Abstract Coherent State Transforms Over Homogeneous Spaces of Compact Groups
Tóm tắt
This paper presents theoretical aspects of a unified generalization for the abstract theory of coherent state/voice transforms over homogeneous spaces of compact groups using operator theory. Let G be a compact group and H be a closed subgroup of G. Let G/H be the left coset space of H in G and
$$\mu $$
be the normalized G-invariant measure on G/H associated to the Weil’s formula with respect to the probability measures of G, H. Let
$$(\pi ,\mathcal {H}_\pi )$$
be a continuous unitary representation of G with non-zero mean over H. In this article, we introduce the generalized notion of coherent state/voice transform associated to
$$\pi $$
on the Hilbert function
$$L^2(G/H,\mu )$$
. We then study basic analytic properties of these transforms.
Tài liệu tham khảo
Ali, S.T., Antoine, J.P., Gazeau, J.P.: Coherent States, Wavelets and Their Generalizations. Springer, New York (2000)
Ali, S.T., De Bivre, S.: Coherent State and Quantization on Homogeneous Spaces, Group Theoretical Methods in Physics (Varna, 1987). Lecture Notes in Phys., vol. 313, pp. 201–207. Springer, Berlin (1988)
Ali, S.T., Antoine, J.P., Gazeau, J.P.: Square integrability of group representations on homogeneous spaces. II. Coherent and quasi-coherent state. The case of the Poincar group. Ann. Henri Poincaré. 55(4), 857–890 (1991)
Ali, S.T., Antoine, J.P., Gazeau, J.P.: Square integrability of group representations on homogeneous spaces. I. Reproducing triples and frames. Ann. Henri Poincaré. 55(4), 829–855 (1991)
Arefijamaal, A., Zekaee, E.: Signal processing by alternate dual Gabor frames. Appl. Comput. Harmon. Anal. 35(3), 535–540 (2013)
Arefijamaal, A., Kamyabi-Gol, R.A.: On the square integrability of quasi regular representation on semidirect product groups. J. Geom. Anal. 19(3), 541–552 (2009)
Dahlke, S., Fornasier, M., Rauhut, H., Steidl, G., Teschke, G.: Generalized coorbit theory, Banach frames, and the relation to \(\alpha \)-modulation spaces. Proc. Lond. Math. Soc. 96(2), 464–506 (2008)
Folland, G.B.: A Course in Abstract Harmonic Analysis. CRC Press, Boca Raton (1995)
Feichtinger, H.G., Gröchenig, K.H.: Banach spaces related to integrable group representations and their atomic decompositions. I. J. Funct. Anal. 86(2), 307–340 (1989)
Feichtinger, H.G., Gröchenig, K.H.: Banach spaces related to integrable group representations and their atomic decompositions. II. Monatsh. Math. 108(2–3), 129–148 (1989)
Führ, H.: Abstract Harmonic Analysis of Continuous Wavelet Transforms. Springer, New York (2005)
Gazeau, J.P.: Coherent States in Quantum Physics. Wiley, New York (2009)
Ghaani Farashahi, A.: Abstract harmonic analysis of wave packet transforms over locally compact abelian groups. Banach J. Math. Anal. 11(1), 50–71 (2017). doi:10.1215/17358787-3721281
Ghaani Farashahi, A.: Abstract operator-valued Fourier transforms over homogeneous spaces of compact groups. Gr. Geom. Dyn. (to appear)
Ghaani Farashahi, A.: Abstract convolution function algebras over homogeneous spaces of compact groups. Ill. J. Math. 59(4), 1025–1042 (2015). http://projecteuclid.org/euclid.ijm/1488186019
Ghaani Farashahi, A.: Abstract Plancherel (trace) formulas over homogeneous spaces of compact groups. Can. Math. Bull. doi:10.4153/CMB-2016-037-6
Ghaani Farashahi, A.: Abstract harmonic analysis of relative convolutions over canonical homogeneous spaces of semidirect product groups. J. Aust. Math. Soc. (2016). doi:10.1017/S1446788715000798
Ghaani Farashahi, A.: Abstract relative Fourier transforms over canonical homogeneous spaces of semi-direct product groups with Abelian normal factor. J. Korean Math. Soc. (2016). doi:10.4134/JKMS.j150610
Ghaani Farashahi, A.: Abstract non-commutative harmonic analysis of coherent state transforms, Ph.D. thesis, Ferdowsi University of Mashhad (FUM), Mashhad (2012)
Grossmann, A., Morlet, J., Paul, T.: Transforms associated to square integrable group representations I. General results. J. Math. Phys. 26(10), 2473–2479 (1985)
Hewitt, E., Ross, K.A.: Absrtact Harmonic Analysis, vol. 1. Springer, Berlin (1963)
Hewitt, E., Ross, K.A.: Absrtact Harmonic Analysis, vol. 2. Springer, Berlin (1970)
Kisil, V.: The real and complex techniques in harmonic analysis from the point of view of covariant transform. Eurasian Math. J. 5(1), 95–121 (2014)
Kisil, V.: Calculus of operators: covariant transform and relative convolutions. Banach J. Math. Anal. 8(2), 156–184 (2014)
Kisil, V.: Advances in applied analysis. Erlangen program at large: an overview. Trends Math., pp. 1–94. Birkhäuser/Springer Basel AG, Basel (2012)
Kisil, V.: Operator covariant transform and local principle. J. Phys. A 45(24), 244022, 10 pp (2012)
Kisil, V.: Wavelets beyond admissibility, progress in analysis and its applications, pp. 219–225. World Sci. Publ, Hackensack (2010)
Murphy, G.J.: C*-Algebras and Operator theory. Academic Press, INC, London (1990)
Reiter, H., Stegeman, J.D.: Classical Harmonic Analysis, 2nd edn. Oxford University Press, New York (2000)