The Expansion of Self-similar Functions in the Faber–Schauder System

Abstract

Let \(\Omega = A^{N}\) be a space of right-sided innite sequences drawn from a nite alphabet \(A = \{0,1\}\), \(N = \{1,2,\dots\}\). Let $$\label{rho} \rho(\boldsymbol{x},\boldsymbol{y}) =\sum_{k=1}^{\infty}|x_{k} - y_{k}|2^{-k} $$ - be a metric on \(\Omega = A^{N}\), and \(\mu\) - the Bernoulli measure on \(\Omega\) with probabilities \(p_0,p_1>0\), \(p_0+p_1=1\). Denote by \(B(\boldsymbol{x},\omega)\) an open ball of radius \(r\) centered at \(\boldsymbol{\omega}\). The main result of this paper is $$ \mu\left(B(\boldsymbol{\omega},r)\right) =r+\sum_{n=0}^{\infty}\sum_{j=0}^{2^n-1}\mu_{n,j}(\boldsymbol{\omega})\tau(2^nr-j), $$ where \(tau(x) =2\min\{x,1-x\}\), \(0\leq x \leq 1\), \(tau(x) = 0, if x<0 or x>1\), $$mu_{n,j}(\boldsymbol{\omega}) = \left(1-p_{\omega_{n+1}}\right) \prod_{k=1}^n p_{\omega_k\oplus j_k},\ \ j = j_12^{n-1}+j_22^{n-2}+\dots+j_n$$. The family of functions \(1,x,\tau(2^nx-j)\), \(j =0,1,\dots,2^n-1\), \(n=0,1,\dots\) 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 the Lebesgue's singular function, Cezaro curves, and Koch{Peano curves.