A Course in Analysis

The following sections are included:

• Preface
• Introduction
• Contents
• List of Symbols

It is hard to give a precise description of what “Mathematical Analysis” constitutes. However, every reasonable definition will have to mention that it is, in particular, occupied with the investigation of (real-valued) functions. An attempt to write the history of mathematical analysis will have to start in the pre-Newton and pre-Leibniz times, it will discuss the introduction of “the Calculus” by Leibniz and Newton, and highlight the contributions of the Bernoulli family, as well as those of Euler, d’Alembert and Lagrange. But neither Euler, d’Alembert nor Lagrange worked with a precise notion of a function. In fact, only in the late 18^th century and the early 19^th century did mathematicians become aware of serious problems in the fundamental definitions and methods used in analysis and other mathematical disciplines up to that point of time. As an example we just want to mention the notion of “integral”…

For two sequences (a_k)_k≥0 and (b_k)_k≥1 of real numbers we consider the formal term

We want to look more closely at the relations between 2π-periodic functions defined on ℝ and functions defined on the unit circle S¹. We now allow all functions to be complex-valued. A point p ∊ S¹ is given by p = e^it, t ∊ ℝ, i.e. we consider S¹ as a subset of ℂ. Given p ∊ S¹, t is only uniquely determined up to an additive multiple of 2π, i.e. for k ∊ ℤ we have p = e^it = e^i(t+2πk). We can enforce uniqueness in this representation by restricting t to any interval of length 2π, e.g. t ∊ [−π, π) or t ∊ [0, 2π). There is another way to clarify the situation by introducing on ℝ the equivalence relation x ∼_2π y if and only if y = x + 2πk for some k ∊ ℤ. From the algebraic point of view 2πℤ is a normal subgroup of ℝ, adding topology to our considerations, we can identify 2πℤ with a discrete, hence closed subgroup of ℝ and now we may pass to the quotient ℝ/ _∼2π as a group and topological space, thus we get a new Abelian topological group. This group is the one-dimensional torus and is denoted by 𝕋 (or 𝕋¹). We can identify 𝕋 algebraically and topologically with S¹ and we will do this in Appendix I. Since periodicity in n (real) variables is linked to functions defined on the n-dimensional torus 𝕋ⁿ = 𝕋¹ × … 𝕋¹ (n copies), in the context of trigonometric series we prefer to use 𝕋 instead of S¹ as notation for the unit circle, but still we identify 𝕋 with a subset of ℂ…

We want to investigate the question for which points t₀ ∊ [−π, π) does a Fourier series of a given function f converge to f(t₀)? For this question to make sense f must be integrable and pointwisely defined, thus we cannot work in L¹(𝕋) since elements of L₁(𝕋) are equivalence classes of functions being λ⁽¹⁾-almost everywhere equal. For this reason, in this chapter we assume all our functions to be defined on ℝ, to be 2π-periodic and to be Riemann integrable on [−π, π]. By Theorem III.8.11 these functions are also Lebesgue integrable, hence they determine a unique element in L¹(𝕋) and we can apply all our previous results…

In Proposition 3.19 we have proved that the image of L²(𝕋) under the mapping F as defined by (3.20) is a subspace of l₂(ℤ) and for f ∊ L²(𝕋) Bessel’ sinequality

Since their emergence Fourier series have stimulated a lot of research in mathematics: integration theory, functional analysis or most of all set theory, and many more subjects are off-springs of Fourier analysis. The reasons for this are the many surprising results showing up when investigating the convergence of Fourier series or more generally trigonometric series. In this chapter we will discuss some of these results which need more challenging proofs. Some readers may prefer to move immediately to the end of this chapter where we have summarized the results. We start by discussing the integral sine function

In this chapter we will also work with holomorphic functions f : G → ℂ, where G ⊂ ℂ is a region, i.e. an open and connected set. In most cases G will be the unit disc D := B₁(0)⊂ ℂ. Points in ℂ are denoted by z = x + iy and we freely switch from the notation g(z) to g(x, y) (instead of g(x + iy)). This applies in particular when decomposing f into its real and imaginary parts $$f\left(z\right)=u\left(z\right)=iv\left(z\right)=u\left(x,y\right)+iv\left(x,y\right).(7.1)$$…

