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Mathematical Notes

, Volume 69, Issue 5–6, pp 644–651 | Cite as

Comparison of the >L1-Norms of Total and Truncated Exponential Sums

  • S. V. Konyagin
  • M. A. Skopina
Article

Abstract

The paper is concerned with a conjecture stated by S. V. Bochkarev in the seventies. He assumed that there exists a “stability” for the L1-norm of trigonometric polynomials when adding new harmonics. In particular, the validity of this conjecture implies the well-known Littlewood inequality. The disproof of a statement close to Bochkarev's conjecture is given. For this, the following method is used: the L1-norm of a sum of one-dimensional harmonics is replaced by the Lebesgue constant of a polyhedron of sufficiently high dimension.

Littlewood conjecture L1-norm of an exponential sum general orthonormal systems Lebesgue constant of a polyhedron 

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REFERENCES

  1. 1.
    G. H. Hardy and J. E. Littlewood, “A new proof of a theorem of rearrangement,” J. London Math. Soc., 23 (1948), no. 91, 163-168.Google Scholar
  2. 2.
    S. V. Bochkarev, “The averaging method in the theory of orthogonal series and some problems of the basis theory,” Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], vol. 146,Nauka, Moscow,1978.Google Scholar
  3. 3.
    S. V. Konyagin, “On the Littlewood problem,” Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.], 45 (1981), no. 2, 243-265.Google Scholar
  4. 4.
    O. C. McGehee, L. Pigno, and B. Smith, “Hardy's inequality and the L1-norm of exponential sums,” Ann. of Math., 113 (1981), no. 3, 613-618.Google Scholar
  5. 5.
    M. A. Skopina, “On the asymptotic behavior of the Lebesgue constants of the linear methods of summation of multiple Fourier series,” Izv. Vyssh. Uchebn. Zaved. Mat. [Soviet Math. (Iz. VUZ)], (1986), no. 6, 70-71.Google Scholar
  6. 6.
    M. A. Skopina, “The Lebesgue constants of Vallée-Poussin multiple sums,” Zap. Nauchn. Sem. LOMI, 125 (1983), 154-165.Google Scholar

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • S. V. Konyagin
    • 1
  • M. A. Skopina
    • 2
  1. 1.M.~V.~Lomonosov Moscow State UniversityRussia
  2. 2.St. Petersburg State UniversityRussia

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