On Piecewise-Constant Approximation of Continuous Functions of n Variables in Integral Metrics

Abstract

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, ..., δn) = ω1(δ1) + ... + ωn(δn), where ωi(δi) are ordinary moduli of continuity that depend on one variable. In the case where ωi(δi) are convex upward, we obtain exact error estimates in the following cases: (i) in the integral metric L2 for ω(δ1, ..., δn) = ω1(δ1) + ... + ωn(δn); (ii) in the integral metric Lp (p ≥ 1) for ω(δ1, ..., δn) = c1δ1 + ... + cnδn; (iii) in the integral metric L(2, ..., 2, 2r) (r = 2, 3, ...) for ω(δ1, ..., δn) = ω1(δ1) + ... + ωn − 1(δn − 1) + cnδn.