# ABSTRACT HARMONIC ANALYSIS OVER SPACES OF COMPLEX MEASURES ON HOMOGENEOUS SPACES OF COMPACT GROUPS

• Received : 2016.05.08
• Accepted : 2017.02.13
• Published : 2017.07.31

#### Abstract

This paper presents a systematic study of the abstract harmonic analysis over spaces of complex measures on homogeneous spaces of compact groups. Let G be a compact group and H be a closed subgroup of G. Then we study abstract harmonic analysis of complex measures over the left coset space G/H.

#### References

1. A. Arefijamaal, On construction of coherent state associated with homogeneous spaces, Turkish J. Math. 34 (2010), no. 4, 515-521.
2. A. Derighetti, Convolution operators on groups, Lecture Notes of the UnioneMatematica Italiana, 11. Springer, Heidelberg; UMI, Bologna, 2011.
3. A. Derighetti, On the multipliers of a quotient group, Bull. Sci. Math. (2) 107 (1983), no. 1, 3-23.
4. J. Dixmier, C*-Algebras, North-Holland and Publishing company, 1977.
5. G. B. Folland, A course in Abstract Harmonic Analysis, CRC press, 1995.
6. A. Ghaani Farashahi, Abstract non-commutative harmonic analysis of coherent state transforms, Ph.D. thesis, Ferdowsi University of Mashhad (FUM), Mashhad, 2012.
7. A. Ghaani Farashahi, Convolution and involution on function spaces of homogeneous spaces, Bull. Malays. Math. Sci. Soc. (2) 36 (2013) no. 4, 1109-1122.
8. A. Ghaani Farashahi, Abstract harmonic analysis of relative convolutions over canonical homogeneous spaces of semidirect product groups, J. Aust. Math. Soc. 101 (2016), no. 2, 171-187. https://doi.org/10.1017/S1446788715000798
9. A. Ghaani Farashahi, Trigonometric polynomials over homogeneous spaces of compact groups, Adv. Oper. Theory 2 (2017), no. 1, 87-97.
10. A. Ghaani Farashahi, Abstract harmonic analysis of wave-packet transforms over locally compact abelian groups, Banach J. Math. Anal. 11 (2017), no. 1, 50-71. https://doi.org/10.1215/17358787-3721281
11. A. Ghaani Farashahi, Abstract Plancherel (trace) formulas over homogeneous spaces of compact groups, Canadian Math. Bulletin: doi:10.4153/CMB-2016-037-6. https://doi.org/10.4153/CMB-2016-037-6
12. A. Ghaani Farashahi, Abstract relative Fourier transforms over canonical homogeneous spaces of semi-direct product groups with Abelian normal factor, J. Korean Math. Soc. 54 (2017), no. 1, 117-139. https://doi.org/10.4134/JKMS.j150610
13. A. Ghaani Farashahi, A class of abstract linear representations for convolution function algebras over homogeneous spaces of compact groups, Canadian J. Math.: http://dx.doi.org/10.4153/CJM-2016-043-9. https://doi.org/10.4153/CJM-2016-043-9
14. A. Ghaani Farashahi, Abstract convolution function algebras over homogeneous spaces of compact groups, Illinois J. Math., to appear.
15. A. Ghaani Farashahi, Abstract operator-valued Fourier transforms over homogeneous spaces of compact groups, Groups, Geometry, Dynamics, to appear.
16. E. Hewitt and K. A. Ross, Absrtact Harmonic Analysis. Vol 1, 1963.
17. E. Hewitt and K. A. Ross, Absrtact Harmonic Analysis. Vol 2, 1970.
18. V. Kisil, Relative convolutions. I. Properties and applications, Adv. Math. 147 (1999), no. 1, 35-73. https://doi.org/10.1006/aima.1999.1833
19. V. Kisil, Operator covariant transform and local principle, J. Phys. A 45 (2012), no. 24, 244022, 10 pp.
20. V. Kisil, Geometry of Mobius Transformations. Elliptic, Parabolic and Hyperbolic Actions of $SL_2({\mathbb{R}})$, Imperial College Press, London, 2012.
21. V. Kisil, Erlangen program at large: an overview. Advances in applied analysis, 1-94, Trends Math., Birkhauser/Springer Basel AG, Basel, 2012.
22. V. Kisil, Calculus of operators: covariant transform and relative convolutions, Banach J. Math. Anal. 8 (2014), no. 2, 156-184. https://doi.org/10.15352/bjma/1396640061
23. G. J. Murphy, C*-Algebras and Operator theory, Academic Press, INC.
24. H. Reiter and J. D. Stegeman, Classical Harmonic Analysis, 2nd Ed, Oxford University Press, New York, 2000.