Fourier transform and Plancherel Theorem for Nilpotent Lie Group

  IJMTT-book-cover
 
International Journal of Mathematical Trends and Technology (IJMTT)          
 
© 2013 by IJMTT Journal
Volume-4 Issue-11                           
Year of Publication : 2013
Authors : Kahar El-Hussein

MLA

Kahar El-Hussein."Fourier transform and Plancherel Theorem for Nilpotent Lie Group"International Journal of Mathematical Trends and Technology (IJMTT),V4(11):288-294 2013. Published by Seventh Sense Research Group.

Abstract
As will known the connected and simply connected nilpotent Lie group N has an important role in quantum mechanics. In this paper we show how the Fourier transform on the n  dimensional vector Lie group n R can be generalized to N in order to obtain the Plancherel theorem. In addition we define the Fourier transform for the subgroup NA = A  N of the real semi-simple Lie group SL(n,R) to get also the Plancherel formula for NA

References

[1] K. El- Hussein, A Fundamental Solution of an Invariant Differential Operator on the Heisenberg Group, Mathematical Forum, 4, no. 12, 601 - 612. 2009
[2] K. El- Hussein, Eigendistributions for the Invariant Differential operators on the Affine Group. Int. Journal of Math. Analysis, Vol. 3, no. 9, 419-429. 2009
[3] K. El- Hussein, Fourier transform and invariant differential operators on the solvable Lie group G4, in Int. J. Contemp. Maths Sci. 5. No. 5-8, 403-417. 2010
[4] K. El- Hussein, On the left ideals of group algebra on the affine group, in Int. Math Forum, Int, Math. Forum 6, No. 1-4, 193-202. 2011
[5] K. El- Hussein, Note on the Solvability of the Mizohata Operator, International Mathematical Forum, 5, no. 37, 1833 - 1838. 2010
[6] K. El- Hussein, Non Commutative Fourier Transform on Some Lie Groups and Its Application to Harmonic Analysis, International Journal of Engineering Research & Technology (IJERT) Vol. 2 Issue 10, 2429- 2442. 2013.
[7] K. El- Hussein, Abstract Harmonic Analysis and Ideals of Banach Algebra on 3-Step Nilpotent Lie Groups, International Journal of Engineering Research & Technology (IJERT), Vol. 2 Issue 11, November - 2013
[8] S. Helgason, The Abel, Fourier and Radon Transforms on Symmetric Spaces. Indagationes Mathematicae. 16, 531-551. 2005
[9] W.Rudin, Fourier Analysis on Groups, Interscience Publishers, New York, NY. 1962.

Keywords
Nilpotent Lie Group, Semi-simple Lie Group , Fourier Transform and Plancherel Theorem