These Lecture Notes are based on a series of lectures I gave at the Linkoping University Department of Electrical Engineering in 1988. In these lectures 1 tried to give an overview of the theory of representation of compact groups and some applications in the fields of image science and pattern recognition.

The participants in the course had a masters degree in electrical engineering and no deeper knowledge in group theory or functional analysis. The first chapter is thus used to introduce some basic concepts from (algebraic) group theory, topology, functional analysis and topological groups. This chapter contains also some of the main results from these fields that are used in the following chapters. This chapter cannot replace full courses in algebra, topology or functional analysis but it should give the reader an impression of the main concepts, tools and results of these branches of mathematics. Some of the groups that will be studied more extensively in the following chapters are also introduced in this chapter.

The central idea of group representations is then introduced in Chapter 3. The basic theme in this chapter is the study of function spaces that are invariant under a group of transformations such as the group of rotations. It will be demonstrated how the algebraic and topological properties of the group determine the structure of these invariant spaces: if the transformation group is commutative, it will be shown that the minimal, invariant spaces are all one-dimensional and if the group is compact, then these spaces are all finite-dimensional. This result explains, for example, the unique role of the complex exponential function. In this chapter we concentrate on the derivation of qualitative information about the invariant function spaces but (with the exception of the complex exponential function that is connected to the group of real numbers R and the group of 2- D rotations) no such function spaces are actually constructed. The construction of these invariant function spaces for some important groups is demonstrated in Chapter 4. There we use methods from the theory of Lie groups to find the functions connected to certain important transformation groups. These groups include the scaling group, the 2-D and 3-D rotation groups, the group of rigid motions and the special linear group of the 2x2 matrices with complex entries and determinant equal to one.

In Chapter 5 we conclude the mathematical part of these lectures by demonstrating how the concept of a Fourier series can be generalized. The basic idea behind this gen- eralization is the following: Consider a periodic function f(<j>) with period 2т. Usually we think of such a function as a (complex-valued) function on an interval or a circle. In the new interpretation we view фas the angle of a rotation and / becomes thus a func- tion defined on the group of 2-D rotations. The decomposition of / into a Fourier series means thus that a function on the group of 2-D rotations can be decomposed into a series of complex exponentials. In Chapter 5 we demonstrate that a similar decomposition is possible for functions defined on compact groups.