On piecewise-constant approximation of continuous functions of n variables in integral metrics

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) = ω 1(δ1) + ... + ωn(δ n), 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(δ n); (ii) in the integral metric Lp (p ≥ 1) for ω (δ1, . . . , δn) = c1δ 1 + ... + cnδn; (iii) in the integral metric L(2, . . . , 2, 2r) (r = 2, 3, . . .) for ω(δ 1, . . . , δn) = ω1(δ 1) + ... + ωn-1(δn-1) + c nδn. © 2002 Plenum Publishing Corporation.