Zero integrals on circles and characterizations of harmonic and analytic functions
HTML articles powered by AMS MathViewer
- by Josip Globevnik PDF
- Trans. Amer. Math. Soc. 317 (1990), 313-330 Request permission
Abstract:
We determine the kernels of two circular Radon transforms of continuous functions on an annulus and use this to obtain a characterization of harmonic functions in the open unit disc which involves Poisson averages over circles computed at only one point of the disc and to obtain a version of Morera’s theorem which involves only the circles which surround the origin.References
- A. M. Cormack and E. T. Quinto, A Radon transform on spheres through the origin in $\textbf {R}^{n}$ and applications to the Darboux equation, Trans. Amer. Math. Soc. 260 (1980), no. 2, 575–581. MR 574800, DOI 10.1090/S0002-9947-1980-0574800-3
- Andrew Russell Forsyth, Theory of differential equations. 1. Exact equations and Pfaff’s problem; 2, 3. Ordinary equations, not linear; 4. Ordinary linear equations; 5, 6. Partial differential equations, Dover Publications, Inc., New York, 1959. Six volumes bound as three. MR 0123757
- Josip Globevnik, Analyticity on rotation invariant families of curves, Trans. Amer. Math. Soc. 280 (1983), no. 1, 247–254. MR 712259, DOI 10.1090/S0002-9947-1983-0712259-6
- Josip Globevnik, Testing analyticity on rotation invariant families of curves, Trans. Amer. Math. Soc. 306 (1988), no. 1, 401–410. MR 927697, DOI 10.1090/S0002-9947-1988-0927697-0
- Josip Globevnik and Walter Rudin, A characterization of harmonic functions, Nederl. Akad. Wetensch. Indag. Math. 50 (1988), no. 4, 419–426. MR 976525 E. Goursat, Cours d’analyse mathématique, Vol. II, Gauthier-Villars, Paris, 1933.
- Sigurdur Helgason, The Radon transform, 2nd ed., Progress in Mathematics, vol. 5, Birkhäuser Boston, Inc., Boston, MA, 1999. MR 1723736, DOI 10.1007/978-1-4757-1463-0
- Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008 T. Lalesco, Sur l’equation de Volterra, J. Math. (6) 4 (1908), 125-202.
- Wilhelm Magnus, Fritz Oberhettinger, and Raj Pal Soni, Formulas and theorems for the special functions of mathematical physics, Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer-Verlag New York, Inc., New York, 1966. MR 0232968 V. G. Romanov, Integral geometry and inverse problems for hyperbolic equations, Tracts in Natural Philos., Vol. 26, Springer, Berlin, 1974. V. Volterra and J. Pérès, Théorie générale des fonctionelles, Gauthier-Villars, Paris, 1936.
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 317 (1990), 313-330
- MSC: Primary 44A05; Secondary 30E20, 31A05, 45D05
- DOI: https://doi.org/10.1090/S0002-9947-1990-0958892-1
- MathSciNet review: 958892