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## Über dieses Buch

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.

## Inhaltsverzeichnis

### 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 ).

### 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 xΔ. In other words, ||fPє|| < 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 ||fp 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)

### 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 mn. 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 dx1dx 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)

### 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 AB 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 BA, 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., AB 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 AnΓ and all A n mutually disjoint) implies
$$\mu (A) = \sum\nolimits_1^\infty {\mu ({A_n})} .$$
(3)

### 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.

### Chapter 6. Fourier Series of Summable Functions

Abstract
Given fL1(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 fL1(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 fL1(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)

### 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.