We address the optimal constants in the strong and the weak Stechkin inequalities, both in their discrete and continuous variants. These inequalities appear in the characterization of approximation spaces which arise from sparse approximation or have applications to interpolation theory. An elementary proof of a constant in the strong discrete Stechkin inequality given by Bennett is provided, and we improve the constants given by Levin and Stechkin and by Copson. Finally, the minimal constants in the weak discrete Stechkin inequalities and both continuous Stechkin inequalities are presented.

中文翻译：

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关于两侧 Stechkin 不等式中的最优常数

我们在离散和连续变体中讨论强和弱 Stechkin 不等式中的最佳常数。这些不等式出现在由稀疏近似产生的近似空间的表征中，或者应用于插值理论。提供了 Bennett 给出的强离散 Stechkin 不等式中常数的基本证明，我们改进了 Levin 和 Stechkin 以及 Copson 给出的常数。最后，给出了弱离散 Stechkin 不等式和连续 Stechkin 不等式中的最小常数。