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 $\{kb_k\}$

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 $0.2$.