Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


Pointwise convergence of lacunary partial sums of almost periodic Fourier series

Author: Andrew D. Bailey
Journal: Proc. Amer. Math. Soc. 142 (2014), 1757-1771
MSC (2012): Primary 42A75; Secondary 42A24, 42B25
Published electronically: February 19, 2014
MathSciNet review: 3168481
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is a classical result that lacunary partial sums of the Fourier series of functions $ f \in L^p(\mathbb{T})$ converge almost everywhere for $ p \in (1, \infty )$. In 1968, E. A. Bredihina established an analogous result for functions belonging to the Stepanov space of almost periodic functions $ S^2$ whose Fourier exponents satisfy a natural separation condition. Here, the maximal operators corresponding to lacunary partial summation of almost periodic Fourier series are shown to be bounded on the Stepanov spaces $ S^{2^k}$, $ k \in \mathbb{N}$, for functions satisfying the same condition; Bredihina's result follows as a consequence. In the process of establishing these bounds, some general results are obtained which will facilitate further work on operator bounds and convergence issues in Stepanov spaces. These include a boundedness theorem for the Hilbert transform and a theorem of Littlewood-Paley type. An improvement of ``$ S^{2^k}$, $ k \in \mathbb{N}$'' to ``$ S^p$, $ p \in (1, \infty )$'' is also seen to follow from a natural conjecture on the boundedness of the Hilbert transform.

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

  • [1] J. Andres, A. M. Bersani, and R. F. Grande, Hierarchy of almost-periodic function spaces, Rend. Mat. Appl. (7) 26 (2006), no. 2, 121-188. MR 2275292 (2008b:43010)
  • [2] Andrew D. Bailey, Almost everywhere convergence of dyadic partial sums of Fourier series for almost periodic functions, MPhil thesis, University of Birmingham, September 2008 (conferral in July 2009).
  • [3] A. S. Besicovitch, Almost periodic functions, Dover Publications Inc., New York, 1955. MR 0068029 (16,817a)
  • [4] Harald Bohr, Almost periodic functions, Chelsea Publishing Company, New York, 1947. MR 0020163 (8,512a)
  • [5] E. A. Bredihina, Concerning A. N. Kolmogorov's theorem on lacunary partial sums of Fourier series, Sibirsk. Mat. Ž. 9 (1968), 456-461 (Russian). MR 0226285 (37 #1875)
  • [6] Lennart Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135-157. MR 0199631 (33 #7774)
  • [7] Javier Duoandikoetxea, Fourier analysis, Graduate Studies in Mathematics, vol. 29, American Mathematical Society, Providence, RI, 2001. Translated and revised from the 1995 Spanish original by David Cruz-Uribe. MR 1800316 (2001k:42001)
  • [8] John J. F. Fournier and James Stewart, Amalgams of $ L^p$ and $ l^q$, Bull. Amer. Math. Soc. (N.S.) 13 (1985), no. 1, 1-21. MR 788385 (86f:46027),
  • [9] Loukas Grafakos, Classical Fourier analysis, 2nd ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2008. MR 2445437 (2011c:42001)
  • [10] Sumiyuki Koizumi, On an integral inequality of the Stepanoff type and its applications, Proc. Japan Acad. 42 (1966), 896-900. MR 0213828 (35 #4685)
  • [11] A. Kolmogoroff, Une contribution à l'étude de la convergence des séries de Fourier (a contribution to the study of the convergence of Fourier series) (in French), Fundamental Mathematical 5 (1924), 96-97.
  • [12] B. M. Levitan, Počti-periodičeskie funkcii, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1953 (Russian). MR 0060629 (15,700a)
  • [13] Hugh L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics, vol. 84. Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the Amer. Math. Soc., Providence, RI, 1994. MR 1297543 (96i:11002)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2012): 42A75, 42A24, 42B25

Retrieve articles in all journals with MSC (2012): 42A75, 42A24, 42B25

Additional Information

Andrew D. Bailey
Affiliation: School of Mathematics, Watson Building, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom
Address at time of publication: Tessella, 26 The Quadrant, Abingdon Science Park, Abingdon, OX14 3YS, United Kingdom

Received by editor(s): December 19, 2011
Received by editor(s) in revised form: June 26, 2012
Published electronically: February 19, 2014
Additional Notes: The author was supported by an EPSRC doctoral training grant.
This work has appeared previously as part of the author’s MPhil thesis [2]. The author would like to express his gratitude to his MPhil and PhD supervisor, Jonathan Bennett, for all his support and assistance.
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society