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, ..., 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