Applied Functional Analysis

By D.H. Griffel

A stimulating introductory text, this volume examines many important applications of functional analysis to mechanics, fluid mechanics, diffusive growth, and approximation. Detailed enough to impart a thorough understanding, the text is also sufficiently straightforward for those unfamiliar with abstract analysis. Its four-part treatment begins with distribution theory and discussions of Green's functions. Essentially independent of the preceding material, the second and third parts deal with Banach spaces, Hilbert space, spectral theory, and variational techniques. The final part outlines the ideas behind Frechet calculus, stability and bifurcation theory, and Sobolev spaces. 1985 edition. 25 Figures. 9 Appendices. Supplementary Problems. Indexes.

• Language: English
• Publisher: Dover Publications
• Release date: Apr 26, 2012
• ISBN: 9780486141329

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MATHEMATICS

Preface

Aims. This book is intended to be a simple and easy introduction to the subject. It is aimed at undergraduate students of mathematics, and mathematical physics and engineering, though I hope to interest other readers too. I have tried to avoid difficult ideas, as far as possible; the spectral theorem, for example, is discussed for operators with discrete eigenvalues only. But within this limitation I have tried to tell a coherent story, and to give both proofs and motivation for the theory, at least in the central Parts II and III; in Parts I and IV some proofs are omitted.

To illustrate the uses of the theory, a few problems of theoretical mechanics are discussed in some detail, and a chapter is devoted to variational approximation methods. I have tried to give some sort of application of the theoretical ideas as soon as possible after they are introduced; I have found that students are more willing to grapple with compactness, for example, after they have seen the idea at work in fluid mechanics.

Prerequisites. and δ. An acquaintance with uniform convergence and with Sturm-Liouville systems would be helpful, but brief outlines of these subjects are given in the Appendices. Complex contour integration is used in one section, for the calculation of retarded waves, but this material is not used in the rest of the book, and can be omitted.

Conventions. appears after the statement of a theorem, it means that no proof will be given. Terms being defined are printed in bold type. The following rough definitions may be useful: a theorem is an important result; a proposition is a result less significant than a theorem, but still of some interest; a lemma is a result of little intrinsic interest which is stated because it is needed in the proof of a theorem; a corollary is a result which is a (more or less) obvious and immediate consequence of the preceding theorem.

Other Remarks. For an outline of the structure of the book, see the introductions to the four parts into which it is divided.

I have not used a consistent scheme of notation throughout the text. This is only partly laziness; it seems to me that students brought up on a uniform notation sometimes become addicted, and have difficulty coping with a different notation. Hence there is some virtue in inconsistency. Of course, where there is a generally accepted standard notation, I have used it.

A related matter is the question of notation for functions. I assume that my readers have learnt calculus thoroughly enough to understand clearly the distinction between a function ƒand the number ƒ(x), the value of ƒ. I therefore feel free to use convenient expressions like "the function ƒ(x) = ∫K(x,y)g(y) dy instead of the more correct but clumsy the function ƒ defined by ƒ(x) = ∫K(x,y)g(y) dy".

There are plenty of exercises at the ends of the chapters. Hints and answers to some of them are given at the back of the book. Complete solutions are also available; see p. 367.

My thanks are due to many people who have read drafts of some of the text, and to the typists in Bristol, and the staff of Ellis Horwood who have turned it into print. I am afraid that despite all their efforts, some of my mistakes will have survived; I will be grateful to any readers kind enough to point them out.

D. H. Griffel,

School of Mathematics,

University Walk,

Bristol, BS8 1TW.

England.

PREFACE TO SECOND EDITION

Many small amendments have been made in this second edition, and some new problems have been added. I am grateful to many people who have pointed out errors in the first edition, and urge readers to follow their example. I wish also to acknowledge the helpful and friendly collaboration of Ellis Horwood Ltd in the production of this book.

