  Let $\Omega$ be a perfectly normal topological space, let $A$ be a non-empty
$G_\delta$-subset of $\Omega$ and let $B_1(A)$ denote the space of all
functions $A\to\mathbb{R}$ of Baire-one class on $A$. Let also
$\|\cdot\|_\infty$ be the supremum norm. The symbol $\chi_A$ stands for the
characteristic function of $A$. We prove that for every bounded function $f\in
B_1(A)$ there is a sequence $(H_n)$ of both $F_\sigma$- and $G_\delta$-subsets
of $\Omega$ such that the function $\overline{f}\colon\Omega\to\mathbb{R}$
given by the uniformly convergent series on $\Omega$ with the formula:
$\overline{f}:=c\sum_{n=0}^\infty\left(\frac{2}{3}\right)^{n+1}\left(\frac{1}{2}-\chi_{H_n}\right)$
extends $f$ with $\overline{f}\in B_1(\Omega)$ and the condition ($\triangle$)
of the form: $\|f(A)\|_\infty=\|\overline{f}(\Omega)\|_\infty$. We apply the
above series to obtain an extension of $f$ positive to $\overline{f}$ positive
with the condition ($\triangle$). A similar technique allows us to obtain an
extension of Baire-alpha function on $A$ to Baire-alpha function on $\Omega$.
