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1989 | Book

The Space of Continuous Functions Read chapter Read first chapter

Continuity, Integration and Fourier Theory

Author: Adriaan C. Zaanen

Publisher: Springer Berlin Heidelberg

Book Series : Universitext

Included in: Professional Book Archive

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About this book

This book is a textbook for graduate or advanced undergraduate students in mathematics and (or) mathematical physics. It is not primarily aimed, therefore, at specialists (or those who wish to become specialists) in integra­ tion theory, Fourier theory and harmonic analysis, although even for these there might be some points of interest in the book (such as for example the simple remarks in Section 15). At many universities the students do not yet get acquainted with Lebesgue integration in their first and second year (or sometimes only with the first principles of integration on the real line ). The Lebesgue integral, however, is indispensable for obtaining a familiarity with Fourier series and Fourier transforms on a higher level; more so than by us­ ing only the Riemann integral. Therefore, we have included a discussion of integration theory - brief but with complete proofs - for Lebesgue measure in Euclidean space as well as for abstract measures. We give some emphasis to subjects of which an understanding is necessary for the Fourier theory in the later chapters. In view of the emphasis in modern mathematics curric­ ula on abstract subjects (algebraic geometry, algebraic topology, algebraic number theory) on the one hand and computer science on the other, it may be useful to have a textbook available (not too elementary and not too spe­ cialized) on the subjects - classical but still important to-day - which are mentioned in the title of this book.

Table of Contents

Frontmatter
Chapter 1. The Space of Continuous Functions
Abstract
Let k be a natural number, i.e., k is an integer satisfying k ≥ 1. By definition, #x211D; k is the set of all points x = (x1,…, xk where x1…, xk are real numbers. These are called the coordinates of the point x. The point with all coordinates zero is called the origin of ℝ k . For k = 1 the set ℝ k is simply the set ℝ of all real numbers. The set ℝ k is a real vector space with respect to the familiar laws of addition and multiplication by real constants, i.e., if x = (x1,…, x k ), y = (y1,…, y k ) and ⋋ is a real number, then x + y = (x1+y1,…,x k + y k ) and ⋋x = (⋋x1 ,…, ⋋x k ).
Adriaan C. Zaanen
Chapter 2. Theorems of Korovkin and Stone-Weierstrass
Abstract
Let Δ = [a, b] be a closed interval in ℝ. The classical approximation theorem of Weierstrass (1885) asserts that any f ∈ C(Δ) can be approximated uniformly by polynomials, i.e., for any є > 0 there exists a polynomial Pє such that |f(x)–Pє (x) | < є holds for all x ∈ Δ. In other words, ||f–Pє|| < є, where || • || denotes the uniform norm in C(Δ). Equivaiently, we may say that there exists a sequence (p n : n = 1,2,…) of polynomials such that ||f–p n || → 0 as n → ∞. Is it possible to denote explicitly a sequence (p n ) satisfying this condition? The answer is affirmative. For Δ = [0,1] we may choose for p n the n-th Bernstein polynomial B n (f), defined on [0,1] by
$${B_n}(f) = \sum\limits_{m = o}^n {\left( {_m^n} \right)f\left( {\frac{m}{n}} \right)} {x^m}{(1 - x)^{n - m}}.$$
(1)
Adriaan C. Zaanen
Chapter 3. Fourier Series of Continuous Functions
Abstract
We begin the present section with some simple definitions (probably already known to most readers). For m, n integers the so-called Kronecker delta δ mn is defined by δ mn = 1 if m = n and δ mn = 0 if m ≠ n. For our second definition, let (f n : n = 0, ±1, ±2,...) be a set of real or complex functions, defined on the subset Δ of ℝ k . The set (f n ) is called an orthonormal set (or orthonormal system) on Δ if
$$f_n^ - $$
(1)
is the complex conjugate of f n and dx stands for dx1… dx k Of course, the definition makes sense only if the integral of f m
$${f_m}f_n^ - $$
(3)
exists for all m, n. The definition is analogous if m and n are restricted to 0,1,2,… or to 1,2,… . If it is only given that
$$\int {_\Delta } fmf_n^ - dx = 0form \ne n,$$
(4)
then (fn) is said to be an orthogonal system on Δ. We immediately mention an example. For n = 0, ±1, ±2,…, let e n (x) = (2π)-1/2einx on ℝ. The system (e n : n = 0, ±1, ±2,…) is orthonormal on any interval [a, a + 2π], i.e., on any interval of length 2π in ℝ. The proof is immediate by observing that
$$2\pi \left\{ {{e_m}\left( x \right)\overline {{e_n}\left( x \right)} } \right\} = {e^{i\left( {m - n} \right)x + i\sin (m - n)x}}.$$
(5)
Adriaan C. Zaanen
Chapter 4. Integration and Differentiation
Abstract
Let (a1,b1;…;a k ,b k ] be an interval in ℝ k , open on the left and closed on the right. Precisely stated, we assume that aj < bj for j = 1,…,k and the interval consists now of all points (x1,…,x k ) in ℝ k such that aj < xj ≤ bj for j = 1,…,k. We shall call an interval of this kind a cell. For reasons of convenience, the empty set will also be called a cell. Observe now that the collection Γ of all cells is not empty and it has the property that if A and B belong to Γ, then A⋂B belongs to Γ and A\B can be written as a finite disjoint union ⋃C n of cells. In the case that B = A or B ⊃ A, all C n in the finite union ⋃C n are then equal to the empty set. In view of the mentioned properties of Γ the collection Γ is called a semiring of subsets of ℝ k . Note that the collection of all intervals that are closed on the left and open on the right is likewise a semiring. On the other hand, the collection of all open (closed) intervals is not a semiring because the boundaries of the intervals cause difficulties. For any cell A = (a1,b1;…, a k ,b k ] we call the product
$$\prod\nolimits_j^k {_{ = 1}({b_j} - {a_j})} $$
(1)
measure A, and we denote this number by µ(A). Furthermore, we define µ(ф) = 0. Of course, to say that µ(A) is the measure of A is a neutral terminology for what is called the length of A if k = 1, the area of A if k = 2 and the content or volume of A if k = 3. The measure is, therefore, a map from Γ into It ℝ having the following properties:
(i)
µ, is non-negative and µ(ф) = 0,
 