PART I

DISTRIBUTION THEORY AND GREEN’S FUNCTIONS

The central ideas of our subject, the theories of Banach and Hilbert spaces, are contained in Parts II and III. Part IV can be regarded as a coda, and Part I as a prelude. The theory of Parts II and III is independent of the distribution theory developed in Part I, which can be omitted if desired (though an acquaintance with Green’s functions would be helpful). In Part IV, however, the two theories will be unified.

The theory of distributions is essentially a new foundation for mathematical analysis; that is, a new structure, replacing functions of a real variable by new objects, defined quite differently, but having many of the same properties and usable for many of the same purposes as ordinary functions. This theory gives a useful technique for analysing linear problems in applied mathematics, but is less useful for nonlinear problems (which is why it is not used in Parts II and III). It leads naturally to the introduction of Green’s functions, which are essential to the application of functional analysis to differential equations. And it forms a useful prelude to Parts II and III because it introduces fairly abstract ideas in a fairly concrete setting.

Our treatment of distribution theory is intended to be detailed enough to give a good general understanding of the subject, but it does not aim at completeness. Many details and many worthwhile topics are omitted for the sake of digestibility. A full account would take a full-length book; see the references given in the last section of each chapter.

The plan of Part I is as follows. In Chapter 1 we set out the basic theory of distributions. Chapter 2 discusses ordinary differential equations from the distributional point of view, and introduces Green’s functions. Chapter 3 begins with a discussion of the Fourier transform from both the classical and distributional points of view, and then uses it to obtain Green’s functions for Laplace’s equation and for the wave equation.

The following sections form a short account of Green’s functions, which may be useful to readers who do not wish to learn distribution theory: 1.1; 2.3 as far as Example 2.9; 2.4; 2.5 omitting the proof of 2.15; 3.1; the second half of 3.4; 3.5 ; 3.6.

Chapter 1

Generalised Functions

In this chapter we lay the theoretical foundations for the treatment of differential equations in Chapters 2 and 3. We begin in section 1.1 by discussing the physical background of the delta function, which was the beginning of distribution theory. In section 1.2 we set out the basic theory of generalised functions or distributions (we do not distinguish between these terms), and in sections 1.3 and 1.4 we define the operations of algebra and calculus on generalised functions. The ideas and definitions of the theory are more elaborate than those of ordinary calculus; this is the price paid for developing a theory which is in many ways simpler as well as more comprehensive. In particular, the theorems about convergence and differentiation of series of generalised functions are simpler than in ordinary analysis. This is illustrated by examples in sections 1.5 and 1.6. References to other accounts of this subject are given in section 1.7.

1.1 The Delta Function

The theory of generalised functions was invented in order to give a solid theoretical foundation to the delta function, which had been introduced by Dirac (1930) as a technical device in the mathematical formulation of quantum mechanics. But the idea of Dirac’s delta function can easily be understood in classical terms, as follows.

Consider a rod of nonuniform thickness. In order to describe how its mass is distributed along its length, one introduces a ‘mass-density function’ ρ(x); this is defined physically as the mass per unit length of the rod at the point x, and defined mathematically as a function such that the total mass of the section of the rod from a to b . This is a satisfactory description of continuous mass-distributions; dynamical properties such as its centre of mass and moment of inertia can be expressed in terms of the function ρ.

But if the mass is concentrated at a finite number of points instead of being distributed continuously, then the above description breaks down. Consider, for instance, a wire of negligible mass, with a small but heavy bead attached to its mid-point, x = 0. Suppose that the bead has unit mass and is so small that it is reasonable to represent it mathematically as a point. Then the total mass in the interval (a,b) is zero if 0 is outside the interval, and is one if zero is inside the interval. There is no function p that can represent this mass-distribution. If there were, then we would have ρ(x) = 0 for all x ≠ 0, since the mass per unit length is zero except at x = 0. But if a function vanishes everywhere except at a single point, it is easy to prove that its integral over any interval must be zero, so that integrating it over an interval including the origin cannot give the correct value, 1. From the physical point of view, the mass-density is zero everywhere except at x = 0, where it is infinite because a finite mass is concentrated in zero length; and it is so infinitely large there that the integral is non-zero even though the integrand is positive over an ‘infinitesimally small’ region only. This makes good physical sense, though it is mathematically absurd. Dirac therefore introduced a function δ(x) having just those properties:

(1.1)

If one uses the function δ as the mass-density function in any calculation or theoretical work involving continuous distributions on a line, one is led to the corresponding result for a point particle, and thus the two cases of continuous and discrete distributions can be included in a single formalism if the delta function is used. It can be considered as a technical trick, or short cut, for obtaining results for discrete point particles from the continuous theory, results which can always be verified if desired by working out the discrete case from first principles.

