Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T17:42:10.876Z Has data issue: false hasContentIssue false
  • English
  • Français

Vector valued mean-periodic functions on groups

Part of: Abstract harmonic analysis Harmonic analysis in one variable

Published online by Cambridge University Press:  09 April 2009

P. Devaraj  and
Inder K. Rana
P. Devaraj
Affiliation:
Department of Mathematics, Indian Institute of Technology, Powai, Mumbai-76, PIN-400076 India e-mail: devaraj@math.iitb.ac.in
Inder K. Rana
Affiliation:
Department of Mathematics, Indian Institute of Technology, Powai, Mumbai-76, PIN-400076 India e-mail: ikr@math.iitb.ac.in
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let G be a locally compact Hausdorif abelian group and X be a complex Banach space. Let C(G, X) denote the space of all continuous functions f: G → X, with the topology of uniform convergence on compact sets. Let X′ denote the dual of X with the weak* topology. Let Mc(G, X′) denote the space of all X′-valued compactly supported regular measures of finite variation on G. For a function f ∈ C(G, X) and μ ∈ Mc(G, X′), we define the notion of convolution f * μ. A function f ∈ C(G, X) is called mean-periodic if there exists a non-trivial measure μ ∈ Mc(G, X′) such that f * μ = 0. For μ ∈ Mc(G, X′), let M P(μ) = {f ∈ C(G, X): f * μ = 0} and let M P(G, X) = ∪μ M P(μ). In this paper we analyse the following questions: Is M P(G, X) ≠ 0? Is M P(G, X) ≠ C(G, X)? Is M P(G, X) dense in C(G, X)? Is M P(μ) generated by ‘exponential monomials’ in it? We answer these questions for the groups G = ℝ, the real line, and G = T, the circle group. Problems of spectral analysis and spectral synthesis for C(ℝ, X) and C(T, X) are also analysed.

Keywords

Convolution of vector valued functionsspectrumvector valued mean-periodic functionsspectral synthesis

MSC classification

Secondary: 43A45: Spectral synthesis on groups, semigroups, etc. 42A75: Classical almost periodic functions, mean periodic functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 72 , Issue 3 , June 2002 , pp. 363 - 388
Copyright
Copyright © Australian Mathematical Society 2002

