On Piecewise-Constant Approximation of Continuous Functions of n Variables in Integral Metrics - Bel'skii, S. A. (Ukrains’kyi Matematychnyi Zhurnal, 2002)
We consider the approximation by piecewise-constant functions for classes of functions of many variables defined by moduli of continuity of the form ω(δ1, ..., δ n ) = ω1(δ1) + ... + ω n (δ n ), where ω i (δ i ) are ordinary moduli of continuity that depend on one variable. In the case where ω i (δ i ) are convex upward, we obtain exact error estimates in the following cases: (i) in the integral metric L 2 for ω(δ1, ..., δ n ) = ω1(δ1) + ... + ω n (δ n ); (ii) in the integral metric L p (p ≥ 1) for ω(δ1, ..., δ n ) = c 1δ1 + ... + c nδ n ; (iii) in the integral metric L (2, ..., 2, 2r) (r = 2, 3, ...) for ω(δ1, ..., δ n ) = ω1(δ1) + ... + ω n − 1(δ n − 1) + c nδ n .