A point particle can be considered as the limit of a sequence of continuous distributions which become more and more concentrated. The delta function can similarly be considered as the limit of a sequence of ordinary functions. Consider, for example,

(1.2)

Then dn(x) → 0 as n → ∞ for any x ≠ 0, and dn(0) → ∞ as n → ∞ (see , and if a < 0 < b, as n → ∞. Thus we might say that ‘in the limit as n → ∞’, dn has the properties of δ(x). The delta function is sometimes defined in this way, but again this is not a proper mathematical definition, since dn(x) does not have a limit for all x as n → ∞. However, when one uses the delta function in practice it usually appears inside an integral at some stage, and it is only integrals of δ(x), multiplied by ordinary functions, that have direct physical significance. If one replaces δ by dn, and then lets n → ∞ at the end of the calculation, the integrals involving dn will generally be well behaved and tend to finite limits as n → ∞, and the mathematical inconsistency is removed. The delta function can be considered as a kind of mathematical shorthand representing that procedure, and results obtained by its use can always be verified if desired by working with dn and then evaluating the limit.

The point of view described above is that of many physicists, engineers, and applied mathematicians who use the delta function. To the pure mathematicians of Dirac’s generation it presented a challenge: an idea which is mathematically absurd, but still works, and gives useful and correct results, must be somehow essentially right. There must be a theory in which δ(x) has a rightful place, instead of being sneaked in by the back door as a mathematical shorthand, to be justified by doing the calculation again without using it. The situation is reminiscent of the use of complex numbers in the 16th century for solving algebraic equations. It proved useful to pretend that −1 has a square root, even though it clearly has not, since one could then use an algorithm involving imaginary numbers for obtaining real roots of cubic equations; any result obtained this way could be verified by directly substituting it in the equation and showing that it really was a root. It was only much later that complex numbers were given a solid mathematical foundation, and then with the development of the theory of functions of a complex variable their applications far transcended the simple algebra which led to their introduction. In the same way, Dirac found it useful to pretend that there exists a function δ satisfying (1.1), even though there does not. The solid foundation was developed by Sobolev (in 1936) and Schwartz (in the 1950s), and again goes far beyond merely propping up the delta function. The theory of generalised functions that they developed can be used to replace ordinary analysis, and is in many ways simpler. Every generalised function is differentiable, for example; and one can differentiate and integrate series term by term without worrying about uniform convergence. The theory also has limitations: that is, it shows clearly what you cannot do with the delta function as well as what you can – namely, you cannot multiply it by itself, or by a discontinuous function. The other disadvantage of the theory is that it involves a certain amount of formal machinery.

which approximate to the delta function.

1.2 Basic Distribution Theory

There are several ways of generalising the idea of a function in order to include Dirac’s δ. We shall follow the method of Schwartz, who called his generalised functions ‘distributions’; the idea is to formalise Dirac’s idea of the delta function as something which makes sense only under an integral sign, possibly multiplied by some other function φ Given any interval, a mass-density function allows one to calculate the mass of that interval; more generally, given any weighting function φ, it allows one to calculate a weighted average of the mass, such as is needed for calculating centres of gravity, etc. We shall define a distribution as a rule which, given any function φ, provides a number; the number may be thought of as a weighted average, with weight function φ, of a corresponding mass-distribution. However, we must be careful about what functions we allow as weighting functions. The definitions below may at first seem arbitrary and needlessly complicated; but they are carefully framed, as you will see, to make the resulting theory as simple as possible. The reader unfamiliar with the notation of set theory should consult Appendix A.

