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Automorphic forms and differentiability properties
Author(s):
Fernando
Chamizo
Journal:
Trans. Amer. Math. Soc.
356
(2004),
1909-1935.
MSC (2000):
Primary 42A16, 11F12, 28A80
Posted:
July 24, 2003
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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:
-
- 1.
- H. Brézis, Analyse fonctionnelle: théorie et applications, Masson, Paris, 1983. MR 85a:46001
- 2.
- P. I. Butzer and E. I. Stark, ``Riemann's example'' of a continuous nondifferentiable function in the light of two letters (1865) of Christoffel to Prym, Bull. Soc. Math. Belg. 38 (1986), 45-73. MR 88d:01007
- 3.
- F. Chamizo and A. Córdoba, Differentiablity and dimension of some fractal Fourier series, Adv. in Math. 142 (1999), 335-354. MR 2000d:42002
- 4.
- K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, 1986. MR 88d:28001
- 5.
- J. R. Ford, Fractions, Amer. Math. Monthly 45 (1938), 586-601.
- 6.
- J. Gerver, The differentiability of the Riemann function at certain rational multiples of
 , Amer. J. Math. 92 (1970), 33-55. MR 42:434 - 7.
- G. H. Hardy, Weierstrass's non-differentiable functions, Trans. Amer. Math. Soc. 17 (1916), 301-325.
- 8.
- G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, fifth edition, Oxford University Press, 1979. MR 81i:10002
- 9.
- D. Husemöller, Elliptic curves, Graduate Texts in Mathematics, vol. 111, Springer-Verlag, 1987. MR 88h:11039
- 10.
- H. Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics, vol. 17, American Mathematical Society, Providence, RI, 1997. MR 98e:11051
- 11.
- W. C. W. Li, Number theory with applications, Series on University Mathematics, vol. 7, World Scientific, Singapore, 1996. MR 98b:11001
- 12.
- M. V. Melián and D. Pestana, Geodesic excursions into cusps in finite-volume hyperbolic manifolds, Michigan Math. J. 40 (1993), 77-93. MR 94d:53067
- 13.
- W. Rudin, Real and complex analysis, third edition, McGraw-Hill, New York, 1987. MR 35:1420 (1st ed.); MR 88k:00002
- 14.
- J.-P. Serre and H. Stark, Modular forms of weight
 , Lecture Notes in Mathematics 627, Springer-Verlag, 1977, 29-68. MR 57:12400 - 15.
- G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton University Press, 1971. MR 47:3318
- 16.
- 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.
- 17.
- A. Zygmund, Trigonometric series I, II, second edition, latest reprint, Cambridge University Press, 1990. MR 58:29731, etc.
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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
DOI:
10.1090/S0002-9947-03-03349-X
PII:
S 0002-9947(03)03349-X
Received by editor(s):
May 14, 2002
Received by editor(s) in revised form:
March 27, 2003
Posted:
July 24, 2003
Copyright of article:
Copyright
2003,
American Mathematical Society
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