Abstract
Under what conditions can one conclude that a continuous function on a plane domain Ω is holomorphic, given that its restrictions to a collection of Jordan curves in Ω which cover Ω admit holomorphic extensions? We survey progress on this problem over the past 40 years, with an emphasis on recent results.
An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-88-470-1947-8_24
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References
Agranovsky, M.L.: The Fourier transform on SL 2(ℝ) and theorems of Morera type. Sov. Math. Dokl. 19, 1522–1525 (1978)
Agranovsky, M.L.: CR foliations, the strip-problem and Globevnik–Stout conjecture. C. R. Acad. Sci. Paris 343, 91–94 (2006)
Agranovsky, M.L.: Propagation of boundary CR-foliations and Morera type theorems for manifolds with attached analytic discs. Adv. Math. 211, 284–326 (2007)
Agranovsky, M.L.: Characterization of polyanalytic functions by meromorphic extensions in chains of circles. J. Anal. Math. 113, 305–329 (2011)
Agranovsky, M.L.: Complex dimensions of real manifolds, attached analytic discs and parametric argument principle. arXiv:0704.2871v2 [math]
Agranovsky, M.L., Globevnik, J.: Analyticity on circles for rational and real-analytic functions of two real variables. J. Anal. Math. 91, 31–65 (2003)
Agranovsky, M.L., Narayanan, E.K.: Isotopic families of contact manifolds for elliptic PDE. Proc. Am. Math. Soc. 134, 2117–2123 (2006)
Agranovsky, M.L., Valsky, R.E.: Maximality of invariant subalgebras of functions. Sib. Math. J. 12, 1–7 (1971)
Agranovsky, M.L., Volchkov, V.V., Zalcman, L.: Conical uniqueness sets for the spherical Radon transform. Bull. Lond. Math. Soc. 31, 231–236 (1999)
Berenstein, C.A., Zalcman, L.: Pompeiu’s problem on spaces of constant curvature. J. Anal. Math. 30, 113–130 (1976)
Berenstein, C.A., Zalcman, L.: Pompeiu’s problem on symmetric spaces. Comment. Math. Helv. 55, 593–621 (1980)
Brown, L., Schreiber, B.M., Taylor, B.A.: Spectral synthesis and the Pompeiu problem. Ann. Inst. Fourier (Grenoble) 23(3), 125–154 (1973)
Ehrenpreis, L.: Three problems at Mount Holyoke. In: Radon Transforms and Tomography. Contemp. Math., vol. 278, pp. 123–130 (2001)
Ehrenpreis, L.: The Universality of the Radon Transform. Clarendon, Oxford (2003)
Ehrenpreis, L.: The strip problem for PDE. Inverse Probl. Theor. Imaging 4(4), III–IV (2010)
Fridman, B.L., Ma, D.: Testing holomorphy on curves. arXiv:1012.4329v2
Globevnik, J.: Analyticity on rotation invariant families of circles. Trans. Am. Math. Soc. 280, 247–254 (1983)
Globevnik, J.: Testing analyticity on rotation invariant families of curves. Trans. Am. Math. Soc. 306, 401–410 (1988)
Globevnik, J.: Holomorphic extensions and rotation invariance. Complex Var. Theory Appl. 24, 49–51 (1994)
Globevnik, J.: Holomorphic extensions from open families of circles. Trans. Am. Math. Soc. 355, 1921–1931 (2003)
Globevnik, J.: Meromorphic extensions from small circles and holomorphic extensions from spheres. arXiv:1101.0136v2
Hu, S.T.: Homotopy Theory. Academic Press, San Diego (1959)
Tumanov, A.: A Morera type theorem in the strip. Math. Res. Lett. 11, 23–29 (2004)
Tumanov, A.: Testing analyticity on circles. Am. J. Math. 129, 785–790 (2007)
Volchkov, V.V.: Integral Geometry and Convolution Equations. Kluwer Academic, Dordrecht (2003)
Volchkov, V.V., Volchkov, V.V.: Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group. Springer, London (2009)
Zalcman, L.: Analyticity and the Pompeiu problem. Arch. Ration. Mech. Anal. 47, 237–254 (1972)
Zalcman, L.: Mean values and differential equations. Isr. J. Math. 14, 339–352 (1973)
Zalcman, L.: A bibliographic survey of the Pompeiu problem. In: Approximation by Solutions of Partial Differential Equations, pp. 185–194. Kluwer Academic, Dordrecht (1992)
Zalcman, L.: Supplementary bibliography to “A bibliographic survey of the Pompeiu problem”. In: Radon Transforms and Tomography. Contemp. Math., vol. 278, pp. 69–74 (2001)
Acknowledgements
This work was partially supported by Israel Science Foundation Grants 688/08 and 395/07 and is a part of the European Science Foundation Networking Programme HCAA.
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We dedicate this paper to the memory of our friend Leon Ehrenpreis. Leon was fascinated by the strip problem, contributed to its solution [13], and led the way in generalizing it from a result concerning analytic functions to solutions of elliptic equations [14]. Indeed, one of his last major addresses, the opening lecture of the conference Integral Geometry and Tomography, delivered at Stockholm University on August 12, 2008, was entitled “The Strip Theorem for PDE”; see [15, II–IV].
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Agranovsky, M., Zalcman, L. (2012). Analyticity on Curves. In: Sabadini, I., Struppa, D. (eds) The Mathematical Legacy of Leon Ehrenpreis. Springer Proceedings in Mathematics, vol 16. Springer, Milano. https://doi.org/10.1007/978-88-470-1947-8_4
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DOI: https://doi.org/10.1007/978-88-470-1947-8_4
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