(ii)
µ is monotone, i.e., A ⊂ B in Γ implies µ(A) ≤ µ(B),
 
(iii)
µ is σ-additive, i.e., A =
$$\bigcup\nolimits_1^\infty {{A_n}} $$
(2)
(with A , all An ∈ Γ and all A n mutually disjoint) implies
$$\mu (A) = \sum\nolimits_1^\infty {\mu ({A_n})} .$$
(3)
 
Adriaan C. Zaanen
Chapter 5. Spaces L p and Convolutions
Abstract
Before discussing an important special class of measurable functions we prove a fundamental inequality, known as Hölder’s inequality. There are two variants, one for sums and one for integrals. The original variant for integrals of continuous functions or Riemann integrable functions was extended to measurable functions without additional difficulties.
Adriaan C. Zaanen
Chapter 6. Fourier Series of Summable Functions
Abstract
Given f ∈L1(π,µ)the Fourier coefficients (c n : n = 0, ±1, ±2,…) of f were introduced in Definition 8.1 by defining
$${c_n} = {(2\pi )^{ - 1}}\int\limits_\Delta {f(x){e^{ - inx}}} dx,$$
(1)
where Δ is any interval of length 2π. To indicate that the Fourier coeffi­cients are those of the function f, the notation c n (f) does sometimes occur. Frequently the notation fˆ(n) instead of cn(f) is also used. The sequence (fˆ(n) : n = 0, ±1, ±2,…) is then denoted by fˆ. For any f ∈ L1(ℝ,µ) there is an analogous notion, although now it is not a sequence of numbers but again a function defined on the whole of ℝ. Precisely formulated, for f ∈ L1(ℝ,µ) the Fourier transform fˆ of f is the function, defined for any x ∈ ℝ by
$${{f}^{{\left( x \right)}}} = \int\limits_{\mathbb{R}} {f(y){{e}^{{ - ixy}}}} dy.$$
(2)
Adriaan C. Zaanen
Chapter 7. Fourier Integral
Abstract
For use in next sections we shall discuss here how to compute some inte­grals. To this end we need generalizations of the theorems on integration of monotone sequences and on dominated convergence; the discrete parameter n in these theorems will be replaced by a continuous parameter ⋋. Let first µ, be a σ-finite measure in the (non-empty) point set X.
Adriaan C. Zaanen
Chapter 8. Additional Results
Abstract
In 1898 the physicist A. Michaelson, by means of a mechanical instrument, tried to draw graphs of the partial sums (up to the eightieth term) of the Fourier series of a real function f. He observed that if, for example, f is a 2π-periodic sawtooth function, the graph of the partial sum s n , for large n, does not behave as expected near a jump of f. At a downward jump of f the graph of instead of attaching itself closely to the graph of f until very near the jump and then steeply going downwards, starts to oscillate before diving down. An explanation of this phenomenon was discovered and explained already earlier by H. Wilbraham (1848), but this was forgotten for a long time.
Adriaan C. Zaanen
Backmatter
Metadata
Title
Continuity, Integration and Fourier Theory
Author
Adriaan C. Zaanen
Copyright Year
1989
Publisher
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-73885-2
Print ISBN
978-3-540-50017-9
DOI
https://doi.org/10.1007/978-3-642-73885-2