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 L 1-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 L 1-norm of a sum of one-dimensional harmonics is replaced by the Lebesgue constant of a polyhedron of sufficiently high dimension.
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Konyagin, S.V., Skopina, M.A. Comparison of the >L 1-Norms of Total and Truncated Exponential Sums. Mathematical Notes 69, 644–651 (2001). https://doi.org/10.1023/A:1010253609303
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DOI: https://doi.org/10.1023/A:1010253609303