2019
Том 71
№ 11

All Issues

Bel'skii S. A.

Articles: 1
Article (Russian)

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

Bel'skii S. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2002. - 54, № 3. - pp. 293-303

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 ) = ω11) + ... + ω n n ), where ω i i ) are ordinary moduli of continuity that depend on one variable. In the case where ω i 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 ) = ω11) + ... + ω n 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 ) = ω11) + ... + ω n − 1 n − 1) + c nδ n .