×

Wilson’s functional equations on groups. (English) Zbl 0830.39016

From the author’s summary: We study properties of solutions \(f,g,h \in C(G)\) of the functional equation \[ \int_K f(xk \cdot y) \overline {\chi (k)}dk = g(x)h(y), \quad x,y \in G \] and the special case \[ \int_K f(xk \cdot y) \overline {\chi (k)}dk = g(x)f(y), \quad x,y \in G, \] where \(G\) is a locally compact group, \(K\) a compact subgroup of \(\operatorname{Aut} (G)\) and \(\chi\) a character on \(K\). We show that \(g\) and \(h\) are associated to certain \(K\)-spherical functions and use that to compute the complete set of solutions in special examples; in particular in the case of \(G = \mathbb{R}^n\).

MSC:

39B52 Functional equations for functions with more general domains and/or ranges
43A90 Harmonic analysis and spherical functions
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Aczél, J.,Vorlesungen über Funktionalgleichungen und ihre Anwendungen. Birkhäuser, Basel-Stuttgart, 1961. · Zbl 0096.09102
[2] Aczél, J. andDhombres, J.,Functional equations in several variables. Cambridge University Press, Cambridge–New York–New Rochelle–Melbourne–Sydney, 1989.
[3] d’Alembert, J.,Mémoire sur les principes de mécanique. Hist. Acad. Sci. Paris1769, 278–286.
[4] d’Alembert, J.,Recherches sur la courbe que forme une corde tendue mise en vibration, I. Hist. Acad. Berlin1747, 214–219.
[5] d’Alembert, J.,Recherches sur la courbe que forme une corde tendue mise en vibration, II. Hist. Acad. Berlin1747, 220–249.
[6] Andrade, J.,Sur l’équation fonctionelle de Poisson. Bull. Soc. Math. France28 (1900), 58–63. · JFM 31.0403.03
[7] Benson, C., Jenkins, J. andRatcliff, G.,Bounded K-spherical functions on Heisenberg groups. J. Funct. Anal.105 (1992), 409–443. · Zbl 0763.22007 · doi:10.1016/0022-1236(92)90083-U
[8] Chevalley, C.,Theory of Lie groups, I. Princeton University Press, Princeton, 1946. · Zbl 0063.00842
[9] Corovei, I.,The cosine functional equation for nilpotent groups. Aequationes Math.15 (1977), 99–106. · Zbl 0348.39004 · doi:10.1007/BF01837878
[10] Gajda, Z.,Unitary representations of topological groups and functional equations. J. Math. Anal. Appl.152 (1990), 6–19. · Zbl 0721.39006 · doi:10.1016/0022-247X(90)90089-X
[11] Gajda, Z.,On functional equations associated with characters of unitary representations of groups. Aequationes Math.44 (1992), 109–121. · Zbl 0772.39007 · doi:10.1007/BF01834209
[12] Helgason, S.,Groups and geometric analysis. Academic Press, Orlando–San Diego–San Francisco–New York–London–Toronto–Montreal–Sydney–Tokyo–São Paulo, 1984. · Zbl 0543.58001
[13] Hewitt, E. andRoss, K. A.,Abstract harmonic analysis. I. Springer-Verlag, Berlin-Göttingen–Heidelberg, 1963.
[14] Hewitt, E. andRoss, K. A.,Abstract harmonic analysis. II. Springer-Verlag, Berlin–Heidelberg–New York, 1970.
[15] Hochschild, G.,The structure of Lie groups, Holden-Day, San Francisco–London–Amsterdam, 1965. · Zbl 0131.02702
[16] Hörmander, L.,The analysis of linear partial differential operators I. Springer-Verlag, Berlin–Heidelberg–New York–Tokyo, 1983. · Zbl 0521.35001
[17] Jensen, J. L. W. V.,Om Fundamentalligningers Opløsning ved elementoere Midler. Tidsskrift for Mathematik (4)2 (1878), 149–155.
[18] Kaczmarz, S.,Sur l’équation fonctionelle f(x) + f(x + y) = {\(\phi\)}(y)f(x + (y/2)). Fund. Math.6 (1924), 122–129.
[19] Kannappan, P.,The functional equation f(xy) + f(xy = 2f(x)f(y) for groups. Proc. Amer. Math. Soc.19 (1968), 69–74. · Zbl 0169.48102
[20] Litvinov, G. L. andLomonosov, V. I.,Density theorems in locally convex spaces and applications. Selecta Math. Soviet.8 (1989), 323–339. Translated to English from Trudy Sem. Vector. Tenzor. Anal.20 (1981), 210–227.
[21] Ljubenova, E. T.,On d’Alembert’s functional equation on an Abelian group. Aequationes Math.22 (1981), 54–55. · Zbl 0462.39007 · doi:10.1007/BF02190159
[22] O’Connor, Thomas A.,A solution of d’Alembert’s functional equation on a locally compact Abelian group. Aequationes Math.15 (1977), 235–238. · Zbl 0384.43009 · doi:10.1007/BF01835653
[23] Penney, R. C. andRukhin, A. L.,D’Alembert’s functional equation on groups. Proc. Amer. Math. Soc.77 (1979), 73–80.
[24] Stetkaer, H.,Scalar irreducibility of certain eigenspace representations. J. Funct. Anal.61 (1985), 295–306. · Zbl 0593.22003 · doi:10.1016/0022-1236(85)90024-2
[25] Stetkaer, H.,Complete irreducibility and X-spherical representations. J. Funct. Anal.113 (1993), 413–425. · Zbl 0789.22025 · doi:10.1006/jfan.1993.1056
[26] Stetkaer, H.,D’Alembert’s equations and spherical functions. Aequationes Math.48 (1994), 220–227. · Zbl 0810.39008 · doi:10.1007/BF01832986
[27] Székelyhidi, L.,Almost periodicity and functional equations. Aequationes Math.26 (1983), 163–175. · Zbl 0566.39010 · doi:10.1007/BF02189679
[28] Wilson, W. H.,On certain related functional equations. Bull. Amer. Math. Soc.26 (1919–20), 300–312. · JFM 47.0320.01 · doi:10.1090/S0002-9904-1920-03310-0
[29] Wilson, W. H.,Two general functional equations. Bull. Amer. Math. Soc.31 (1925), 330–334. · JFM 51.0311.01 · doi:10.1090/S0002-9904-1925-04045-8
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.