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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Automorphic forms and differentiability properties
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by Fernando Chamizo PDF
Trans. Amer. Math. Soc. 356 (2004), 1909-1935 Request permission

Abstract:

We consider Fourier series given by a type of fractional integral of automorphic forms, and we study their local and global properties, especially differentiability and fractal dimension of the graph of their real and imaginary parts. In this way we can construct fractal objects and continuous non-differentiable functions associated with elliptic curves and theta functions.
References
  • Haïm Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree], Masson, Paris, 1983 (French). Théorie et applications. [Theory and applications]. MR 697382
  • Paul L. Butzer and Eberhard L. Stark, “Riemann’s example” of a continuous nondifferentiable function in the light of two letters (1865) of Christoffel to Prym, Bull. Soc. Math. Belg. Sér. A 38 (1986), 45–73 (1987). MR 885523
  • Fernando Chamizo and Antonio Córdoba, Differentiability and dimension of some fractal Fourier series, Adv. Math. 142 (1999), no. 2, 335–354. MR 1680194, DOI 10.1006/aima.1998.1792
  • K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR 867284
  • J. R. Ford, Fractions, Amer. Math. Monthly 45 (1938), 586-601.
  • Joseph Gerver, The differentiability of the Riemann function at certain rational multiples of $\pi$, Amer. J. Math. 92 (1970), 33–55. MR 265525, DOI 10.2307/2373496
  • G. H. Hardy, Weierstrass’s non-differentiable functions, Trans. Amer. Math. Soc. 17 (1916), 301-325.
  • G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR 568909
  • Dale Husemoller, Elliptic curves, Graduate Texts in Mathematics, vol. 111, Springer-Verlag, New York, 1987. With an appendix by Ruth Lawrence. MR 868861, DOI 10.1007/978-1-4757-5119-2
  • Henryk Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics, vol. 17, American Mathematical Society, Providence, RI, 1997. MR 1474964, DOI 10.1090/gsm/017
  • W. C. Winnie Li, Number theory with applications, Series on University Mathematics, vol. 7, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1390759, DOI 10.1142/2716
  • María V. Melián and Domingo Pestana, Geodesic excursions into cusps in finite-volume hyperbolic manifolds, Michigan Math. J. 40 (1993), no. 1, 77–93. MR 1214056, DOI 10.1307/mmj/1029004675
  • Reinhold Baer, Nets and groups, Trans. Amer. Math. Soc. 46 (1939), 110–141. MR 35, DOI 10.1090/S0002-9947-1939-0000035-5
  • J.-P. Serre and H. M. Stark, Modular forms of weight $1/2$, Modular functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976) Lecture Notes in Math., Vol. 627, Springer, Berlin, 1977, pp. 27–67. MR 0472707
  • Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures, No. 1, Iwanami Shoten Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Publications of the Mathematical Society of Japan, No. 11. MR 0314766
  • K. Weierstrass, Über continuirliche Functionen eines reellen Arguments, die für keinen Werth des Letzteren einen bestimmten Differentialquotienten besitzen (1872); English translation included in: Classics on Fractals (Ed., G.A. Edgar), Addison-Wesley Publishing Company, 1993.
  • A. Zygmund, Trigonometric series. Vol. I, II, Cambridge University Press, Cambridge-New York-Melbourne, 1977. Reprinting of the 1968 version of the second edition with Volumes I and II bound together. MR 0617944
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Additional Information
  • Fernando Chamizo
  • Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Ciudad Universitaria de Cantoblanco, Madrid 28049, Spain
  • Email: fernando.chamizo@uam.es
  • Received by editor(s): May 14, 2002
  • Received by editor(s) in revised form: March 27, 2003
  • Published electronically: July 24, 2003
  • © Copyright 2003 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 356 (2004), 1909-1935
  • MSC (2000): Primary 42A16, 11F12, 28A80
  • DOI: https://doi.org/10.1090/S0002-9947-03-03349-X
  • MathSciNet review: 2031046