On approximation of continuous functions by piecewise-constant ones in integral metrics

Abstract

We obtain upper bound of the best approximation of the classes $$$H^{\omega} [a, b]$$$ by piecewise-constant functions over uniform split in metrics of $$$L_{\varphi}[a, b]$$$ spaces, which are generated by continuous non-decreasing functions $$$\varphi$$$ that are equal to zero in zero. We study the classes of functions $$$\varphi$$$, for which the obtained bound is exact for all convex moduli of continuity.