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Mat. Zametki, 2014, Volume 95, Issue 4, Pages 590–604 (Mi mz10288)  

This article is cited in 2 scientific papers (total 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.

DOI: https://doi.org/10.4213/mzm10288

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

Bibliographic databases:

UDC: 517.518
Received: 14.04.2013

Citation: J. V. Tikhanov, “On the Rate of Approximation of Singular Functions by Step Functions”, Mat. Zametki, 95:4 (2014), 590–604; Math. Notes, 95:4 (2014), 530–543

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. J. V. Tikhonov, S. V. Shaposhnikov, I. A. Sheipak, “On the Singularity of Functions and the Quantization of Probability Measures”, Math. Notes, 102:4 (2017), 587–590  mathnet  crossref  crossref  mathscinet  isi  elib
    2. Yu. V. Tikhonov, “On the relationship between the dimension of the Lebesgue–Stieltjes measure and the rate of approximation of a function by step functions”, Dokl. Math., 97:2 (2018), 157–160  mathnet  crossref  crossref  zmath  isi  elib  scopus
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