September  2007, 6(3): 549-567. doi: 10.3934/cpaa.2007.6.549

A theoretical framework for wavelet analysis in a Hermitean Clifford setting

1. 

Clifford Research Group, Department of Mathematical Analysis, Faculty of Engineering, Ghent University, Galglaan 2, 9000 Gent, Belgium, Belgium, Belgium

Received  January 2006 Revised  September 2006 Published  June 2007

Hermitean Clifford analysis focusses on monogenic functions taking values in a complex Clifford algebra or in a complex spinor space. Here monogenicity is expressed by means of two complex mutually adjoint Dirac operators, which are invariant under the action of a representation of the unitary group. In this paper we have further developed the Hermitean theory by introducing so-called zonal functions and by studying plane wave null solutions of the Hermitean Dirac operators. Moreover we have defined new Hermite polynomials in this Hermitean setting and expressed them in terms of the former Clifford-Hermite polynomials and of the one-dimensional Laguerre polynomials. These Hermitean Hermite polynomials are good candidates for mother wavelets in a Hermitean continuous wavelet transformation theory yet to be developed.
Citation: Fred Brackx, Hennie De Schepper, Frank Sommen. A theoretical framework for wavelet analysis in a Hermitean Clifford setting. Communications on Pure & Applied Analysis, 2007, 6 (3) : 549-567. doi: 10.3934/cpaa.2007.6.549
[1]

R. Wong, L. Zhang. Global asymptotics of Hermite polynomials via Riemann-Hilbert approach. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 661-682. doi: 10.3934/dcdsb.2007.7.661

[2]

Bin Han, Qun Mo. Analysis of optimal bivariate symmetric refinable Hermite interpolants. Communications on Pure & Applied Analysis, 2007, 6 (3) : 689-718. doi: 10.3934/cpaa.2007.6.689

[3]

Hilde De Ridder, Hennie De Schepper, Frank Sommen. The Cauchy-Kovalevskaya extension theorem in discrete Clifford analysis. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1097-1109. doi: 10.3934/cpaa.2011.10.1097

[4]

P. Cerejeiras, M. Ferreira, U. Kähler, F. Sommen. Continuous wavelet transform and wavelet frames on the sphere using Clifford analysis. Communications on Pure & Applied Analysis, 2007, 6 (3) : 619-641. doi: 10.3934/cpaa.2007.6.619

[5]

Akram Kheirabadi, Asadollah Mahmoudzadeh Vaziri, Sohrab Effati. Solving optimal control problem using Hermite wavelet. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 101-112. doi: 10.3934/naco.2019008

[6]

Michael Boshernitzan, Máté Wierdl. Almost-everywhere convergence and polynomials. Journal of Modern Dynamics, 2008, 2 (3) : 465-470. doi: 10.3934/jmd.2008.2.465

[7]

Elisavet Konstantinou, Aristides Kontogeorgis. Some remarks on the construction of class polynomials. Advances in Mathematics of Communications, 2011, 5 (1) : 109-118. doi: 10.3934/amc.2011.5.109

[8]

Nathaniel D. Emerson. Dynamics of polynomials with disconnected Julia sets. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 801-834. doi: 10.3934/dcds.2003.9.801

[9]

Jayadev S. Athreya, Gregory A. Margulis. Values of random polynomials at integer points. Journal of Modern Dynamics, 2018, 12: 9-16. doi: 10.3934/jmd.2018002

[10]

Shrihari Sridharan, Atma Ram Tiwari. The dependence of Lyapunov exponents of polynomials on their coefficients. Journal of Computational Dynamics, 2019, 6 (1) : 95-109. doi: 10.3934/jcd.2019004

[11]

Shu-Lin Lyu. On the Hermite--Hadamard inequality for convex functions of two variables. Numerical Algebra, Control & Optimization, 2014, 4 (1) : 1-8. doi: 10.3934/naco.2014.4.1

[12]

Irene I. Bouw, Sabine Kampf. Syndrome decoding for Hermite codes with a Sugiyama-type algorithm. Advances in Mathematics of Communications, 2012, 6 (4) : 419-442. doi: 10.3934/amc.2012.6.419

[13]

Shujuan Lü, Zeting Liu, Zhaosheng Feng. Hermite spectral method for Long-Short wave equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 941-964. doi: 10.3934/dcdsb.2018255

[14]

Janos Kollar. Polynomials with integral coefficients, equivalent to a given polynomial. Electronic Research Announcements, 1997, 3: 17-27.

[15]

Nur Fadhilah Ibrahim. An algorithm for the largest eigenvalue of nonhomogeneous nonnegative polynomials. Numerical Algebra, Control & Optimization, 2014, 4 (1) : 75-91. doi: 10.3934/naco.2014.4.75

[16]

Jean-François Biasse, Michael J. Jacobson, Jr.. Smoothness testing of polynomials over finite fields. Advances in Mathematics of Communications, 2014, 8 (4) : 459-477. doi: 10.3934/amc.2014.8.459

[17]

Anca Radulescu. The connected Isentropes conjecture in a space of quartic polynomials. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 139-175. doi: 10.3934/dcds.2007.19.139

[18]

Ricardo García López. A note on L-series and Hodge spectrum of polynomials. Electronic Research Announcements, 2009, 16: 56-62. doi: 10.3934/era.2009.16.56

[19]

Vladimir Dragović, Katarina Kukić. Discriminantly separable polynomials and quad-equations. Journal of Geometric Mechanics, 2014, 6 (3) : 319-333. doi: 10.3934/jgm.2014.6.319

[20]

Domingo González, Gamaliel Blé. Core entropy of polynomials with a critical point of maximal order. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 115-130. doi: 10.3934/dcds.2019005

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]