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Automatic Control and Computer Sciences

, Volume 51, Issue 7, pp 586–591 | Cite as

Expansion of Self-Similar Functions in the Faber–Schauder System

  • E. A. Timofeev
Article

Abstract

Let Ω = AN be a space of right-sided infinite sequences drawn from a finite alphabet A = {0,1}, N = {1,2,…}. Let ρ(x, yk=1|x k y k |2k be a metric on Ω = AN, and μ the Bernoulli measure on Ω with probabilities p0, p1 > 0, p0 + p1 = 1. Denote by B(x,ω) an open ball of radius r centered at ω. The main result of this paper \(\mu (B(\omega ,r))r + \sum\nolimits_{n = 0}^\infty {\sum\nolimits_{j = 0}^{{2^n} - 1} {{\mu _{n,j}}} } (\omega )\tau ({2^n}r - j)\), where τ(x) = 2min {x,1 − x}, 0 ≤ x ≤ 1, (τ(x) = 0, if x < 0 or x > 1 ), \({\mu _{n,j}}(\omega ) = (1 - {p_{{\omega _{n + 1}}}})\prod _{k = 1}^n{p_{{\omega _k}}} \oplus {j_k}\), \(j = {j_1}{2^{n - 1}} + {j_2}{2^{n - 2}} + ... + {j_n}\). The family of functions 1, x, τ(2 n rj), j = 0,1,…, 2 n − 1, n = 0,1,…, is the Faber–Schauder system for the space C([0,1]) of continuous functions on [0, 1]. We also obtain the Faber–Schauder expansion for Lebesgue’s singular function, Cezaro curves, and Koch–Peano curves. Article is published in the author’s wording.

Keywords

Faber–Schauder system Haar wavelet self-similar Lebesgue’s function Cezaro curves Koch–Peano curves 

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References

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Copyright information

© Allerton Press, Inc. 2017

Authors and Affiliations

  1. 1.Demidov Yaroslavl State UniversityYaroslavlRussia

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