We show that Hölder continuity of the gradient is not only a sufficient condition, but also a necessary condition for the existence of a global upper bound on the error of the first-order Taylor approximation. We also relate this global upper bound to the Hölder constant of the gradient. This relation is expressed as an interval, depending on the Hölder constant, in which the error of the first-order Taylor approximation is guaranteed to be. We show that, for the Lipschitz continuous case, the interval cannot be reduced. An application to the norms of quadratic forms is proposed, which allows us to derive a novel characterization of Euclidean norms.

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关于Hölder连续梯度函数一阶逼近的质量

我们证明梯度的 Hölder 连续性不仅是一阶泰勒近似误差的全局上界存在的充分条件，而且是必要条件。我们还将这个全局上限与梯度的 Hölder 常数联系起来。这种关系表示为一个区间，取决于 Hölder 常数，其中保证一阶泰勒近似的误差为。我们表明，对于 Lipschitz 连续情况，间隔不能减少。提出了对二次形式范数的应用，这使我们能够推导出欧几里得范数的新特征。