References

[1] Anselone, P. M. and Korevaar, J., ‘Translation invariant subspaces of finite dimension', Proc. Amer. Math. Soc. 15 (1964), 747–752.Google Scholar
[2] Bagchi, S. C. and Sitaram, A., ‘Spherical mean-periodic functions on semi simple Lie groups', Pacific J. Math. 84 (1979), 241–250.CrossRefGoogle Scholar
[3] Berenstein, C. A. and Taylor, B. A., ‘Mean-periodic functions', Internat. J. Math. Math. Sei. 3 (1980), 199–235.Google Scholar
[4] Brown, L., Schreiber, B. M. and Taylor, B. A., ‘Spectral synthesis and the Pompeiu problem', Ann. Inst. Fourier (Grenoble) 23 (1973), 125–154.Google Scholar
[5] Delsarte, J., ‘Les fonctions moyenne-périodiques', J. Math. Pures Appl. 14 (1935), 403–453.Google Scholar
[6] Devaraj, P. and Rana, I. K., ‘Relation between Pompeiu groups and mean-periodic groups', preprint, 2000.Google Scholar
[7] Dickson, D. G., ‘Analytic mean-periodic functions', Trans. Amer. Math. Soc. 14 (1972), 361–374.Google Scholar
[8] Diestel, J. and Uhl, J. J. Jr, Vector measures. Math. Surveys Monographs 15 (Amer. Math. Soc., Providence RI, 1977).Google Scholar
[9] Dugunji, J., Topology (Prentice-Hall, New Delhi, 1975).Google Scholar
[10] Ehrenpreis, L., ‘Appendix to the paper ‘Mean-periodic functions I”, Amer. J. Math. 11 (1955), 731–733.Google Scholar
[11] Ehrenpreis, L., ‘Mean-periodic functions, Part I. Varieties whose annihilator ideals are principal', Amer. J. Math. 11 (1955), 293–328.Google Scholar
[12] Elliott, R. J., ‘Some results in spectral synthesis', Proc. Camb. Phil. Soc. 61 (1965), 395–424.Google Scholar
[13] Elliott, R. J., ‘Two notes on spectral synthesis for discrète abelian groups', Proc. Camb. Phil. Soc. 61 (1965), 617–620.CrossRefGoogle Scholar
[14] Engert, M., ‘Finite dimensional translation invariant subspaces', Pacific J. Math. 32 (1970), 333–343.Google Scholar
[15] Gilbert, J. E., ‘Spectral synthesis problems for invariant subspaces on groups II', in: Proc. Int. Sym. on Function Algebras at Tulane Univ. (1965) pp. 257–264.Google Scholar
[16] Gilbert, J. E., ‘Spectral synthesis problems for invariant subspaces on groups I', Amer. J. Math. 88 (1966), 626–635.Google Scholar
[17] Gurevich, D. L., ‘Counter examples to a problem of L. Schwanz'. Funct. Anal. Appl. 9 (1975), 116–120.CrossRefGoogle Scholar
[18] Hille, E. and Phillips, R. S., Functional analysis and semi-groups. Amer. Math. Soc. Colloq. Publ. 21 (Amer. Math. Soc, Providence, RI, 1957).Google Scholar
[19] Kahane, J. P., Lectures on mean-periodic functions (Tata Institute, 1957).Google Scholar
[20] Koosis, P., ‘On functions which are mean-periodic on a half-line', Comm. Pure Appl. Math. 10 (1957), 133–149.CrossRefGoogle Scholar
[21] Ladas, G. E. and Lakshmikantham, V., Differential équations in abstract spaces (Académie Press, New York, 1972).Google Scholar
[22] Laird, P. G., ‘Some properties of mean-periodic fonctions', J. Austral. Math. Soc. 14 (1972), 424–432.Google Scholar
[23] Laird, P. G., ‘Functional differential équations and continuous mean-periodic functions', J. Math. Anal. Appl. 47 (1974), 406–423.Google Scholar
[24] Laird, P. G., ‘Entire mean-periodic functions', Canad. J. Math. 17 (1975), 805–818.Google Scholar
[25] Laird, P. G., ‘On characterisations of exponential polynomials', Pacific J. Math. 80 (1979), 503–507.Google Scholar
[26] Lefranc, M., ‘L’ analysis harmonique dans Zn, C. R. Acad. Sei. Paris 246 (1958), 1951-1953.Google Scholar
[27] Levitan, B. M. and Zhikov, V. V., Almost periodic functions and differential équations (Cambridge University Press, Cambridge, 1982).Google Scholar
[28] Malgrange, B., ‘Sur quelques propriétés des équations des convolution', C. R. Acad. Sei. Paris 238 (1954), 2219-2221.Google Scholar
[29] Meril, A., ‘Analytic functions with unbounded carriers and mean-periodic functions', Trans. Amer. Math. Soc. 278 (1983), 115–136.Google Scholar
[30] Meyer, Y., ‘Harmonie analysis of mean-periodic functions', in: Studies in harmonie analysis, MAA Stud. Math. 13 (Math. Assoc. Amer., Washington D.C., 1976) pp. 151–160.Google Scholar
[31] Novak, E. and Rana, I. K., ‘On the unsmoothing of functions on the real une', Proc. Nede. Acad. Sei. Ser. A 89 (1986), 201–207.Google Scholar
[32] Rana, I. K., ‘Unsmoothing over balls via plane wave décomposition'. Rend. Cire. Mat. Palermo (2) 34 (1990), 217–234.Google Scholar
[33] Rana, I. K. and Gowri, N., ‘Integrable mean-periodic functions on locally compact abelian groups', Proc. Amer. Math. Soc. 117 (1993), 405–410.Google Scholar
[34] Schmets, J., Spaces ofvector valued continuous functions. Lecture Notes in Math. 1003 (Springer, 1983).Google Scholar
[35] Schwanz, L., ‘Theorie générale des fonctions moyenne-périodiques', Ann. of Math. (2) 48 (1947), 857–929.Google Scholar
[36] Shapiro, H. S., ‘The expansions of mean-periodic functions in séries of exponentials', Comm. Pure Appl. Math. 11 (1958), 1–21.Google Scholar
[37] Székelyhidi, L., ‘The Fourier transform of mean-periodic functions', Utilitas Math. 29 (1986), 43–48.Google Scholar
[38] Székelyhidi, L., Convolution type functional équations on topological abelian groups (World Scientific Publishing, Singapore, 1991).Google Scholar
[39] Weit, Y., ‘On Schwartz theorem for the motion group', Ann. Inst. Fourier (Grenoble) 30 (1980), 91–107.Google Scholar