Positive Operator-Valued Measures and Frames

Abstract

This chapter, and the three succeeding it, constitute a mathematical interlude, preparing the ground for the formal definition of a coherent state in Chapter 7 and the subsequent development of the general theory. As should be clear already, from a look at the last chapter, in order to define CS mathematically and obtain a synthetic overview of the different contexts in which they appear, it is necessary to understand a bit about positive operator-valued (POV) measures on Hilbert spaces and their close connection with certain types of group representations. In Chapter 2, we have also encountered examples of reproducing kernels and reproducing kernel Hilbert spaces, which in turn are intimately connected with the notion of POV measures and, hence, coherent states. In this chapter, we gather together the relevant mathematical concepts and results about POV measures. In the next chapter, we will do the same for the theory of groups and group representations. Chapters 5 and 6 will then be devoted to a study of reproducing kernel Hilbert spaces. The treatment is necessarily condensed, but we give ample reference to more exhaustive literature. Although the mathematically initiated reader may wish to skip these four mathematical chapters, the discussion of many of the topics here is sufficiently different from their treatment in standard texts to warrant at least a cursory glance at it.