Laplace Operators on Fractals and Related Functional Equations

We give an overview over the application of functional equations, namely the classical Poincar\'e and renewal equations, to the study of the spectrum of Laplace operators on self-similar fractals. We compare the techniques used to those used in the euclidean situation. Furthermore, we use the obtained information on the spectral zeta function to define the Casimir energy of fractals. We give numerical values for this energy for the Sierpi\'nski gasket

Divide-and-Conquer Recurrences, Functional Equations and Their Asymptotic Analysis

Introduction and formulation of main results There are several important problems, stemming from the theory of algorithms, probability theory etc., leading to the study of the asymptotic behaviour of the sequence ff n g defined by means of the recurrence relation [2 n \Gamma (a 1 + a 0 )]f n = a 1 n\Gamma1 X k=0 ` n k ' ff n\Gammak f k + b 1 fi n ; n ? l; (1.1) supplemented by given initial values f 0 ; f 1 ; : : : ; f l : (1.2) Here ff; fi ? 0, a 0 ; a 1 ; b 1 2 R are given parameters. Suppose

ASYMPTOTICS OF THE POINCARÉ FUNCTIONS

The asymptotic behaviour of the solutions of of Poincaré’s functional equation f(λz) = P(f(z)) (λ ∈ C, |λ |> 1) for P a polynomial of degree ≥ 2 is studied in different regions of the complex plane