Integral Operators

Abstract

In this chapter we introduce a certain type of integral operators that shall play an important role later on. The simplest ones in this class are Laplace operators, while some of the others will be Ecalle’s acceleration operators, that will be studied in detail in Chapter 11. All of them will be used in the next chapter to define some summability methods that, in the terminology common in this field, are called moment methods. As we shall prove, most of these methods are equivalent in the sense that they sum the same formal power series to the same holomorphic functions. The reason for investigating all these equivalent methods is that for particular formal power series it will be easier to check applicability of a particular method. Thus having all of them at our disposal gives a great deal of flexibility. Moreover, it also is of theoretical interest to know what properties of the methods are needed to sum a certain class of formal power series. It should, however, be noted that statements on methods being equivalent are here to be understood for summation of power series in interior points of certain regions, while the methods may be inequivalent when studying the same series at some boundary point.