Integration with Respect to Linearly Balanced Measures

In applications that imply an iterated process, often it is necessary to define some object used to study limit properties of the process and in many cases self-similarity is one of these properties. This is expressed as a balancing law that has to be respected at different scales. The self-similarity property defines fractals, and in this chapter we will consider measures with a linear balancing property, thus a noticeable example of fractal measures. These measures are well defined as a fixed point of a contraction on the set of probability measures (see ¶2). Moments and the coefficients of the orthogonal polynomials with respect to these measures can be computed by recursively, as we will review in ¶4. A new result is presented in section ¶4.2 where we explore a nested relation that allows to calculate also the modified moments. As an application, in section 5 we develop quadrature rules for numerical integration with respect to linearly balanced measures. In order to construct general formulae, null rules are presented and their use for the construction of general interpolatory rules is described in sections ¶5.1 and ¶5.2 . This technique uses the definition of modified moments, therefor it is possible to use it in this framework in virtue of the recurrence relation presented before. The procedure can be quite widely applied, but is not optimal in some cases where other properties can be used. For this reason we conclude our work with a section on the construction of the Gauss rules (and its modifications) and the Clenshaw-Curtis rules.