September  2007, 6(3): 775-787. doi: 10.3934/cpaa.2007.6.775

Refinable functions on the Heisenberg group

1. 

LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, China

2. 

Department of Mathematics and Mechanics, University of Science and Technology, Beijing 100083, China

Received  October 2005 Revised  August 2006 Published  June 2007

The orthonormal wavelets associated with a multiresolution analysis are mainly determined by the corresponding refinable function. In this paper, we study the continuity of refinable functions on the Heisenberg group. The characterization of Lipschitz continuous refinable functions is given in terms of the uniform joint spectral radius. We also give an investigation of the refinable functions in the generalized Lipschitz spaces.
Citation: Heping Liu, Yu Liu. Refinable functions on the Heisenberg group. Communications on Pure & Applied Analysis, 2007, 6 (3) : 775-787. doi: 10.3934/cpaa.2007.6.775
[1]

Fausto Ferrari, Qing Liu, Juan Manfredi. On the characterization of $p$-harmonic functions on the Heisenberg group by mean value properties. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2779-2793. doi: 10.3934/dcds.2014.34.2779

[2]

Sébastien Gouëzel. An interval map with a spectral gap on Lipschitz functions, but not on bounded variation functions. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1205-1208. doi: 10.3934/dcds.2009.24.1205

[3]

Jean-Francois Bertazzon. Symbolic approach and induction in the Heisenberg group. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1209-1229. doi: 10.3934/dcds.2012.32.1209

[4]

C. A. Micchelli, Q. Sun. Interpolating filters with prescribed zeros and their refinable functions. Communications on Pure & Applied Analysis, 2007, 6 (3) : 789-808. doi: 10.3934/cpaa.2007.6.789

[5]

Isabeau Birindelli, J. Wigniolle. Homogenization of Hamilton-Jacobi equations in the Heisenberg group. Communications on Pure & Applied Analysis, 2003, 2 (4) : 461-479. doi: 10.3934/cpaa.2003.2.461

[6]

Xiongping Dai, Yu Huang, Mingqing Xiao. Realization of joint spectral radius via Ergodic theory. Electronic Research Announcements, 2011, 18: 22-30. doi: 10.3934/era.2011.18.22

[7]

Xinjing Wang, Pengcheng Niu, Xuewei Cui. A Liouville type theorem to an extension problem relating to the Heisenberg group. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2379-2394. doi: 10.3934/cpaa.2018113

[8]

L. Brandolini, M. Rigoli and A. G. Setti. On the existence of positive solutions of Yamabe-type equations on the Heisenberg group. Electronic Research Announcements, 1996, 2: 101-107.

[9]

Pablo Ochoa. Approximation schemes for non-linear second order equations on the Heisenberg group. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1841-1863. doi: 10.3934/cpaa.2015.14.1841

[10]

Luis F. López, Yannick Sire. Rigidity results for nonlocal phase transitions in the Heisenberg group $\mathbb{H}$. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2639-2656. doi: 10.3934/dcds.2014.34.2639

[11]

Patrizia Pucci. Critical Schrödinger-Hardy systems in the Heisenberg group. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 375-400. doi: 10.3934/dcdss.2019025

[12]

Victor Kozyakin. Iterative building of Barabanov norms and computation of the joint spectral radius for matrix sets. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 143-158. doi: 10.3934/dcdsb.2010.14.143

[13]

J. M. Peña. Refinable functions with general dilation and a stable test for generalized Routh-Hurwitz conditions. Communications on Pure & Applied Analysis, 2007, 6 (3) : 809-818. doi: 10.3934/cpaa.2007.6.809

[14]

Houda Mokrani. Semi-linear sub-elliptic equations on the Heisenberg group with a singular potential. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1619-1636. doi: 10.3934/cpaa.2009.8.1619

[15]

Pablo Ochoa, Julio Alejo Ruiz. A study of comparison, existence and regularity of viscosity and weak solutions for quasilinear equations in the Heisenberg group. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1091-1115. doi: 10.3934/cpaa.2019053

[16]

Pavel Krejčí, Thomas Roche. Lipschitz continuous data dependence of sweeping processes in BV spaces. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 637-650. doi: 10.3934/dcdsb.2011.15.637

[17]

Anton Petrunin. Harmonic functions on Alexandrov spaces and their applications. Electronic Research Announcements, 2003, 9: 135-141.

[18]

Feng Luo. Geodesic length functions and Teichmuller spaces. Electronic Research Announcements, 1996, 2: 34-41.

[19]

Miklós Horváth. Spectral shift functions in the fixed energy inverse scattering. Inverse Problems & Imaging, 2011, 5 (4) : 843-858. doi: 10.3934/ipi.2011.5.843

[20]

Yuri Berest, Alimjon Eshmatov, Farkhod Eshmatov. On subgroups of the Dixmier group and Calogero-Moser spaces. Electronic Research Announcements, 2011, 18: 12-21. doi: 10.3934/era.2011.18.12

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]