V.P. Havin – författare

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.

Classical harmonic analysis is an important part of modern physics and mathematics, comparable in its significance with calculus. Created in the 18th and 19th centuries as a distinct mathematical discipline it continued to develop, conquering new unexpected areas and producing impressive applications to a multitude of problems. It is widely understood that the explanation of this miraculous power stems from group theoretic ideas underlying practically everything in harmonic analysis. This book is an unusual combination of the general and abstract group theoretic approach with a wealth of very concrete topics attractive to everybody interested in mathematics. Mathematical literature on harmonic analysis abounds in books of more or less abstract or concrete kind, but the lucky combination as in this volume can hardly be found.

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.