It is primarily a text for a course at the advanced undergraduate level, but i hope it will also be useful as a reference for people who have taken such a course and continue to use fourier analysis in their later work. The reader is presumed to have i a solid back- ground in calculus of one and several variables, ii knowledge of the elementary theory of linear ordinary differential equations i. In addition, the theory of analytic functions power series, contour integrals, etc. I have written the book so that lack of knowledge of complex analysis is not a serious impediment; at the same time, for the beneft of those who do know the subject, it would be shame not to use it when it arises naturally. In particular, the Laplace transform without analytic functions is like Popeye without his spinach. At any rate the facts from complex analysis that are used here are summarized in appendix 2 The subject of this book is the whole circle of ideas that includes Fourier series, Fourier and Laplace transforms, and eigenfunction expansions for differ ential operators.
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It is primarily a text for a course at the advanced undergraduate level, but i hope it will also be useful as a reference for people who have taken such a course and continue to use fourier analysis in their later work. The reader is presumed to have i a solid back- ground in calculus of one and several variables, ii knowledge of the elementary theory of linear ordinary differential equations i.
In addition, the theory of analytic functions power series, contour integrals, etc. I have written the book so that lack of knowledge of complex analysis is not a serious impediment; at the same time, for the beneft of those who do know the subject, it would be shame not to use it when it arises naturally.
In particular, the Laplace transform without analytic functions is like Popeye without his spinach. At any rate the facts from complex analysis that are used here are summarized in appendix 2 The subject of this book is the whole circle of ideas that includes Fourier series, Fourier and Laplace transforms, and eigenfunction expansions for differ ential operators.
Since i thereby hope to please both the pure and the ap plied factions but run the risk of pleasing neither, i should give some explanation of what i am trying to do and why i am trying to do it First, this book deals almost exclusively with those aspects of fourier analysis that are useful in physics and engineering rather than those of interest only in pure mathematics. On the other hand, it is a book on applicable mathematics rather than applied mathematics: the principal role of the physical applications herein is to illustrate and illuminate the mathematics, not the other way around I have refrained from including many applications whose principal conceptual content comes from Subject X rather than Fourier analysis, or whose appreciation requires specialized knowledge from Subject X; such things belong more properly in a book on Subject x where the background can be more fully explained.
Many of my favorite applications come from quantum physics, but in accordance with this principle I have mentioned them only briefly. Similarly, I have not worried too much about the physical details of the applications studied here. For example, when i think about the 1-dimensional heat equation i usually envision a long thin rod, but one who prefers to envision a 3-dimensional slab whose temperature varies only along one axis is free to do so; the mathematics is the same vi Preface Second, there is the question of how much emphasis to lay on the theoretical aspects of the subject as opposed to problem-solving techniques.
I firmly believe that theory meaning the study of the ideas underlying the subject and the reasoning behind the techniques - is of intellectual value to everyone, applied or pure. On the other hand, i do not take"theory "to be synonymous with"logical rigor. Of course, where to draw the line is a matter of judgment, and I suppose nobody will be wholly satisfied with my choices.
But those instructors who wish to include more details in their lectures are free to do so, and readers who tire of a formal argument have only to skip to the end-of-proof sign L. Thus, the book should be fairly flexible with regard to the level of rigor its users wish to adopt.
One feature of the theoretical aspect of this book deserves special mention The development of lebesgue integration and functional analysis in the period has led to enormous advances in our understanding of the concepts underlying Fourier analysis.
For example the completeness of L and the shift from pointwise convergence to norm convergence or weak convergence simplifies much of the discussion of orthonormal bases and the validity of series expansions These advances have usually not found their way into application-oriented books because a rigorous development of them necessitates the building of too much machinery.
However, most of this machinery can be ignored if one is willing to take a few things on faith, as one takes the intermediate value theorem on faith in freshman calculus. I then proceed to use these facts without further ado.
