Geodesic
New Inequalities in the Littlewood--Paley Theory and Estimates of the~$L_1$ Norm of Trigonometric Series and Polynomials
Informatics and Automation, Studies on function theory and differential equations, Tome 248 (2005), pp. 64-73.

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Extensions of the Littlewood–Paley theorem to the limit values $p=1$ and $p=\infty$ are applied in order to obtain new lower bounds for the $L_1$ norms of trigonometric series and polynomials. Applications of the method to the estimation of polynomials with quadratic spectrum are suggested. 
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S. V. Bochkarev. New Inequalities in the Littlewood--Paley Theory and Estimates of the~$L_1$ Norm of Trigonometric Series and Polynomials. Informatics and Automation, Studies on function theory and differential equations, Tome 248 (2005), pp. 64-73. http://geodesic.mathdoc.fr/item/TRSPY_2005_248_a6/

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