An efficient and reliable quadrature algorithm for integration with respect to binomial measures

Abstract

In this work numerical methods for integration with respect to binomial measures are considered. Binomial measures are examples of fractal measures and arise when multifractal properties are investigated. Interpolatory quadrature rules are considered. An automatic integrator with local quadrature rules that generalize the five points Newton Cotes formula and error estimates based on null rules is then described. Numerical tests are performed to verify the efficiency and accuracy of the method. These tests confirm that the automatic integrator turns out to be as good as one of the best known quadrature algorithms with respect to the Lebesgue measure. (2000): 28A25, 60G18, 65D30, 65D32, 68M15. AMS subject classification

An evaluation of Clenshaw–Curtis quadrature rule for integration w.r.t. singular measures

“…In general we will be able to do this for balanced measures, that are particular invariant measures, see [10,Section 4]. Relevance of these measures for applications has been highlighted in [1,18]. In this section we will also see the connection with the recently introduced linear refinable functionals [14].…”

Simple Monte Carlo integration with respect to Bernoulli convolutions

“…We selected two functions from [4], namely 1 H 1 (y) = sin(πy) + 1 4 cos(8πy) and H 2 (y) = |y|. We deliberately chose H 2 not of class C 1 in order to evaluate convergence when our bounds are uninformative.…”

“…However, other ways of computing the value of these integrals may be considered (such as the adaptive procedure presented in [4]), giving similar numbers.…”

“…In [4], Calabrò and Corbo Esposito have proposed a number of quadrature formulae for binomial measures, although extending their work to general Bernoulli convolutions is not straightforward. Another possible approach to integration with respect to these distributions is given by Monte Carlo procedures.…”

Refinable functions, functionals, and iterated function systems

“…The scope of this paper is to outline as simply as possible the translation language between the two approaches, and to expose the main results that can be useful to carry from one side to the other, like e.g the techniques for the evaluation of the integrals [8,9,11,32,35]. Some previous work in this direction can be found in [34,40].…”