On the Fourier Spectra of Distributions in Clifford Analysis

Abstract

In recent papers by Brackx, Delanghe and Sommen, some fundamental higher dimensional distributions have been reconsidered in the framework of Clifford analysis, eventually leading to the introduction of four broad classes of new distributions in Euclidean space. In the current paper we continue the in-depth study of these distributions, more specifically the study of their behaviour in frequency space, thus extending classical results of harmonic analysis.

Keywords

Fourier spectra / Distributions / Clifford analysis / 30G35 / 46F10

References

[1] Brackx, F., De Knock, B., De Schepper, H. and Sommen, F., Distributions in Clifford Analysis: an Overview, Clifford Analysis and Applications, S.-L. Eriksson (ed.) (Proceedings of the Summer School, Tampere, August 2004), Tampere University of Technology, Institute of Mathematics, Research Report, No. 82, 2006, 59–73.

[2] Brackx, F., Delanghe, R. and Sommen, F., Clifford Analysis, Pitman Publishers, Boston-London-Melbourne, 1982.

[3] Brackx, F., Delanghe, R. and Sommen, F., Spherical means and distributions in Clifford analysis, Advances in Analysis and Geometry: New Developments Using Clifford Algebra, T. Qian, Th. Hempfling, A. McIntosch and F. Sommen (eds.), Trends in Mathematics, Birkhäuser, Basel, 2004, 65–96.

[4] Brackx Chin. Ann. Math., 2003, 24B: 133

[5] Delanghe Complex Var. Theory Appl., 2002, 47: 653

[6] Brackx Complex Var. Theory Appl., 2004, 49: 1079

[7] Brackx Chin. Ann. Math., 2005, 26B: 1

[8] Brackx Math. Methods Appl. Sci., 2005, 28: 2173

[9] Delanghe, R., Sommen, F. and Souček, V., Clifford Algebra and Spinor-Valued Functions, Kluwer Academic Publ., Dordrecht, 1992.

[10] Gelfand, I. M. and Shilov, G. E., Generalized Functions, Vol. 1, Academic Press, New York, 1964.

[11] Gilbert, J. and Murray, M., Clifford Algebra and Dirac Operators in Harmonic Analysis, Cambridge Univ. Press, Cambridge, 1991.

[12] Gürlebeck, K. and Sprößig, W., Quaternionic and Clifford calculus for physicists and engineers, Mathematical Methods in Practice, Wiley, Chichester, 1997.

[13] Horváth Trans. Amer. Math. Soc., 1950, 82: 52

[14] Kou Methods and Appl. Anal., 2002, 9: 273

[15] Mitrea, M., Clifford Wavelets, Singular Integrals and Hardy Spaces, Lecture Notes in Mathematics, 1575, Springer-Verlag, Berlin, 1994.

[16] Qian, T., Hempfling, Th., McIntosh, A. and Sommen, F., Advances in Analysis and Geometry: New Developments Using Clifford Algebras, Birkh¨auser Verlag, Basel Boston Berlin, 2004.

[17] Ryan Cubo Math. Educ., 2000, 2: 226

[18] Ryan, J. and Struppa, D., Dirac Operators in Analysis, Addison Wesley Longman Ltd., Harlow, 1998.

[19] Schwartz, L., Théorie des Distributions, Hermann, Paris, 1966.

[20] Stein, E. and Weiß, G., Introduction to Fourier Analysis on Euclidean spaces, Princeton Univ. Press, Princeton, 1971.

[21] Sommen, F., Spin groups and spherical means, Clifford Algebras and Their Applications in Mathematical Physics, JSR Chisholm and AK Common (eds.), Nato ASI Series, D. Reidel Publ. Co., Dordrecht, 1985.