Inequalities of the Kolmogorov type for norms of Riesz derivatives of multivariate functions and some of their applications |
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Authors: | Vladislav F. Babenko Natal’ya V. Parfinovich |
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Affiliation: | 1. O. Gonchar Dnepropetrovsk National University, 72, Gagarin Ave., Dnepropetrovsk, 49010, Ukraine
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Abstract: | We obtained new exact inequalities that estimate the L ?? -norm of the Riesz derivative D ?? f of a function f defined on $ {mathbb{R}^m} $ in terms of the uniform norm of the function itself and the L s -norm of the function acted by the Laplace operator. On a class of functions f such that ||??f||s ?? 1, we solved the problem of approximation of an unbounded operator D ?? by bounded ones and the problem of optimal recovery of the operator D ?? on elements of this class given with known error. |
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