Kolmogorov-type inequalities for norms of Riesz derivatives of functions of several variables with Laplacian bounded in L∞ and related problems

• Published in: Mathematical Notes Vol. 95; no. 1-2; pp. 3 - 14
• Main Authors: Babenko, V. F., Parfinovich, N. V., Pichugov, S. A.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 2014
• Subjects: Mathematics; Mathematics and Statistics; optimal recovery problem for operators; Riesz derivative; Kolmogorov-type inequality; Banach space; Laplacian; Stechkin approximation problem
• ISSN: 0001-4346; 1573-8876
• DOI: 10.1134/S0001434614010015

Let L ∞,∞ Δ (ℝ m ) be the space of functions f ∈ L ∞ (ℝ m ) such that Δ f ∈ L ∞ (ℝ m ). We obtain new sharp Kolmogorov-type inequalities for the L ∞ -norms of the Riesz derivatives D α f of the functions f ∈ L ∞,∞ Δ (ℝ m ) and solve the Stechkin problem of approximating an unbounded operator D α by bounded operators on the class f ∈ L ∞ (ℝ m ) such that ‖Δ f ‖ ∞ ≤ 1, and also the problem of the best recovery of the operator D α from elements of this class given with error δ .