This chapter applies the theory of linear ordinary differential equations to certain boundary-value problems for partial differential equations.

Section 1 briefly introduces some notation and defines the three partial differential equations of principal interest—the heat equation, Laplace’s equation, and the wave equation.

Section 2 is a first exposure to solving partial differential equations, working with boundary-value problems for the three equations introduced in Section 1. The settings are ones where the method of “separation of variables” is successful. In each case the equation reduces to an ordinary differential equation in each independent variable, and some analysis is needed to see when the method actually solves a particular boundary-value problem. In simple cases Fourier series can be used. In more complicated cases Sturm’s Theorem, which is stated but not proved in this section, can be helpful.

Section 3 returns to Sturm’s Theorem, giving a proof contingent on the Hilbert–Schmidt Theorem, which itself is proved in Chapter II. The construction within this section finds a Green’s function for the second-order ordinary differential operator under study; the Green’s function defines an integral operator that is essentially an inverse to the second-order differential operator.

This chapter proves a first version of the Spectral Theorem and shows how it applies to complete the analysis in Sturm’s Theorem of Section I.3.

Section 1 introduces compact linear operators from a Hilbert space into itself and characterizes them as the limits in the operator norm topology of the linear operators of finite rank. The adjoint of a compact operator is compact.

Section 2 proves the Spectral Theorem for compact self-adjoint operators on a Hilbert space, showing that such operators have orthonormal bases of eigenvectors with eigenvalues tending to 0.

Section 3 establishes two versions of the Hilbert–Schmidt Theorem concerning self-adjoint integral operators with a square-integrable kernel. The abstract version gives an $L^{2}$ expansion of the members of the image of the operator in terms of eigenfunctions, and the concrete version, valid when the kernel is continuous and the space is compact metric, proves that the eigenfunctions are continuous and the expansion in terms of eigenfunctions is uniformly convergent.

Section 4 introduces unitary operators on a Hilbert space, establishing the equivalence of three conditions that may be used to define them.

Section 5 studies compact linear operators on an abstract Hilbert space, with special attention to two kinds—the Hilbert–Schmidt operators and the operators of trace class. All three sets of operators—compact, Hilbert–Schmidt, and trace-class—are ideals in the algebra of all bounded linear operators and are closed under the operation of adjoint. Trace-class implies Hilbert–Schmidt, which implies compact. The product of two Hilbert–Schmidt operators is of trace class.

This chapter takes up several independent topics in Euclidean Fourier analysis, all having some bearing on the subject of partial differential equations.

Section 1 elaborates on the relationship between the Fourier transform and the Schwartz space, the subspace of $L^{1}(\mathbb{R}^{N})$ consisting of smooth functions with the property that the product of any iterated partial derivative of the function with any polynomial is bounded. It is possible to make the Schwartz space into a metric space, and then one can consider the space of continuous linear functionals; these continuous linear functionals are called “tempered distributions.” The Fourier transform carries the space of tempered distributions in one-one fashion onto itself.

Section 2 concerns weak derivatives, and the main result is Sobolev’s Theorem, which tells how to recover information about ordinary derivatives from information about weak derivatives. Weak derivatives are easy to manipulate, and Sobolev’s Theorem is therefore a helpful tool for handling derivatives without continually having to check the validity of interchanges of limits.

Sections 3–4 concern harmonic functions, those functions on open sets in Euclidean space that are annihilated by the Laplacian. The main results of Section 3 are a characterization of harmonic functions in terms of a mean-value property, a reflection principle that allows the extension to all of Euclidean space of any harmonic function in a half space that vanishes at the boundary, and a result of Liouville that the only bounded harmonic functions in all of Euclidean space are the constants. The main result of Section 4 is a converse to properties of Poisson integrals for half spaces, showing that harmonic functions in a half space are given as Poisson integrals of functions or of finite complex measures if their $L^{p}$ norms over translates of the bounding Euclidean space are bounded.

Sections 5–6 concern the Calderón–Zygmund Theorem, a far-reaching generalization of the theorem concerning the boundedness of the Hilbert transform. Section 5 gives the statement and proof, and two applications are the subject of Section 6. One of the applications is to Riesz transforms, and the other is to the Beltrami equation, whose solutions are “quasiconformal mappings.”

Sections 7–8 concern multiple Fourier series for smooth periodic functions. The theory is established in Section 7, and an application to traces of integral operators is given in Section 8.

