The best Lp approximation of the Laplace operator by linear bounded operators in the classes of functions of two and three variables

Abstract

Close two-sided estimates are obtained for the best approximation in the space L p(ℝ m), m = 2 and 3, 1 ≤ p ≤ ∞, of the Laplace operator by linear bounded operators in the class of functions for which the second power of the Laplace operator belongs to the space L p(ℝ m). We estimate the best constant in the corresponding Kolmogorov inequality and the error of the optimal recovery of values of the Laplace operator on functions from this class given with an error. We present an operator whose deviation from the Laplace operator is close to the best. © 2012 Pleiades Publishing, Ltd.