We give a solution of the problem on the best uniform approximation on the number axis of the first-order differentiation operator on the class of functions with bounded second derivative by linear operators bounded in the space L₂. This is one of the few cases of the exact solution of the problem on the approximation of the differentiation operator in some space with the use of approximating operators that are bounded in another space. We obtain a related exact inequality between the uniform norm of the derivative of a function, the variation of the Fourier transform of the function, and the L_∞-norm of its second derivative. This inequality can be regarded as a nonclassical variant of the Hadamard-Kolmogorov inequality.

中文翻译：

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空间L 2上界的算子的微分算子的最佳一致逼近

我们给出了关于一阶微分算子的数轴上的最佳均匀逼近的一个问题的解决方案，该函数类别具有以空间L _2为边界的线性算子为边界的二阶导数。这是在某些空间中使用在另一个空间中有界的逼近算子逼近问题的精确解决方案的少数情况之一。我们获得该衍生物的函数的均匀范数之间的相关确切不等式，傅立叶变化变换的功能，和大号_∞其二次导数的范数。该不等式可以看作是Hadamard-Kolmogorov不等式的非经典变体。