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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
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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. Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. MR 0482275
  • 2. P. Billard, Sur la convergence presque partout des séries de Fourier-Walsh des fonctions de l’espace 𝐿²(0,1), Studia Math. 28 (1966/1967), 363–388. MR 0217510
  • 3. A. P. Calderón and A. Zygmund, A note on the interpolation of linear operations, Studia Math. 12 (1951), 194–204. MR 0046567
  • 4. Lennart Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135–157. MR 0199631
  • 5. Charles Fefferman, Pointwise convergence of Fourier series, Ann. of Math. (2) 98 (1973), 551–571. MR 0340926
  • 6. Richard A. Hunt, Almost everywhere convergence of Walsh-Fourier series of 𝐿² functions, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 655–661. MR 0511000
  • 7. Svante Janson, On interpolation of multilinear operators, Function spaces and applications (Lund, 1986) Lecture Notes in Math., vol. 1302, Springer, Berlin, 1988, pp. 290–302. MR 942274, 10.1007/BFb0078880
  • 8. Michael Lacey and Christoph Thiele, 𝐿^{𝑝} estimates on the bilinear Hilbert transform for 2<𝑝<∞, Ann. of Math. (2) 146 (1997), no. 3, 693–724. MR 1491450, 10.2307/2952458
  • 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. Per Sjölin, An inequality of Paley and convergence a.e. of Walsh-Fourier series., Ark. Mat. 7 (1969), 551–570. MR 0241885
  • 12. Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
  • 13. Christoph M. Thiele and Lars F. Villemoes, A fast algorithm for adapted time-frequency tilings, Appl. Comput. Harmon. Anal. 3 (1996), no. 2, 91–99. MR 1385046, 10.1006/acha.1996.0009
  • 14. J. L. Walsh, A closed set of normal orthogonal functions, Amer. J. Math. 45 (1923), 5-24.

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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