This chapter pursues three lines of investigation in the subject of functional analysis—one involving smooth functions and distributions, one involving fixed-point theorems, and one involving spectral theory.

Section 1 introduces topological vector spaces. These are real or complex vector spaces with a Hausdorff topology in which addition and scalar multiplication are continuous. Examples include normed linear spaces, spaces given by a separating family of countably many seminorms, and weak and weak-star topologies in the context of Banach spaces. Various general properties of topological vector spaces are proved, and it is proved that the quotient of a topological vector space by a closed vector subspace is Hausdorff and is therefore a topological vector space.

Section 2 introduces a topology on the space $C^{\infty}(U)$ of smooth functions on an open subset of $\mathbb{R}^{N}$. The support of a continuous linear functional on $C^{\infty}(U)$ is defined and shown to be a compact subset of $U$. Accordingly, the continuous linear functionals are called distributions of compact support.

Section 3 studies weak and weak-star topologies in more detail. The main result is Alaoglu’s Theorem, which says that the closed unit ball in the weak-star topology on the dual of a normed linear space is compact. In an earlier chapter a preliminary form of this theorem was used to construct elements in a dual space as limits of weak-star convergent subsequences.

Section 4 follows Alaoglu’s Theorem along a particular path, giving what amounts to a first example of the Gelfand theory of Banach algebras. The relevant theorem, known as the Stone Representation Theorem, says that conjugate-closed uniformly closed subalgebras containing the constants in $B(S)$ are isomorphic via a norm-preserving algebra isomorphism to the space of all continuous functions on some compact Hausdorff space. The compact space in question is the space of multiplicative linear functionals on the subalgebra, and the proof of compactness uses Alaoglu’s Theorem.

Sections 5–6 return to the lines of study toward distributions and fixed-point theorems. Section 5 studies the relationship between convexity and the existence of separating linear functionals. The main theorem makes use of the Hahn–Banach Theorem. Section 6 introduces locally convex topological vector spaces. Application of the basic separation theorem from the previous section shows the existence of many continuous linear functionals on such a space.

Section 7 specializes to the line of study via smooth functions and distributions. The topic is the introduction of a certain locally convex topology on the space $C^{\infty}_{\mathrm{com}}(U)$ of smooth functions of compact support on $U$. This is best characterized by a universal mapping property introduced in the section.

Sections 8–9 pursue locally convex spaces along the other line of study that split off in Section 5. Section 8 gives the Krein–Milman Theorem, which asserts the existence of a supply of extreme points for any nonempty compact convex set in a locally convex topological vector space. Section 9 relates compact convex sets to the subject of fixed-point theorems.

Section 10 takes up the abstract theory of Banach algebras, with particular attention to commutative $C^{*}$ algebras with identity. Three examples are the algebras characterized by the Stone Representation Theorem, any $L^{\infty}$ space, and any adjoint-closed commutative Banach algebra consisting of bounded linear operators on a Hilbert space and containing the identity.

Section 11 continues the investigation of the last of the examples in the previous section and derives the Spectral Theorem for bounded self-adjoint operators and certain related families of operators. Powerful applications follow from a functional calculus implied by the Spectral Theorem. The section concludes with remarks about the Spectral Theorem for unbounded self-adjoint operators.

This chapter makes a detailed study of distributions, which are continuous linear functionals on vector spaces of smooth scalar-valued functions. The three spaces of smooth functions that are studied are the space $C^{\infty}_{\mathrm{com}}(U)$ of smooth functions with compact support in an open set $U$, the space $C^{\infty}(U)$ of all smooth functions on $U$, and the space of Schwartz functions $\mathcal{S}(\mathbb{R}^{N})$ on $\mathbb{R}^{N}$. The corresponding spaces of continuous linear functionals are denoted by $\mathcal{D}^{\prime}(U)$, $\mathcal{E}^{\prime}(U)$, and $\mathcal{S}^{\prime}(\mathbb{R}^{N})$.

Section 1 examines the inclusions among the spaces of smooth functions and obtains the conclusion that the corresponding restriction mappings on distributions are one-one. It extends from $\mathcal{E}^{\prime}(U)$ to $\mathcal{D}^{\prime}(U)$ the definition given earlier for support, it shows that the only distributions of compact support in $U$ are the ones that act continuously on $C^{\infty}(U)$, it gives a formula for these in terms of derivatives and compactly supported complex Borel measures, and it concludes with a discussion of operations on smooth functions.

