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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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http://mi.mathnet.ru/eng/mz10288https://doi.org/10.4213/mzm10288 http://mi.mathnet.ru/eng/mz/v95/i4/p590
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This publication is cited in the following articles:
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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
 -
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
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