Abstract:
Let
$\Omega = {\mathcal A}^{{\mathbb N}}$ be a space of right-sided infinite sequences
drawn from a finite alphabet ${\mathcal A} = \{0,1\}$,
${\mathbb N} = \{1,2,\dots \} $.
Let
$$
\rho(\mathbf{x},\mathbf{y}) =
\sum_{k=1}^{\infty}|x_{k} - y_{k}|2^{-k}
$$
be a metric on $\Omega = {\mathcal A}^{{\mathbb N}}$,
and $\mu$ the Bernoulli measure on $\Omega$ with probabilities
$p_0,p_1>0$, $p_0+p_1=1$.
Denote by
$B(\mathbf{x},\omega)$
an open ball
of radius $r$ centered at $\mathbf{\omega}$.
The main result of this paper is
$$
\mu\left(B(\mathbf{\omega},r)\right)
=
r+\sum_{n=0}^{\infty}\sum_{j=0}^{2^n-1}\mu_{n,j}(\mathbf{\omega})\tau(2^nr-j),
$$
where $\tau(x) =2\min\{x,1-x\}$, $0\leq x \leq 1$, ($\tau(x) = 0$, if $x<0$ or $x>1$),
$$
\mu_{n,j}(\mathbf{\omega}) = \left(1-p_{\omega_{n+1}}\right)
\prod_{k=1}^n p_{\omega_k\oplus j_k},\ \ j = j_12^{n-1}+j_22^{n-2}+\dots+j_n.
$$
The family of functions $1,x,\tau(2^nr-j)$, $j =0,1,\dots,2^n-1$, $n=0,1,\dots$, 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.
Keywords:Faber–Schauder system, Haar wavelet, self-similar, Lebesgue's function.
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