Kolmogorov’s problem on the class of multiply monotone functions. (English) Zbl 1325.41007
Let \(X\) be the class of \(r\)-monotone functions on the negative half-line \(\mathbb{R}_{-}\). The authors present a complete solution of Kolmogorov’s problem for the class \(X\) which can be formulated as follows: find necessary and sufficient conditions for the system of positive numbers \(M_{k_1}\), \(M_{k_2}, \dots\), \(M_{k_d}\), \(0 \leq k_1<\ldots < k_d \leq r\), to guarantee the existence of a function \(f\in X\) such that \(\|f^{(k_i)}\|_{\infty}= M_{k_i}\), \(i =1, 2, \dots, d\). Applications to some versions of the moment problem, the problem of the smoothest Hermite-Birkhoff interpolation, the Hermite-Birkhoff interpolation by perfect splines (with free knots and higher derivatives that take only two values, one of which is zero), and extremal properties of interpolating splines are given. Some extremal problems on the sets of solutions of Kolmogorov’s problem are also considered.
Reviewer: Yuri A. Farkov (Moscow)
MSC:
| 41A17 | Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) |
| 41A10 | Approximation by polynomials |
| 26D10 | Inequalities involving derivatives and differential and integral operators |
| 41A30 | Approximation by other special function classes |
| 41A44 | Best constants in approximation theory |
Keywords:
multiply monotone; sharp inequalities; derivatives; Kolmogorov’s problem; moment problem; Hermite-Birkhoff interpolation; extremal problemsReferences:
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