Distribution Theory and Transform Analysis: An Introduction to Generalized Functions, with Applications (Dover Books on Mathematics) by A.H. Zemanian (PDF)

Description

Distribution theory, a relatively recent mathematical approach to classical Fourier analysis, not only opened up new areas of research but also helped promote the development of such mathematical disciplines as ordinary and partial differential equations, operational calculus, transformation theory, and functional analysis. This text was one of the first to give a clear explanation of distribution theory; it combines the theory effectively with extensive practical applications to science and engineering problems.Based on a graduate course given at the State University of New York at Stony Brook, this book has two objectives: to provide a comparatively elementary introduction to distribution theory and to describe the generalized Fourier and Laplace transformations and their applications to integrodifferential equations, difference equations, and passive systems.After an introductory chapter defining distributions and the operations that apply to them, Chapter 2 considers the calculus of distributions, especially limits, differentiation, integrations, and the interchange of limiting processes. Some deeper properties of distributions, such as their local character as derivatives of continuous functions, are given in Chapter 3. Chapter 4 introduces the distributions of slow growth, which arise naturally in the generalization of the Fourier transformation. Chapters 5 and 6 cover the convolution process and its use in representing differential and difference equations. The distributional Fourier and Laplace transformations are developed in Chapters 7 and 8, and the latter transformation is applied in Chapter 9 to obtain an operational calculus for the solution of differential and difference equations of the initial-condition type. Some of the previous theory is applied in Chapter 10 to a discussion of the fundamental properties of certain physical systems, while Chapter 11 ends the book with a consideration of periodic distributions.Suitable for a graduate course for engineering and science students or for a senior-level undergraduate course for mathematics majors, this book presumes a knowledge of advanced calculus and the standard theorems on the interchange of limit processes. A broad spectrum of problems has been included to satisfy the diverse needs of various types of students.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I am trying to teach myself this subject which was never taught in school for me.Cons:#1) this is a “dover” publication. which means for those of you who are difficult learners (i am), all of the whitespace has been expunged from the book in the interests of saving costs. (Nowhere to pause, write notes, glean emphasis etc etc).#2) the author completely ignores the mystery and beauty of the massive subject of functionals and distributions, making do instead with a non stop dry, prosaic, solely mathematical exposition of “facts”. Unfair? Chap 1, Line 1: author asserts it’s better to view physical variables not as classical functions of time, but as functionals evaluated on INTERVALS……(then straight on to spaces of test fxns etc). Whoa? not even one allusion to/about the related concept of “action” in physics! further, discussing the “distributional derivative in Ch2. not even ONE mention of the vast and foundational applications in Physics and Engrg of “calculus of variations/distributions” as in the Euler-Lagrange principle! I’m not suggesting the author fully branch into that, but at least BRING UP the name in a footnote so someone can go look into it later! then there’s the most dry approach to pseudo-functions known to man….enough to put into fuzzy coma/stupor.Pros:I have gleaned there are two other books in this field: Kanwal’s “Generalized Functions” and Lighthill’s book. The latter is an author who is a little “too brilliant” for my (dimwitted) taste, and the former is even more dry than Zemanian. So Zemanian is the defaulting best choice, but keep GOOGLE open as you read it if you are teaching it to yourselfs, this subject is TOO important to miss.

⭐Good Book.

⭐This is an good book for those who are struggling to understand generalized functions and distributions in particle physics and other areas.

⭐Just a note for readers of these reviews: there are two delta functions used in signal processing. In digital signal processing the Kronecker delta is used, which is simple. In continuous-time signal processing the Dirac delta is used, which is a generalized function and causes sophomores no end of headaches. Zemanian’s book will be helpful for those wishing to know more about generalized functions such as the Dirac delta. Those just wishing to know a little more might find the articles on Wikipedia about Dirac delta more helpful.

⭐This is a good second or third book on generalized functions which are otherwise known as “distributions”. A good first book is “Fourier Analysis and Generalised Functions” by Lighthill. This book is based on a graduate course and provides a good introduction to distribution theory and generalized Fourier analysis. You need to have a good background in advanced calculus and be comfortable with Lebesgue theorems concerning interchangeable limit processes. The examples are quite good, but the notation is intense and the way that it is packed in small font onto a page doesn’t help. The writing is compact and thorough. I recommend this book to anyone who has had the standard digital signal processing course where the Delta function (distribution) is used freely and who has wondered what it really means.

⭐Muy bueno.This book covers, from a rigorous perspective, the basic theory of distributions. It contains a good mixture of theory and applications. It also extends from the basics in a nice way, covering topics one doesn’t usually encounter in such introductions. Namely, ultradistributions and periodic distributions.I would say that this book is perfect for an analyst, or somebody with an interest in gaining a deeper insight into linear differential (ordinary and partial) equations. It is equally well suited to a physicist seeking to gain deeper technical knowledge of generalized functions.

⭐Really nice

⭐Tudo certo e no prazo.

⭐

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