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)



The quartile operator and pointwise convergence of Walsh series

Author: Christoph Thiele
Journal: Trans. Amer. Math. Soc. 352 (2000), 5745-5766
MSC (2000): Primary 42A50, 42A20, 42C10
Published electronically: August 3, 2000
MathSciNet review: 1695038
Full-text PDF

Abstract | References | Similar Articles | Additional Information


The bilinear Hilbert transform is given by

\begin{displaymath}H(f,g)(x):= p.v. \int f(x-t)g(x+t)\frac{dt}{t}. \end{displaymath}

It satisfies estimates of the type

\begin{displaymath}\Vert H(f,g)\Vert _r\le C(s,t)\Vert f\Vert _s \Vert g\Vert _t.\end{displaymath}

In this paper we prove such estimates for a discrete model of the bilinear Hilbert transform involving the Walsh Fourier transform. We also reprove the well-known fact that the Walsh Fourier series of a function in $L^p[0,1]$, with $1<p$ converges pointwise almost everywhere. The purpose of this exposition is to clarify the connection between these two results and to present an easy approach to recent methods of time-frequency analysis.

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

  • 1. J. Bergh and J. Löfström, Interpolation spaces, an introduction, Springer-Verlag, Berlin, Heidelberg, New York, 1976. MR 58:2349
  • 2. P. Billard, Sur la convergence presque partout des series de Fourier Walsh des fonctions de l'espace $L^2(0,1)$, Studia Math. 28 (1967), 363-388. MR 36:599
  • 3. A. P. Calderon and A. Zygmund, A note on the interpolation of linear operators, Studia Math. 12 (1951), 194-204. MR 13:754e
  • 4. L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135-157. MR 33:7774
  • 5. C. Fefferman, Pointwise convergence of Fourier series, Ann. of Math. 98 (1973), 551-571. MR 49:5676
  • 6. R. Hunt, Almost everywhere convergence of Walsh-Fourier series of $L^2$-functions, Actes, Congrès Internat. Math., vol. 2, 1970, pp. 655-661. MR 58:23330
  • 7. S. Janson, On interpolation of multi-linear operators, (Proceedings Lund, 1986), Springer Lecture Notes in Math., vol. 1302, 1988. MR 89g:46114
  • 8. M. Lacey and C. Thiele, $L^p$ estimates on the bilinear Hilbert transform for $2<p<\infty$, Ann. of Math. (2) 146 (1997), 693-724. MR 99b:42014
  • 9. M. Lacey and C. Thiele, Bounds for the bilinear Hilbert transform on $L^p$, Proc. Nat. Acad. Sci. 94 (1997), 33-35.
  • 10. R.E.A.C. Paley, A remarkable series of orthogonal functions (I), Proc. London Math. Soc. 34 (1932), 241-264.
  • 11. P. Sjölin, An inequality of Paley and convergence a.e. of Walsh-Fourier series, Ark. Mat. 7 (1968), 551-570. MR 39:3222
  • 12. E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, N.J., 1970. MR 44:7280
  • 13. C. M. Thiele and L. F. Villemoes, A fast algorithm for adapted time frequency tilings, Appl. Comput. Harmon. Anal. 3 (1996), 91-99. MR 98g:65134
  • 14. J. L. Walsh, A closed set of normal orthogonal functions, Amer. J. Math. 45 (1923), 5-24.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 42A50, 42A20, 42C10

Retrieve articles in all journals with MSC (2000): 42A50, 42A20, 42C10

Additional Information

Christoph Thiele
Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06511
Address at time of publication: Department of Mathematics, University of California, Los Angeles, California 90095-1555

Keywords: Bilinear Hilbert transform, convergence of Fourier series, Walsh functions
Received by editor(s): September 25, 1997
Published electronically: August 3, 2000
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society