Best Approximation of a Differentiation Operator on the Set of Smooth Functions with Exactly or Approximately Given Fourier Transform

Let \(Y^n,\) , be the set of continuous bounded functions on the numerical axis with the following two properties: (1) the Fourier transform of a function is a function of bounded variation on the axis (in particular, a summable function); (2) a function is \(n-1\) times continuously differentiable, its derivative of order \(n-1\) is locally absolutely continuous, and the nth order derivative is bounded, more exactly, belongs to the space \(L_\infty \). In the space \(Y^n,\) consider the class \(\mathcal {Q}^n\) of functions, for which the \(L_\infty \)-norm of the nth order derivative is bounded by a constant, for example, by 1. The following two approximation problems are discussed: the best approximation of the differentiation operator \(D^k\) of order k, , by bounded operators on the class \(\mathcal {Q}^n\) and the optimal calculation of the differentiation operator \(D^k\) on functions from the class \(\mathcal {Q}^n\) under the assumption that their Fourier transform is given with a known error in the space of functions of bounded variation, in particular, in the space L of functions summable on the axis. In interrelation with these two problems, we discuss the exact Kolmogorov type inequality in the space \(Y^n\) between the uniform norm of the kth order derivative of a function, the variation of the Fourier transform of the function, and the \(L_\infty \)-norm of its derivative of order n.