Geodesic
On the Rate of Approximation of Singular Functions by Step Functions
Matematičeskie zametki, Tome 95 (2014) no. 4, pp. 590-604.

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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, Cantor function
Mots-clés : Lebesgue–Stieltjes measure, Lebesgue measure. 
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J. V. Tikhanov. On the Rate of Approximation of Singular Functions by Step Functions. Matematičeskie zametki, Tome 95 (2014) no. 4, pp. 590-604. http://geodesic.mathdoc.fr/item/MZM_2014_95_4_a9/

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