Abstract
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 \(0.2\).
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Funding
The research of A. Yu. Popov was supported by the Russian Foundation for Basic Research under grant 20-01-00584.
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Alferova, E.D., Popov, A.Y. 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 \(\{kb_k\}\). Math Notes 108, 471–476 (2020). https://doi.org/10.1134/S0001434620090199
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DOI: https://doi.org/10.1134/S0001434620090199