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".
Erscheinungsdatum: 26.11.2013

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nexusstc/Integral Transformations, Operational Calculus, and Generalized Functions/d000f48a0b09b76640b52a2defe97754.pdf

zlib/Mathematics/R. G. Buschman (auth.)/Integral Transformations, Operational Calculus, and Generalized Functions_2296686.pdf

Buschman, R.G.

Springer US : Imprint : Springer

Kluwer Academic Publishers

Mathematics and its applications (Kluwer Academic Publishers), v. 377, Dordrecht ; Boston, ©1996

Mathematics and its applications (D. Reidel Publishing Company), 1st ed. 1996, Dordrecht, 1996

Mathematics and Its Applications, 377, Boston, MA, 1996

United States, United States of America

1996, 2013

lg1127977

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It is not the object of the author to present comprehensive covƯ 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 transformaƯ tion is taken as the model type of an integral transformation and a number of its properties are developed; later, the Fourier transforƯ mation is introduced. The operational calculus of Mikusinski is preƯ 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 analƯ ogous to the construction of the reals from the rationals by means of Cauchy sequences. A chapter with sections on a variety of transƯ 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 approƯ 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 actuƯ ally exist for such entities as the "Dirac delta-function."

Various Related Ideas Are Explored In This Introductory Survey, But No Topic Is Treated Thoroughly, Since Monographs Are Available On Each Topic Separately. 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. There Are More Examples Than Proofs, Including Applications In Which The Computations Do Not Overwhelm The Ideas. A Calculus Background And A Scattering Of Other Mathematical Ideas Is Needed For Self Study. Deeper Or Lengthier Problems Are Inserted Along With Simpler Exercises. Several Short Tables Of Useful Transform Pairs Are Included. Audience: Upper Level Undergraduates And Beginning Graduate Students In Mathematics, Physics And Engineering. By R. G. Buschman.

Front Matter....Pages i-xiii
Laplace Transformations....Pages 1-61
Mikusiński Operators....Pages 63-80
Fourier Transformations....Pages 81-106
Generalized Functions....Pages 107-125
Other Transformations....Pages 127-166
Back Matter....Pages 167-240

2014-01-18