Geodesic
Two-Sided Estimates of the $L^\infty$-Norm of the Sum of a Sine Series with Monotone Coefficients~$\{b_k\}$ via the $\ell^\infty$-Norm of the Sequence~$\{kb_k\}$
Matematičeskie zametki, Tome 108 (2020) no. 4, pp. 483-489.

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We refine the classical boundedness criterion for sums of sine series with monotone coefficients $b_k$: the sum of a series is bounded on $\mathbb R$ if and only if the sequence $\{kb_k\}$ is bounded. We derive a two-sided estimate of the Chebyshev norm of the sum of a series via a special norm of the sequence $\{kb_k\}$. The resulting upper bound is sharp, and the constant in the lower bound differs from the exact value by at most $0.2$.
Keywords: two-sided estimate of a norm, sine series
Mots-clés : monotone coefficients. 
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     author = {E. D. Alferova and A. Yu. Popov},
     title = {Two-Sided {Estimates} of the $L^\infty${-Norm} of the {Sum} of a {Sine} {Series} with {Monotone} {Coefficients~}$\{b_k\}$ via the $\ell^\infty${-Norm} of the {Sequence~}$\{kb_k\}$},
     journal = {Matemati\v{c}eskie zametki},
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E. D. Alferova; A. Yu. Popov. Two-Sided Estimates of the $L^\infty$-Norm of the Sum of a Sine Series with Monotone Coefficients~$\{b_k\}$ via the $\ell^\infty$-Norm of the Sequence~$\{kb_k\}$. Matematičeskie zametki, Tome 108 (2020) no. 4, pp. 483-489. http://geodesic.mathdoc.fr/item/MZM_2020_108_4_a0/

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