Definition 1.1 The support of a function ƒis {x: ƒ(x) ≠ 0} , written supp(ƒ)¹. A function has bounded support if there are numbers a,b such that sup(ƒ) ⊂ [a,b].

Definition 1.2 A function ƒis said to be n times continuously differentiable if its first n derivatives exist and are continuous. ƒ is said to be smooth or infinitely differentiable if its derivatives of all orders exist and are continuous.

Definition 1.3 A test function .

Example 1.4 is a test function with support (a,b).

This is probably the simplest example of a test function. They are bound to have somewhat complicated forms, for the following reason. If supp (φ) (a,b), then φ(x) = 0 for x ≤ a, so all derivatives of φ vanish at a. Hence the Taylor series of φ about a is identically zero. But φ(x) ≠ 0 for a < x < b, so φ does not equal its Taylor series. Test functions are thus peculiar functions; they are smooth, yet Taylor expansions are not valid. The above example clearly shows singular behaviour at x = a and x = b. Fortunately, we never really need explicit formulas for test functions. They are used for theoretical purposes only, and are well-behaved (smooth etc.) even though their functional forms may be complicated. The following result gives another nice mathematical property.

Proposition 1.5 The sum of two test functions is a test function; the product of a test function with any number is a test function.

Proof. Obvious.

A set of functions with this property is often called a ‘space’, for reasons that will become clear in Chapter 4.

Definition 1.6 A linear functional such that ƒ(aφ + bψ) = aƒ(φ) + bƒ(ψ) for all a,b and φ,ψ

means a rule which, given any φ , produces a corresponding z ; we write z = ƒ(φ). We also speak of the ‘action’ of the functional ƒ on φ producing the number ƒ (φ). The notation φ ƒ(φ) stands for the phrase ‘φ is mapped into the number ƒ(φ)’; see Appendix A.

Examples 1.7 (a) φ φdx is a functional; the integral converges because φ dx is a linear functional for any function ƒ sufficiently well behaved to ensure convergence of the integral.

. The reader will know that more than one meaning can be attached to the phrase ‘a sequence of functions is convergent’. For some purposes pointwise convergence is suitable; for other purposes uniform convergence is needed (an outline of the theory of uniform convergence is given in Appendix B). One of the characteristics of functional analysis is its use of many different kinds of convergence, as demanded by different problems. The most useful for our present purpose is the following.

Definition 1.8 (Convergence) If (φn) is a sequence of test functions and Φ another test function, we say φn if (i) there is an interval [a,b] containing supp(Φ) and supp(φn) for all n; (ii) φn(x) → Φ (x) as n → ∞, uniformly for x ∈ [a,b] ; and (iii) for each kas n → ∞, uniformly for x ∈ [a,b] , where φ(k) denotes the k-th derivative of φ.

This is a stringent definition, much stronger than ordinary convergence. We do not offer an example because specific examples are never needed: test functions are only the scaffolding upon which the main part of the theory is built.

Definition 1.9 A functional ƒ is continuous , that is, if ƒ(φn) → ƒ(Φ) whenever φn is called a distribution, or generalised function.

functions is: ƒ > 0 there is a δ > 0 such that |ƒ(x) − ƒ(yx − y < δ. This can be shown to be equivalent to the condition that ƒ of the modulus of a number which appears in the other definition (in Chapter 4 we shall consider this question further).

Notation 1.10 We shall use bold type to signify a distribution, and 〈ƒ,φ〉 to denote the ‘action’ of the distribution ƒ on the test function φ; in other words, 〈ƒ,φ〉 is the number into which ƒ maps φ. The reason for using this odd-looking notation, rather than ƒ(φ), will appear shortly.

Example 1.11 The delta distribution δ is defined by

〈δ,φ〉 = φ(0)

for all φ . This is the functional defined in Example 1.7(a), using different notation. To justify calling it a distribution, we must show that the functional is continuous, i.e. that (δ,φn〉 is a convergent sequence of numbers whenever (φn. In fact it follows immediately from Definition 1.8 that φnΦ(0) if φn , so δ is indeed a distribution.