Sections 2-3 introduce operations on distributions and study properties of these operations. Section 2 briefly discusses distributions given by functions, and it goes on to work with multiplications by smooth functions, iterated partial derivatives, linear partial differential operators with smooth coefficients, and the operation $(\cdot)^{\vee}$ corresponding to $x \mapsto -x$. Section 3 discusses convolution at length. Three techniques are used—the realization of distributions of compact support in terms of derivatives of complex measures, an interchange-of-limits result for differentiation in one variable and integration in another, and a device for localizing general distributions to distributions of compact support.

Section 4 reviews the operation of the Fourier transform on tempered distributions; this was introduced in Chapter III. The two main results are that the Fourier transform of a distribution of compact support is a smooth function whose derivatives have at most polynomial growth and that the convolution of a distribution of compact support and a tempered distribution is a tempered distribution whose Fourier transform is the product of the two Fourier transforms.

Section 5 establishes a fundamental solution for the Laplacian in $\mathbb{R}^{N}$ for $N > 2$ and concludes with an existence theorem for distribution solutions to $\Delta u = f$ when $f$ is any distribution of compact support.

This chapter investigates several ways that groups play a role in real analysis. For the most part the groups in question have a locally compact Hausdorff topology.

Section 1 introduces topological groups, their quotient spaces, and continuous group actions. Topological groups are groups that are topological spaces in such a way that multiplication and inversion are continuous. Their quotient spaces by subgroups are of interest when they are Hausdorff, and this is the case when the subgroups are closed. Many examples are given, and elementary properties are established for topological groups and their quotients by closed subgroups.

Sections 2–4 investigate translation-invariant regular Borel measures on locally compact groups and invariant measures on their quotient spaces. Section 2 deals with existence and uniqueness in the group case. A left Haar measure on a locally compact group $G$ is a nonzero regular Borel measure invariant under left translations, and right Haar measures are defined similarly. The theorem is that left and right Haar measures exist on $G$, and each kind is unique up to a scalar factor. Section 3 addresses the relationship between left Haar measures and right Haar measures, which do not necessarily coincide. The relationship is captured by the modular function, which is a certain continuous homomorphism of the group into the multiplicative group of positive reals. The modular function plays a role in constructing Haar measures for complicated groups out of Haar measures for subgroups. Of special interest are “unimodular” locally compact groups $G$, i.e., those for which the left Haar measures coincide with the right Haar measures. Every compact group, and of course every locally compact abelian group, is unimodular. Section 4 concerns translation-invariant measures on quotient spaces $G/H$. For the setting in which $G$ is a locally compact group and $H$ is a closed subgroup, the theorem is that $G/H$ has a nonzero regular Borel measure invariant under the action of $G$ if and only if the restriction to $H$ of the modular function of $G$ coincides with the modular function of $H$. In this case the $G$ invariant measure is unique up to a scalar factor. Section 5 introduces convolution on unimodular locally compact groups $G$. Familiar results valid for the additive group of Euclidean space, such as those concerning convolution of functions in various $L^{p}$ classes, extend to be valid for such groups $G$.

Sections 6–8 concern the representation theory of compact groups. Section 6 develops the elementary theory of finite-dimensional representations and includes some examples, Schur orthogonality, and properties of characters. Section 7 contains the Peter–Weyl Theorem, giving an orthonormal basis of $L^{2}$ in terms of irreducible representations and concluding with an Approximation Theorem showing how to approximate continuous functions on a compact group by trigonometric polynomials. Section 8 shows that infinite-dimensional unitary representations of compact groups decompose canonically according to the irreducible finite-dimensional representations of the group. An example is given to show how this theorem may be used to take advantage of the symmetry in analyzing a bounded operator that commutes with a compact group of unitary operators. The same principle applies in analyzing partial differential operators.

This chapter provides an introduction to partial differential equations, particularly linear ones, beyond the material on separation of variables in Chapter I.

Sections 1–2 give an overview. Section 1 addresses the question of how many side conditions to impose in order to get local existence and uniqueness of solutions at the same time. The Cauchy–Kovalevskaya Theorem is stated precisely for first-order systems in standard form and for single equations of order greater than one. When the system or single equation is linear with constant coefficients and entire holomorphic data, the local holomorphic solutions extend to global holomorphic solutions. Section 2 comments on some tools that are used in the subject, particularly for linear equations, and it gives some definitions and establishes notation.

Section 3 establishes the basic theorem that a constant-coefficient linear partial differential equation $Lu = f$ has local solutions, the technique being multiple Fourier series.

