaternion Fourier transform cannot both be highly concentrated. Two part results are provided, one part is Heisenberg Weyl's uncertainty principle associated with the quaternion Fourier transform. It is formulated by using logarithmic estimates which may be obtained from a sharp of Pitt's inequality; the other part is the uncertainty principle of Donoho and Stark associated with the quaternion Fourier transform. (C) 2014 Elsevier Inc. All rights reserved.
引用
收藏
页码:681 / 700
页数:20
相关论文
共 35 条
[1]  
[Anonymous], 2001, Analysis
[2]   NONORTHOGONAL DOMAINS IN PHASE-SPACE OF QUANTUM OPTICS AND THEIR RELATION TO FRACTIONAL FOURIER-TRANSFORMS [J].
AYTUR, O ;
OZAKTAS, HM .
OPTICS COMMUNICATIONS, 1995, 120 (3-4) :166-170
[3]   An uncertainty principle for quaternion Fourier transform [J].
Bahri, Mawardi ;
Hitzer, Eckhard S. M. ;
Hayashi, Akihisa ;
Ashino, Ryuichi .
COMPUTERS & MATHEMATICS WITH APPLICATIONS, 2008, 56 (09) :2398-2410
[4]  
Bas P, 2003, 2003 IEEE INTERNATIONAL CONFERENCE ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL III, PROCEEDINGS, P521
[5]   Quaternion fourier descriptors for the preprocessing and recognition of spoken words using images of spatiotemporal representations [J].
Bayro-Corrochano, Eduardo ;
Trujillo, Noel ;
Naranjo, Michel .
JOURNAL OF MATHEMATICAL IMAGING AND VISION, 2007, 28 (02) :179-190
[6]   PITTS INEQUALITY AND THE UNCERTAINTY PRINCIPLE [J].
BECKNER, W .
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 1995, 123 (06) :1897-1905
[7]  
Beckner W, 2008, P AM MATH SOC, V136, P1871
[8]  
Bialynicki-Birula I, 2007, AIP CONF PROC, V889, P52
[9]  
BIALYNICKIBIRULA I, 1985, LECT NOTES MATH, V1136, P90
[10]   The uncertainty principles of windowed wave functions [J].
Cohen, L .
OPTICS COMMUNICATIONS, 2000, 179 (1-6) :221-229