Singular points of a self-similar function of spectral order zero: self-similar Stieltjes string

In the present paper, we introduce the notion of self-similar function of spectral order zero and study its properties. Such functions have at most countably many discontinuity points, and these points are discontinuity points of the first kind except possibly for a single point, which is a singular point. We derive a formula for calculating the coordinates of this point from the parameters of the self-similar function. We also study the behavior of the self-similar function near the singular point. A nondecreasing function $f$ of spectral order zero belonging to the space $L_2[0,1]$ generates a self-similar Stieltjes string, namely, a spectral problem of the form
$$ -y''-\lambda\rho y=0,\qquad y(0)=y(1)=0, $$
where $\rho$ is a function from the space $\mathring W_2^{-1}[0,1]$ and $f'=\rho$. Such a function $f$ that is not of a fixed sign leads to the notion of self-similar indefinite Stieltjes string.