Since a large class of physical problems is defined on bounded domains, we focus on integral equations on bounded domains. As we know, a bounded domain in ℝd may be well approximated by a polygonal domain, which is a union of simplexes, cubes and perhaps L-shaped domains. To develop fast Galerkin, Petrov–Galerkin and collocation methods for solving the integral equations, we need multi scale bases and collocation functionals on polygonal domains. Simplexes, cubes or L-shaped domains are typical examples of invariant sets. This chapter is devoted to a description of constructions of multi scale basis functions including multi scale orthogonal bases, interpolating bases and multi scale collocation functionals. The multi scale basis functions that we construct here are discontinuous piecewise polynomials. For this reason, we describe their construction on invariant sets which can turn to bases on a polygon.

To illustrate the idea of the construction, we start with examples on [0, 1], which is the simplest example of invariant sets. This will be done in Section 4.1. Constructions of multi scale basis functions and collocation functionals on invariant sets are based on self-similar partitions of the sets. Hence, we discuss such partitions in Section 4.2. Based on such self-similar partitions, we describe constructions of multi scale orthogonal bases in Section 4.3. For the construction of the multi scale interpolating basis, we require the availability of the multi scale interpolation points. Section 4.4 is devoted to the notion of refinable sets, which are a base for the construction of the multi scale interpolation points. Finally, in Section 4.5, we present the construction of multi scale interpolating bases.

Multi scale functions on the unit interval

This section serves as an illustration of the idea for the construction of orthogonal multi scale piecewise polynomial bases on an invariant set. We consider the simplest invariant set Ω ≔ [0, 1] in this section.