Morera Theorems for Spheres through a Point in CN

  • Eric Liviu Grinberg
  • Eric Todd Quinto

Abstract

We prove Morera theorems for the Radon transform integrating on spheres through a point in ℂn. The proofs use spherical functions and integral equations techniques as well as a support theorem for a generalized Radon transform on hyperplanes.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Ag1987]
    Agranovsky, M. (1978). Fourier transform on SL2(ℝ) and Morera type theorems, Soviet Math. Dokl., Vol. 19, (pages 1522-1526). Google Scholar
  2. [ABC]
    Agranovsky, M., Berenstein, C.A., and Chang, D.C. (1993). Morera Theorem for holomorphic Hp spaces in the Heisenberg group, J. reine angew. Math., Vol. 443, (pages 49–89).MathSciNetMATHGoogle Scholar
  3. [ABCP]
    Agranovsky, M., Berenstein, C.A., Chang, D.C., Pascuas, D. (1991). A Morera type theorem for L2 functions in the Heisenberg group, J. Analyse Math., Vol. 57, (pages 282–296).MathSciNetMATHGoogle Scholar
  4. [Be]
    Berenstein, C. A. (1984). A test for holomorphy in the unit ball ofn, Proc. Amer. Math. Soc., Vol. 90, (pages 88–90).MathSciNetMATHGoogle Scholar
  5. [BCPZ]
    Berenstein, C.A., Chang, D-C., Pasucas, D., and Zalcman, L. (1992). Variations on the theorem of Morera, Contemp. Math., Vol. 137, (pages 63–78).Google Scholar
  6. [BG1986]
    Berenstein, C.A. and Gay, R. (1986). A local version of the two circles theorem, Israel J. Math., Vol. 55, (pages 267–288).MathSciNetMATHCrossRefGoogle Scholar
  7. [BG1988]
    Berenstein, C.A. and Gay, R. (1988). Le Probléme de Pompeiu local, J. Analyse Math., Vol. 52, (pages 133–166).MathSciNetCrossRefGoogle Scholar
  8. [BGY]
    Berenstein, C.A., Guy R., and Yger, A. (1990). Inversion of the local Pompeiu transform, J. Analyse Math., Vol. 54, (pages 259–287).MathSciNetMATHCrossRefGoogle Scholar
  9. [BP]
    Berenstein, C.A. and Pascuas, D. (1994). Morera and Mean-Value Type theorems in the Hyperbolic Disk, Israel J. Math., Vol. 86, (61–106).MathSciNetMATHCrossRefGoogle Scholar
  10. [BZ]
    Berenstein, C.A. and Zalcman, L. (1980). Pompeiu’s problem on symmetric spaces, Comment. Math. Helv., Vol. 55, (pages 593–621).MathSciNetMATHCrossRefGoogle Scholar
  11. [BQ1987]
    Boman, J. and Quinto, E.T. (1987). Support theorems for real analytic Radon transforms, Duke Math. J., Vol. 55, (pages 943–948).MathSciNetMATHCrossRefGoogle Scholar
  12. [BQ1992]
    Boman, J. and Quinto, E.T. (1993). Support theorems for real analytic Radon transforms on line complexes in3, Trans. Amer. Math. Soc., Vol. 335, (pages 877–890).MathSciNetMATHCrossRefGoogle Scholar
  13. [CQ]
    Cormack, A. and Quinto, E.T. (1980). A Radon transform on spheres through the origin innand applications to the Darboux equation, Trans. Amer. Math. Soc., Vol. 260, (pages 575–581).MathSciNetMATHGoogle Scholar
  14. [DL]
    Delsarte, J. and Lions, J.L. (1959). Moyennes généralisées, Comment. Math. Helv., Vol. 33, (pages 59–69 ).MathSciNetMATHCrossRefGoogle Scholar
  15. [Gl1989]
    Globevnik, J. (1989). Integrals over circles passing through the origin and a characterization of analytic functions, J. Analyse Math., Vol. 52, (pages 199–209).MathSciNetMATHCrossRefGoogle Scholar
  16. [Gl1990]
    Globevnik, J. (1990). Zero integrals on circles and characterizations of harmonic and analytic functions, Trans. Amer. Math. Soc., Vol. 317, (pages 313–330).MathSciNetCrossRefGoogle Scholar
  17. [Gl1994]
    Globevnik, J. (1994). Holomorphic functions on rotation invariant families of curves passing through the origin, J. Analyse Math., Vol. 63, (pages 221–229).MathSciNetMATHCrossRefGoogle Scholar
  18. [GlQ]
    Globevnik, J. and Quinto E. (1996). Morera Theorems and microlocal Analysis, J. Geometric Anal., Vol. 6, pages 19–30).MathSciNetMATHGoogle Scholar
  19. [GrQ]
    Grinberg, E.L. and Quinto E., Morera Theorems for complex manifolds, preprint.Google Scholar
  20. [Jo]
    John, F. (1966). Plane Waves and Spherical Means, Interscience, New York.Google Scholar
  21. [Q1983]
    John, F. (1983). The invertibility of rotation invariant Radon transforms, J. Math. Anal. Appl., Vol. 91, (pages 510–522.MathSciNetCrossRefGoogle Scholar
  22. [Q1993]
    Quinto, E.T. (1993). Pompeiu transforms on geodesic spheres in real analytic manifolds, Israel J. Math., Vol. 84, (pages 353–363).MathSciNetCrossRefGoogle Scholar
  23. [Se]
    Seeley, R. (1966). Spherical Harmonics, Amer. Math. Monthly, Vol. 73, (pages 115–121).MathSciNetMATHCrossRefGoogle Scholar
  24. [Za1972]
    Zalcman, L. (1972). Analyticity and the Pompeiu problem, Arch. Rat. Mech. Anal., Vol. 47, (pages 237–254).MathSciNetMATHCrossRefGoogle Scholar
  25. [Za1980]
    Zalcman, L. (1980). Offbeat Integral Geometry, Amer. Math. Monthly, Vol. 87, (pages 161–175).MathSciNetMATHCrossRefGoogle Scholar
  26. [Za1992]
    Zalcman, L. (1992), A bibliographic survey of the Pompeiu problem, Approximation of Solutions of Partial Differential Equations, B. Fuglede, M. Goldstein, W. Haussmann, W. K. Hayman, and L. Rogge, Editors, Vol. 365, Series C: Mathematics and Physical Sciences, NATO ASI Series, Kluwer Academic, Boston, (pages 185–94).Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Eric Liviu Grinberg
    • 1
  • Eric Todd Quinto
    • 2
  1. 1.Department of MathematicsTemple UniversityPhiladelphiaUSA
  2. 2.Department of MathematicsTufts UniversityMedfordUSA

Personalised recommendations