Section 4 proves a maximum principle for solutions of second-order linear elliptic equations $Lu = 0$ with continuous real-valued coefficients under the assumption that $L(1) = 0$.

Section 5 proves that any linear elliptic equation $Lu = f$ with constant coefficients has a “parametrix,” and it shows how to deduce from the existence of the parametrix the fact that the solutions $u$ are as regular as the data $f$. The section also deduces a global existence theorem when $f$ is compactly supported; this result uses the existence of the parametrix and the constant-coefficient version of the Cauchy–Kovalevskaya Theorem.

Section 6 gives a brief introduction to pseudodifferential operators, concentrating on what is needed to obtain a parametrix for any linear elliptic equation with smooth variable coefficients.

This chapter explains how the theory of pseudodifferential operators extends from open subsets of Euclidean space to smooth manifolds, and it gives examples to illustrate the usefulness of generalizing the theory in this way.

Section 1 gives a brief introduction to differential calculus on smooth manifolds. The section defines smooth manifolds, smooth functions on them, tangent spaces to smooth manifolds, and differentials of smooth mappings between smooth manifolds, and it proves a version of the Inverse Function Theorem for manifolds.

Section 2 extends the theory of smooth vector fields and integral curves from open subsets of Euclidean space to smooth manifolds.

Section 3 develops a special kind of quotient space, called an “identification space,” suitable for constructing general smooth manifolds, vector bundles and fiber bundles, and covering spaces out of local data. In particular, smooth manifolds may be defined as identification spaces without knowledge of the global nature of the underlying topological space; the only problem is in addressing the Hausdorff property.

Section 4 introduces vector bundles, including the tangent and cotangent bundles to a manifold. A vector bundle determines transition functions, and in turn the transition functions determine the vector bundle via the construction of the previous section. The manifold structures on the tangent and cotangent bundles are constructed in this way.

Sections 5–8 concern pseudodifferential operators, including aspects of the theory useful in solving problems in other areas of mathematics. The emphasis is on operators on scalar-valued functions. Section 5 introduces spaces of smooth functions and their topologies, and it defines spaces of distributions; the theory has to compensate for the lack of a canonical underlying measure on the manifold, hence for the lack of a canonical way to view a smooth function as a distribution. Section 5 goes on to study linear partial differential equations on the manifold; although the symbol of the differential operator is not meaningful, the principal symbol is intrinsically defined as a function on the cotangent bundle. The introduction of pseudodifferential operators on smooth manifolds requires new results for the theory in Euclidean space beyond what is in Chapter VII. Section 6 addresses this matter. A notion of transpose is needed, and it is necessary to understand the effect of diffeomorphisms on Euclidean pseudodifferential operators. To handle these questions, it is useful to enlarge the definition of pseudodifferential operator for Euclidean space and to redo the Euclidean theory from the new point of view. Once that program has been carried out, Section 7 patches together pseudodifferential operators in Euclidean space to obtain pseudodifferential operators on smooth separable manifolds. The notions of pseudolocal, properly supported, composition, and elliptic extend, and the theorems are what one might expect from the Euclidean theory. Again the principal symbol is well defined as a function on the cotangent bundle. Section 8 contains remarks about extending the theory to handle operators carrying sections of one vector bundle to sections of another vector bundle, about some other continuations of the theory, and about applications outside real analysis. The section concludes with some bibliographical material.

This chapter introduces probability theory as a system of models, based on measure theory, of some real-world phenomena. The models are measure spaces of total measure 1 and usually have certain distinguished measurable functions defined on them.

Section 1 begins by establishing the measure-theoretic framework and a short dictionary for passing back and forth between terminology in measure theory and terminology in probability theory. The latter terminology includes events, random variables, mean, probability distribution of a random variable, and joint probability distribution of several random variables. An important feature of probability is that it is possible to work with random variables without any explicit knowledge of the underlying measure space, the joint probability distributions of random variables being the objects of importance.

Section 2 introduces conditional probability and uses that to motivate the mathematical definition of independence of events. In turn, independence of events leads naturally to a definition of independent random variables. Independent random variables are of great importance in the subject and play a much larger role than their counterparts in abstract measure theory. Examples at the end of the section indicate the extent to which functions of independent random variables can remain independent. The techniques in the examples are of use in the subject of statistical inference, which is introduced in Section 10.

Section 3 states and proves the Kolmogorov Extension Theorem, a foundational result allowing one to create stochastic processes involving infinite sets of times out of data corresponding to finite subsets of those times. A special case of the theorem provides the existence of infinite sets of independent random variables with specified probability distributions.