In this chapter we discuss some topics, not necessarily related to each other, which go beyond a first introduction to Fourier series. Several of these topics will be picked up in later chapters or volumes. In some cases we will need some “auxiliary results” which are best discussed and proved in a different context, as for example the Riesz representation theorem which we prove in full detail in Volume V. In such a case we give a precise statement of the result, a reference to existing literature, but we will postpone the proof…

As indicated in Chapter 1, Fourier series arose from the attempt to obtain and represent “general” solutions of certain (partial) differential equations as a superposition of relatively simple and well understood solutions, i.e. trigonometric functions. A key step was to identify the coefficients of such an expansion as Fourier coefficients which are integrals, and the “orthogonality” of trigonometric functions is crucial in this approach. Recall that orthogonality is a notion depending on the integral. We have seen in Chapter 5 that the L²-theory of Fourier series is in particular satisfactory which is due to the underlying Hilbert space structure.

In this chapter we will outline the concept of orthonormal expansions in separable Hilbert spaces, a topic which we will pick up and investigate in much more detail in Volume V. The trigonometric functions will be replaced by an orthonormal base and the L²-theory of Fourier series will turn out to be just a special case of an orthonormal series expansion. Our principal goal in this chapter is to introduce some orthonormal bases formed by special functions in certain L²-spaces, e.g. Legendre polynomials, etc. These functions are often related to orthogonal curvilinear coordinates as well as to corresponding ordinary differential equations of second order and they reflect certain symmetries. In Part 9 we will discuss these ordinary differential equations. The relation to spectral theory will be investigated in Volume V and the applications to partial differential equations in Volume VI…

As indicated in the Introduction and discussed in more detail in Chapter 1, when dealing with functions defined on ℝ or more generally on ℝⁿ we have to replace Fourier series by Fourier integrals. It was L. Schwartz who in [108] and [109] pointed out that the space $\mathcal{S}$(ℝⁿ), nowadays called the Schwartz space, is best suited to handle the Fourier integral and the Fourier transform. In this chapter we want to introduce this space…

In this chapter we study the Fourier transform in Schwartz space. To a large extent we follow Section 3.1 of [69] which partially used [68]. We start with…

For u ∊ L¹(ℝⁿ) and ξ ∊ ℝⁿ the function x ↦ e ^−ix·ξu(x) is an element in L¹(ℝⁿ)and therefore we can give…

For every bounded Borel measure $\mu \in {ℳ}_b^{+}\left({ℝ}^n\right)$ the integral

In this chapter we will treat a few topics that are independent of one another, but still of interest and with many applications…

Although we have met ordinary differential equations in several places, only Chapter I.27 was completely devoted to differential equations with the aim to discuss in detail the method of separating the variables, see most of all Theorem I.27.1. In this chapter we want to give some orientation and to study some elementary, not necessarily trivial, methods to solve or integrate some classical types of mainly first order equations. The chapter serves for providing examples, and most of all to give indications where problems or interesting questions arise when trying to establish a type of general theory…

In this and the following chapter we want to discuss for explicit first order differential equations existence and uniqueness results. This chapter is devoted to a single equation while the following chapter will handle first order systems, in general non-linear, and we will see that they also cover the case of higher order equations…

In this chapter we want to discuss explicit first order systems and try to obtain solutions for a corresponding initial value problem along the lines of the Picard-Lindelöf theorem. Thus we want to investigate For R : I × D → ℝⁿ, D ⊂ ℝⁿ, the problem

Let I ⊂ ℝ be an interval with endpoints a < b and A : I → M(n; ℝ) be a matrix-valued function. Since we can identify M(n; ℝ) with ${ℝ}^{n^2}$, basic calculus results do hold for such a mapping, the reader may compare with Chapter II.7. For f : I → ℝⁿ we want to discuss the system…

