This article is cited in 2 papers

On the Rate of Approximation of Singular Functions by Step Functions

J. V. Tikhanov

M. V. Lomonosov Moscow State University

Abstract: We consider approximations of a monotone function on a closed interval by step functions having a bounded number of values: the dependence on the number of values of the rate of approximation in the norm of the spaces $L_p$ is studied. A criterion for the singularity of the function in terms of the rate of approximation is obtained. For self-similar functions, we obtain sharp estimates of the rate of approximation in terms of the self-similarity parameters. Functions with arbitrarily fast and arbitrarily slow (down to the theoretic limit) rate of approximation are constructed.

Keywords: approximations of monotone functions by step functions, the space $L_p$, self-similar function, criterion for the singularity of functions, Hölder's inequality, Lebesgue–Stieltjes measure, Cantor function, Lebesgue measure.

UDC: 517.518

Received: 14.04.2013

DOI: 10.4213/mzm10288

 English version: Mathematical Notes, 2014, 95:4, 530–543

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