×

zbMATH — the first resource for mathematics

Structural transition between \(L^{p}(G)\) and \(L^{p}(G/H)\). (English) Zbl 1314.43005
This paper is devoted to present a systematic and straightforward extension of results derived in [H. Reiter and J. D. Stegeman, Classical harmonic analysis and locally compact groups. 2nd ed. Oxford: Clarendon Press (2000; Zbl 0965.43001)], [A. G. Farashahi, Bull. Malays. Math. Sci. Soc. (2) 36, No. 4, 1109–1122 (2013; Zbl 1291.43003)] and [A. Ghaani Farashahi and R. A. Kamyabi-Gol, “Frames and homogeneous spaces” (English, Persian summary), J. Sci. Islam. Repub. Iran 22, No. 4, 355–361 (2011)], when the measure on the homogeneous space is not totally invariant under the group action. The idea of their work originated from a natural normalization of the linear operator \(T=T_H\), which is called \(T_p\), with respect to the rho-function and each \(L^p\)-space.
The only interesting results of the article, which are not straightforward from the results of the above mentioned papers, are Theorem 3.4 and Proposition 4.3. Theorem 3.4 presents \(L^\infty\)-extension of the linear operator \(P=P_H\) and Proposition 4.3 gives a characterization for the adjoint operator \(T_p^\ast\).
MSC:
43A85 Harmonic analysis on homogeneous spaces
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
46B25 Classical Banach spaces in the general theory
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] O. Christensen, Frames and Bases: An Introductory Course , Springer, Boston, 2008. · Zbl 1152.42001
[2] C.H. Chu and A.T.M. Lau, Jordan structures in harmonic functions and Fourier algebras on homogeneous spaces , Math. Ann. 336 (2006), no. 4, 803-840. · Zbl 1108.43009 · doi:10.1007/s00208-006-0013-y
[3] F. Esmaeelzadeh and R.A. Kamyabi-Gol, Admissible wavelets on groups and their homogeneous spaces , Pure Math. Sci. 3 (2014), no. 1, 1-8.
[4] G.B. Folland, A Course in Abstract Harmonic Analysis , CRC Press, Boca Raton, 1995. · Zbl 0857.43001
[5] B.E. Forrest, Fourier analysis on coset space , Rocky Mountain J. Math. 28 (1998), no. 1. · Zbl 0922.43007 · doi:10.1216/rmjm/1181071828 · math.la.asu.edu
[6] B.E. Forrest, E. Samei and N. Spronk Convolutions on compact groups and Fourier algebras of coset spaces , Studia Math. 196 (2010), no. 3, 223-249. · Zbl 1210.43003 · doi:10.4064/sm196-3-2 · arxiv:0705.4277
[7] A. Ghaani Farashahi and R.A. Kamyabi-Gol, Frames and homogeneous spaces , J. Sci. Islam. Repub. Iran 22 (2011), no. 4, 355-361.
[8] R.A. Kamyabi-Gol and N. Tavallaei, Wavelet transforms via generalized quasi-regular representations , Appl. Comput. Harmon. Anal. 26 (2009), no. 3, 291-300 · Zbl 1184.42031 · doi:10.1016/j.acha.2008.07.001
[9] E. Kaniuth, Weak spectral synthesis in Fourier algebras of coset spaces , Studia Math. 197 (2010), no. 3, 229-246. · Zbl 1196.43005 · doi:10.4064/sm197-3-2
[10] V.V. Kisil, Calculus of operators: covariant transform and relative convolutions , Banach J. Math. Anal. 8 (2014), no. 2, 156-184. · Zbl 1305.43009 · doi:10.15352/bjma/1396640061 · euclid:bjma/1396640061 · arxiv:1304.2792
[11] K. Parthasarathy and N. Shravan Kumar, Fourier algebras on homogeneous spaces , Bull. Sci. Math. 135 (2011), no. 2, 187-205. · Zbl 1219.43004 · doi:10.1016/j.bulsci.2010.03.003
[12] H. Reiter and J.D. Stegeman, Classical Harmonic Analysis and Locally Compact Groups , Oxford Science Publications, Clarendon Press, New York, 2000. · Zbl 0965.43001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.