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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     author = {J. V. Tikhanov},
     title = {On the {Rate} of {Approximation} of {Singular} {Functions} by {Step} {Functions}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {590--604},
     publisher = {mathdoc},
     volume = {95},
     number = {4},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2014_95_4_a9/}
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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/