Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



On cubic lacunary Fourier series

Author: Joseph L. Gerver
Journal: Trans. Amer. Math. Soc. 355 (2003), 4297-4347
MSC (2000): Primary 42A55, 26A27
Published electronically: July 2, 2003
MathSciNet review: 1990754
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For $2<\beta <4$, we analyze the behavior, near the rational points $x=p\pi /q$, of $\sum^\infty_{n=1}n^{-\beta }\exp (ixn^{3})$, considered as a function of $x$. We expand this series into a constant term, a term on the order of $(x-p\pi /q)^{(\beta -1)/3}$, a term linear in $x-p\pi /q$, a ``chirp" term on the order of $(x-p\pi /q)^{(2\beta -1)/4}$, and an error term on the order of $(x-p\pi /q)^{\beta /2}$. At every such rational point, the left and right derivatives are either both finite (and equal) or both infinite, in contrast with the quadratic series, where the derivative is often finite on one side and infinite on the other. However, in the cubic series, again in contrast with the quadratic case, the chirp term generally has a different set of frequencies and amplitudes on the right and left sides. Finally, we show that almost every irrational point can be closely approximated, in a suitable Diophantine sense, by rational points where the cubic series has an infinite derivative. This implies that when $\beta \le (\sqrt {97}-1)/4=2.212\dots $, both the real and imaginary parts of the cubic series are differentiable almost nowhere.

References [Enhancements On Off] (What's this?)

  • 1. P.L. Butzer and E.L. Stark, ``Riemann's example'' of a continuous nondifferentiable function in the light of two letters (1865) of Cristoffel to Prym, Bull. Soc. Math. Belg. Sér. A, 38 (1986) 45-73. MR 88d:01007
  • 2. J.J. Duistermaat, Self-similarity of ``Riemann's nondifferentiable function'', Nieuw Arch. Wisk. (4), 9 (1991) 303-337. MR 93h:26009
  • 3. P. Erdös, On the distribution of the convergents of almost all real numbers, J. Number Theory, 2 (1970) 425-441. MR 42:5941
  • 4. C.F. Gauss, Disquisitiones Arithmeticae, Lipsiae, 1801.
  • 5. J. Gerver, The differentiability of the Riemann function at certain rational multiples of $\pi $, Amer. J. Math. 92 (1970) 33-55. MR 42:434
  • 6. J. Gerver, More on the differentiability of the Riemann function, Amer. J. Math. 93 (1971) 33-41. MR 43:2169
  • 7. G.H. Hardy, Weierstrass's non-differentiable function, Trans. Amer. Math. Soc. 17 (1916) 301-325.
  • 8. G.H. Hardy and J.E. Littlewood, A new solution of Waring's problem, Quarterly J. Math. 48 (1920) 272-293.
  • 9. D.R. Heath-Brown and S.J. Patterson, The distribution of Kummer sums at prime arguments, J. Reine Angew. Math. 310 (1979) 111-130. MR 81e:10033
  • 10. M. Holschneider et P. Tchamitchian, Pointwise analysis of Riemann's nondifferentiable function, Inventiones Math. 105 (1991) 157-175. MR 93g:26010
  • 11. S. Itatsu, Differentiability of Riemann's function, Proc. Japan Acad. Ser. A, 57 (1981) 492-495. MR 83e:42006
  • 12. S. Jaffard, The spectrum of singularities of Riemann's function, Rev. Mat. Iberoamericana, 12 (1996) 441-460. MR 97g:26006
  • 13. S. Jaffard and Y. Meyer, Wavelet methods for pointwise regularity and local oscillations of functions, Mem. Amer. Math. Soc. 123 (1996), no. 587. MR 97d:42005
  • 14. W. Luther, The differentiability of Fourier gap series and ``Riemann's example'' of a continuous, nondifferentiable function, J. Approximation Theory, 48 (1986) 303-321. MR 88d:42018
  • 15. E. Mohr, Wo ist die Riemannsche funktion nicht differenzierbar? Ann. Mat. Pura Appl. 123 (1980) 93-104. MR 81i:26005
  • 16. E. Neuenschwander, Riemann's example of a continuous nondifferentiable function, Math. Intelligencer, 1 (1978) 40-44. MR 58:15962a
  • 17. H. Queffelec, Dérivabilité de certaines sommes de séries de Fourier lacunaires, C.R. Acad. Sci. Paris, 273 (1971) 291-293. MR 44:4159

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 42A55, 26A27

Retrieve articles in all journals with MSC (2000): 42A55, 26A27

Additional Information

Joseph L. Gerver
Affiliation: Department of Mathematics, Rutgers University, Camden, New Jersey 08102

Received by editor(s): October 18, 1999
Published electronically: July 2, 2003
Article copyright: © Copyright 2003 American Mathematical Society