There is a class of ‘regular’ distributions, corresponding to ordinary functions of the following kind.

Definition 1.12 A function ƒis locally integrable exists for all numbers a,b.

Any continuous function is locally integrable; so is any piecewise continuous function, as defined below.

Definition 1.13 A function ƒis called piecewise continuous if it is continuous except at a set of points xi such that any finite interval contains only finitely many xi, and if the left and right hand limits of the function exist at each xi.

x x a for −1 < a x a for a ≤ −1, are not. Note that the behaviour of f(x) as x → ∞ is irrelevant to whether it is locally integrable.

Theorem 1.14 (Regular Distributions) To every locally integrable function ƒ there corresponds a distribution ƒ defined by

The distribution ƒ is said to be generated by the function f.

Proof We must first show that the integral exists. Since any φ has bounded support, contained in [a,b, which exists since the product of an integrable function and a continuous function is an integrable function. Hence ƒ . Linearity is obvious. To prove that ƒ is a distribution, suppose φn . Then there is an interval [a,b] containing the supports of φn ∫p (x)q(x) dx ≤ ∫ p(x)q(xdx p(x)∫ q(xdx, we have

because φn → Φ uniformly (by Definition 1.8). Hence 〈ƒ,φn〉 → 〈ƒ,Φ〉 so f is a continuous functional, that is, a distribution.

The reason for the notation 〈ƒ,φ〉 is that in other branches of analysis (ƒ,g(see section 7.1), and this theorem identifies 〈ƒ,φ〉 with (ƒ,φ), at least when ƒ is locally integrable and φ is real.

Definition 1.15 A distribution which is generated by a locally integrable function is called regular. All other distributions are called singular.

Thus the class of distributions contains objects which correspond to ordinary functions as well as singular distributions which do not. In the next section we shall define operations on distributions analogous to the operations of ordinary algebra and calculus applied to functions. This will justify calling distributions ‘generalised functions’, and will allow us to use distributions for most of the purposes for which ordinary functions are used.

1.3 Operations on Distributions

Definition 1.16 (Addition) If ƒ and g are distributions and a and b are complex numbers, we define the distribution aƒ + bg to be the functional φ a 〈ƒ,φ〉 + b 〈g,φ〉 for all φ .

, but that is very easy and is left to the reader. The same applies to other definitions in this section.

Definition 1.17 (Multiplication) If ƒ is a distribution and h function (cf. Definition 1.2), we define the product of ƒ and h to be the distribution hƒ: φ 〈ƒ,hφ〉 for all φ .

Note that if φ is a test function and h is smooth, then hφ is a test function, and therefore 〈ƒ,hφ〉 is well-defined. If h is not smooth, then neither is hφ, hence h , and Definition 1.17 does not work. Thus we cannot define the product of a distribution with a function which is discontinuous or has a discontinuous derivative. Distributions cannot in general be multiplied. The difficulty can be seen in the following

Example 1.18 The Heaviside function H is defined by

and an almost identical function H1 is defined by

H and H. There is no distinction between H1 and H in generalised function theory, and this reflects the fact that from the point of view of the physicist the distinction between them is artificial: one could never distinguish between H1 and H experimentally. If Definition 1.17 were extended to discontinuous functions, we would have 〈H1δ,φNow, any definition of the product of two distributions would have to agree with Definition 1.17 in the case that one of the distributions was regular. Since H1 and H generate the same distribution H, we would have two different values for 〈Hδ,φ〉, and thus an inconsistent theory. This shows that it is impossible to define the product of δ with a discontinuous function. However, one can extend Definition 1.17 so as to allow multiplication by functions which are not smooth; see Problem 1.4.

Definition 1.17 has the property that if ƒ is a regular distribution generated by a function ƒ, and h is smooth, then the distribution hf is the same as the distribution generated by the locally integrable function hