This article is cited in 5 papers

Comparison of the $L^1$-Norms of Total and Truncated Exponential Sums

S. V. Konyagin^a, M. A. Skopina<sup>b

a</sup> M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
^b Saint-Petersburg State University

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.

UDC: 517.5

Received: 23.02.2000

DOI: 10.4213/mzm533

 English version: Mathematical Notes, 2001, 69:5, 644–651

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