The dominated convergence theorem, it should be noted is a wonderful tool even in the context of Riemann integrable functions. Later, in Chapter 9, I develop the theory of distributions as linear functionals on test functions, the motivation being that the value of a distribution on a test function is a smeared-out version of the value of a function at a point.
Discussion of functional-analytic technicalities which are largely irrelevant at the elementary level is reduced to a minimum With the exception of the prerequisites and the facts about Lebesgue integra- tion mentioned above, this book is more or less logically self-contained.
First, there are some minor dependences that are not shown in the diagram For example, a few paragraphs of text and a few exercises in Sections 6. Hence, one could cover Sections I have taught a one-quarter ten-week course from Chapters and a sequel to it from Chapters , omitting a few items here and there One further point that instructors should keep in mind is the following.
Most of the book deals with rather concrete ideas and techniques, but there are two viii Preface places where concepts of a more general and abstract nature are discussed in a serious way: Chapter 3 L spaces, orthogonal bases, Sturm- Liouville problems and Chapter 9 functions as linear functionals, generalized functions. These parts are likely to be difficult for students who have had little experience with abstract mathematics, and instructors should plan their courses accordingly ourier analysis and its allied subjects comprise an enormous amount of mathematics about which there is much more to be said than is included in this book.
I hope that my readers will find this fact exciting rather than dismaying Accordingly, I have included a sizable although not exhaustive bibliography of books and papers to which the reader can refer for more information on things that are touched on lightly here Most of these references should be reasonably accessible to the students for whom this book is primarily intended, but a few of them are of a considerably more advanced nature this is inevitable; the topics in this book impinge on a lot of sophisticated material and the full story on some of the things discussed here singular Sturm-Liouville problems, for instance cannot be told without going to a deeper level.
Fourier analysis is a powerful tool for many problems, and especially for solving various differential equations of interest in science and engineering.
The purpose of this introductory chapter is to provide some background concerning partial differential equations Specifically, we introduce some of the basic equations of mathematical physics that will provide examples and motivation throughout the book and we discuss a technique for solving them that leads directly to problems in Fourier analysis At the outset, let us present some notations that will be used repeatedly.
The real and complex number systems will be denoted by R and C, respectively We shall be working with functions of one or several real variables x1,. Similarly, a function of n real variables is said to be of class c on a set dc Rn if all of its partial derivatives of order k exist and are continuous on D. If the function possesses continuous derivatives of all orders.
Overture 1. If u is a function of the real variables xI, More precisely, xl,,,Xn are the coordinates of a point x in the medium; t is the time; c is the speed of propagation of waves in the medium; and u x, t is the amplitude of the wave at position x and time t The wave equation provides a reasonable mathematical model for a number of physical processes, such as the following a vibrations of a stretched string, such as a guitar string b vibrations of a column of air, such as an organ pipe or clarinet c Vibrations of a stretched membrane, such as a drumhead.
In a , c ,and d ,u represents the transverse displacement of the string, membrane, or fluid surface; in b and e ,u represents the lon gitudinal displacement of the air; and in f , u is any of the components of the electromagnetic field We shall not attempt to derive the wave equation from physical principles here, since each of the preceding examples involves different physics.
We should point out, however, that in most cases the derivation involves making some simplifying assumptions Hence, the wave equation gives only an approximate description of the actual physical process, and the validity of the approximation will depend on whether Numbers in brackets refer to the bibliography at the end of the book.
ISBN 13: 9780534170943
About this title This book presents the theory and applications of Fourier series and integrals, Laplace Transforms, eigenfunction expansions, and related topics. It deals almost exclusively with those aspects of Fourier analysis that are useful in physics and engineering. Using ideas from modern analysis, it discusses the concepts and reasoning behind the techniques without getting bogged down in the technicalities of rigorous proofs. A wide variety of applications are included, in addition to discussions of integral equations and signal analysis.
Fourier Analysis and Its Applications