We study orthogonal polynomials with respect to self-similar measures,
focusing on the class of infinite Bernoulli convolutions, which are defined by iterated
function systems with overlaps, especially those defined by the Pisot, Garsia, and
Salem numbers. By using an algorithm of Mantica, we obtain graphs of the coefficients
of the 3-term recursion relation defining the orthogonal polynomials. We use
these graphs to predict whether the singular infinite Bernoulli convolutions belong
to the Nevai class. Based on our numerical results, we conjecture that all infinite
Bernoulli Convolutions with contraction ratios greater than or equal to 1/2 belong
to Nevai's class, regardless of the probability weights assigned to the self-similar
For Laplacians defined by measures on a bounded domain in R^n, we prove analogs of the classical eigenvalue estimates for the standard Laplacian: lower bound of sums of eigenvalues by Li and Yau, and gaps of consecutive eigenvalues by Payne, Polya and Weinberger. This work is motivated by the study of spectral gaps for Laplacians on fractals.
We study the heat kernel expansion of the Laplacian on n-forms defined on a subgraph of a directed complete graph. We derive two expressions for the subgraph heat kernel on 0-forms and compute the coefficients of the expansion. We also obtain the subgraph heat kernel of the Laplacian on 1-forms.