Fourier Series and Orthogonal Functions

Harry F. Davis

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486140735
• Topic: Mathématiques
• Subtopic: Analyse mathématique

This incisive text deftly combines both theory and practical example to introduce and explore Fourier series and orthogonal functions and applications of the Fourier method to the solution of boundary-value problems. Directed to advanced undergraduate and graduate students in mathematics as well as in physics and engineering, the book requires no prior knowledge of partial differential equations or advanced vector analysis. Students familiar with partial derivatives, multiple integrals, vectors, and elementary differential equations will find the text both accessible and challenging.
The first three chapters of the book address linear spaces, orthogonal functions, and the Fourier series. Chapter 4 introduces Legendre polynomials and Bessel functions, and Chapter 5 takes up heat and temperature. The concluding Chapter 6 explores waves and vibrations and harmonic analysis. Several topics not usually found in undergraduate texts are included, among them summability theory, generalized functions, and spherical harmonics.
Throughout the text are 570 exercises devised to encourage students to review what has been read and to apply the theory to specific problems. Those preparing for further study in functional analysis, abstract harmonic analysis, and quantum mechanics will find this book especially valuable for the rigorous preparation it provides. Professional engineers, physicists, and mathematicians seeking to extend their mathematical horizons will find it an invaluable reference as well.

CHAPTER 1

LINEAR SPACES

1.1

FUNCTIONS

We begin by stating certain conventions that will be used throughout this book.

The words number and scalar, when not otherwise qualified, may be regarded as synonyms for real number. On the other hand, if they are consistently interpreted to mean complex number, most of the statements made are still valid.

Similarly, the beginner who wishes to do so can take the word function to mean real-valued function, unless indicated otherwise, and in this case complex conjugates can be ignored. Simply read ƒ(x) for

) and α² for |α|², since for real functions

and for real numbers |α|² = α².

The term domain of a function is used for brevity instead of domain of definition of a function. The reader must distinguish between functions for which no domain is specified, and those for which a domain is specified. If no domain is specified, and no indication given to the contrary, the reader should assume that ƒ(x) is defined for every real number x. To carry matters a little further, suppose we ask the reader to consider a function ƒ with the property ƒ(−x) = −ƒ(x). Then the reader may consider the function sin x, or any other function he chooses (having this property) but not csc x, since csc x is not defined when x is an integral multiple of π.

If the domain is specified, then the reader must regard the function as defined only within this domain. This will be clarified later by examples.

A distinguishing characteristic of advanced mathematics is that a function is regarded as a single object, just as points, numbers, and vectors are regarded as single objects. We usually emphasize this by using a single letter to denote a function. For example, if we specify that ƒ(x) = x² for every x, then we have specified a function ƒ. The single letter ƒ denotes the function; ƒ(x) refers to a number, namely the value the function corresponds with x. In this example, if x = 3, then ƒ(x) = 9.

We shall not be pedantic in this matter, however. To avoid circumlocution, we will sometimes refer to “the function x²” or ”the function sin x” where we should more properly say “the function ƒ defined by the requirement that for every real number x the value ƒ(x) is x²” or “the function g where g(x) = sin x for every x.”

Throughout this book function always means single-valued function. For any x in the domain of a function ƒ, ƒ(x) is a single number which is not ambiguous; we do not discuss multiple-valued functions.

If two functions ƒ and g have the same domain, we say they are equal and write ƒ = g if and only if ƒ(x) = g(x) for every x in their domain. Functions having different domains of definition are never regarded as equal.

The sum of two functions is defined only when the functions have the same domain. If ƒ and g are functions with the same domain of definition D, their sum ƒ + g is, by definition, the function having domain D whose values are the sum of the corresponding values of ƒ and g. That is, the values of ƒ + g are given by

(ƒ + g)(x) = ƒ(x) + g(x)

for every x in D.

The same qualification applies to the product of two functions, which is defined by

(ƒg)(x) = [ƒ(x)][g(x)].

This is sometimes called the pointwise product to distinguish it from the convolution product th...