Commutative Harmonic Analysis III

The theory of generalized functions is a general method that makes it possible to consider and compute divergent integrals, sum divergent series, differentiate discontinuous functions, perform the operation of integration to any complex power and carry out other such operations that are impossible in classical analysis. Such operations are widely used in mathematical physics and the theory of differential equations, where the ideas of generalized func­ tions first arose, in other areas of analysis and beyond. The point of departure for this theory is to regard a function not as a mapping of point sets, but as a linear functional defined on smooth densi­ ties. This route leads to the loss of the concept of the value of function at a point, and also the possibility of multiplying functions, but it makes it pos­ sible to perform differentiation an unlimited number of times. The space of generalized functions of finite order is the minimal extension of the space of continuous functions in which coordinate differentiations are defined every­ where. In this sense the theory of generalized functions is a development of all of classical analysis, in particular harmonic analysis, and is to some extent the perfection of it. The more general theories of ultradistributions or gener­ alized functions of infinite order make it possible to consider infinite series of generalized derivatives of continuous functions.

This EMS volume shows the great power provided by modern harmonic analysis, not only in mathematics, but also in mathematical physics and engineering. Aimed at a reader who has learned the principles of harmonic analysis, this book is intended to provide a variety of perspectives on this important classical subject. The authors have written an outstanding book which distinguishes itself by the authors' excellent expository style.
It can be useful for the expert in one area of harmonic analysis who wishes to obtain broader knowledge of other aspects of the subject and also by graduate students in other areas of mathematics who wish a general but rigorous introduction to the subject.