We now want to study linear systems with variable coefficients

In this chapter we discuss a class of differential equations which looks rather special, namely equations of the type

This chapter is devoted to some first observations on boundary value and eigenvalue problems. These are problems which, however, are better dealt with when a certain functional analytic framework is at our disposal and we will do so in Volume V. Here we want to introduce some first problems and ideas of how to solve them…

We have encountered the hypergeometric differential equation

We want to return to the general theory and to continue our studies of systems

So far we have been searching for methods from analysis to solve and to investigate differential equations. However, every differential equation has in some sense a very natural geometric meaning, and indeed geometry is crucial to understand some of the qualitative properties of solutions or the set of all solutions of a given differential equation. We just need to think about stability problems which we will continue to discuss further below. This chapter and the next one are devoted to some geometric aspects of the theory of ordinary differential equation, but at the moment we are limited in our investigation since we do not yet have the theory of differentiable manifolds at our disposal…

We now want to use our geometric tools, e.g. tangent spaces or vector fields, to study a first order system of ordinary differential equations from a more geometric point of view. We restrict ourselves to autonomous systems, i.e. systems of the type…

Many scientists will endorse the statement “Nature is governed by extremal principles”. Taking more modern epistemology into account and adding a bit of Popper’s ideas, the statement “It seems that extremal principles are convenient to model many natural phenomena” is the more agreeable hypothesis. Classical examples are the…

Let F : [a, b] × ℝⁿ × ℝⁿ → ℝ, (t, x, y)↦ F(t, x, y), be a C¹-Lagrange function such that F_yj, j = 1,. …., n, is a C¹-function too. If u ∊ C²([a, b];ℝⁿ) is a C²- minimiser of $ℱ\left(u\right)={\displaystyle {\int }_a^{b}F}\left(t,u\left(t\right),\dot{u}\left(t\right)\right)\text{d}t$ under the boundary conditions u(a) = A and u(b) = B then u is a solution of the Euler-Lagrange equations $$\frac{\text{d}}{\text{d}t}{F}_{yi}\left(t,u\left(t\right),u^{\prime }\left(t\right)\right)-F_{x_j}\left(t,u\left(t\right),u^{\prime }\left(t\right)\right)=0,j=1,\cdots ,n(27.1)$$ or more explicitly

We want to find further necessary, and maybe sufficient, criteria for a minimum of $\mathcal{F}$(u) to exist. For this we take up the considerations of Chapter 26 and discuss

The calculus of variations is naturally linked to the theory of first order partial differential equations. Historically, the link was established in analytical mechanics or Lagrange-Hamilton-Jacobi mechanics as well as in the discussion of the relation of geometric optics to wave optics. The latter is related to the propagation of singularities of solutions of higher order partial differential equations and this is a topic we will handle in Volume VI. Thus, this chapter will serve us to prepare some further investigations on the calculus of variations and, at the same time, we discuss results which we will employ in Volume VI. In Chapter 30 we will return to the calculus of variations…

D’Alembert’s principle of virtual work led, in the hands of Lagrange, to the introduction of the function nowadays called the Lagrange function and the Lagrange equations (Euler-Lagrange equations) for a system of N particles. Using the Lagrange function and guided by his own work on optics, Hamilton stated his basic principle for mechanics and made this a starting point for investigations which culminated in the Hamilton-Jacobi theory which we first of all shall look at as a theory in analytical mechanics. However, for a mathematician it is also a part of the calculus of variations and the theory of partial differential equations of first order…

The following sections are included:

• Appendix I: Harmonic Analysis on Locally Compact Abelian Groups
• Appendix II: Convergence of Measures
• Appendix III: Generating Functions, Orthonormal Polynomials
• Appendix IV: On Brouwer’s Fixed Point Theorem

The following sections are included:

• Solutions to Problems of Part 8
• Solutions to Problems of Part 9
• Solutions to Problems of Part 10

The following sections are included:

• References
• Mathematicians Contributing to Analysis (Continued)
• Subject Index