Geodesic
Comparison of the $L^1$-Norms of Total and Truncated Exponential Sums
Matematičeskie zametki, Tome 69 (2001) no. 5, pp. 699-707

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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. 
@article{MZM_2001_69_5_a5,
     author = {S. V. Konyagin and M. A. Skopina},
     title = {Comparison of the $L^1${-Norms} of {Total} and {Truncated} {Exponential} {Sums}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {699--707},
     publisher = {mathdoc},
     volume = {69},
     number = {5},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2001_69_5_a5/}
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S. V. Konyagin; M. A. Skopina. Comparison of the $L^1$-Norms of Total and Truncated Exponential Sums. Matematičeskie zametki, Tome 69 (2001) no. 5, pp. 699-707. http://geodesic.mathdoc.fr/item/MZM_2001_69_5_a5/