Integral Transformations, Operational Calculus, and Generalized Functions: 377 (Mathematics and Its Applications, 377) - Hardcover

Buschman, R.G.

Hardcover

ISBN 10: 079234183X ISBN 13: 9780792341833

Publisher: Springer, 1996

Synopsis

It is not the object of the author to present comprehensive cov­ erage of any particular integral transformation or of any particular development of generalized functions, for there are books available in which this is done. Rather, this consists more of an introductory survey in which various ideas are explored. The Laplace transforma­ tion is taken as the model type of an integral transformation and a number of its properties are developed; later, the Fourier transfor­ mation is introduced. The operational calculus of Mikusinski is pre­ sented as a method of introducing generalized functions associated with the Laplace transformation. The construction is analogous to the construction of the rational numbers from the integers. Further on, generalized functions associated with the problem of extension of the Fourier transformation are introduced. This construction is anal­ ogous to the construction of the reals from the rationals by means of Cauchy sequences. A chapter with sections on a variety of trans­ formations is adjoined. Necessary levels of sophistication start low in the first chapter, but they grow considerably in some sections of later chapters. Background needs are stated at the beginnings of each chapter. Many theorems are given without proofs, which seems appro­ priate for the goals in mind. A selection of references is included. Without showing many of the details of rigor it is hoped that a strong indication is given that a firm mathematical foundation does actu­ ally exist for such entities as the "Dirac delta-function".

About this title

Synopsis

Various related ideas are explored in this introductory survey text of integral transformations, operational calculus and generalized functions. The laplace transformation serves as a model for the developments of the operational properties of several other transformations. Further, it provides a connection to the Mikusinski calculus and generalized functions. The development of the Fourier transformation is followed by a different treatment of generalized functions (distributions). These introductions to generalized functions parallel the constructive approaches to the rational and real number systems.

Edition details

• Publisher: Springer
• Publication date: 1996
• Language: English
• ISBN 10: 079234183X
• ISBN 13: 9780792341833
• Binding: Hardcover
• Number of pages: 254