Section 4 establishes the celebrated Strong Law of Large Numbers, which says that the Cesàro sums of a sequence of identically distributed independent random variables with finite mean converge almost everywhere to a constant random variable, the constant being the mean. This is a theorem that is vaguely known to the general public and is widely misunderstood. The proof is based on Kolmogorov’s inequality.

Sections 5–8 provide background for the Central Limit Theorem, whose statement and proof are in Section 9. Section 5 discusses three successively weaker kinds of convergence for random variables—almost sure convergence, convergence in probability, and convergence in distribution. Convergence in distribution will be the appropriate kind for the Central Limit Theorem. Section 6 contains the Portmanteau Lemma, which gives some equivalent formulations of convergence in distribution, Section 7 introduces characteristic functions as Fourier transforms of probability distributions, and Section 8 proves the Lévy Continuity Theorem, which formulates convergence in distribution in terms of characteristic functions.

Section 9 contains the statement and proof of the Central Limit Theorem, followed by some simple examples. This theorem is the most celebrated result in probability theory and has many applications in mathematics and other fields.

Section 10 is a brief introduction to the subject of statistical inference, showing how the Central Limit theorem plays a role in practice through the $t$ test of W. S. Gosset.

This chapter introduces the relatively recent subject of wavelets, which is an outgrowth of Fourier analysis in mathematics and signal processing in engineering. Except in one case, construction of examples of wavelets tends to be difficult. Much of the chapter is devoted to construction of some of the better known examples and lists of their most important properties.

Section 1 defines wavelets and discusses three features of traditional Fourier analysis: the Uncertainty Principle, Gibbs phenomenon, and the Shannon Sampling Theorem. It ends with a brief essay on the need for wavelets in various applications.

Section 2 establishes that the Haar system is an orthonormal basis of $L^{2}(\mathbb {R})$. The Haar wavelet predates the general theory of wavelets by many decades but provides a prototype for some of the development. The section ends with some discussion of convergence of one-sided Haar expansions for function spaces besides $L^{2}(\mathbb {R})$.

Section 3 begins the general theory of wavelets, introducing the notion of multiresolution analysis to abstract the construction in Section 2 of the Haar wavelet. The ingredients of a multiresolution analysis are a scaling function, traditionally called $\varphi$, and an increasing sequence of closed subspaces $V_{j}$ of $L^{2}(\mathbb {R})$ with certain properties. The wavelet that is constructed is traditionally called $\psi$.

Section 4 introduces the Shannon wavelet, whose construction is immediate from the theory of multiresolution analyses. The new ingredient here, beyond the ideas used for the Haar wavelet, is the careful use of the generating function of the scaling function to obtain a formula for the wavelet.

Section 5 supplements the theory of Section 3 by showing how to build a multiresolution analysis out of a candidate for the scaling function.

Section 6 introduces the Meyer wavelets, each of which is smooth and has Fourier transform of a prescribed order of differentiability. The full theory of Sections 3 and 5 is used in their construction.

Section 7 introduces splines, examines one example, and sees the need for more theory. It develops one further aspect of the general theory, showing how to replace a “Riesz system” with an orthonormal set. It therefore allows one to relax the conditions needed in Section 5 for a function to be a scaling function. In addition, it uses elementary complex analysis to prove a series expansion for $\pi^{2}/\sin^{2}\pi z$ that is needed in Section 8.

Section 8 continues the discussion of the role of splines in the theory of wavelets, introducing the Battle–Lemarié wavelets. As with the Meyer wavelets each is smooth and has Fourier transform of a prescribed order of differentiability.

Section 9 develops the Daubechies wavelets. These have compact support, but except for the first one, neither they nor their Fourier transforms have known formulas in closed form. The construction begins by pinpointing necessary conditions on the generating function.

Section 10 deals with smoothness questions. It contains three results. The first gives an estimate for the decay of the Fourier transform of the Daubechies scaling function of each order. The second deduces a certain amount of differentiability of a scaling function from the estimate in the first result. The third shows in the converse direction that a Daubechies wavelet can never be of class $C^{∞}$. The section concludes with a table summarizing properties of the specific wavelets that have been constructed in Sections 2–9.

Section 11 gives a quick introduction to applications. It discusses the discrete wavelet transform and its use in storage and compression of data, it identifies some applications of wavelets in one and two dimensions, and it makes brief remarks about some of the applications.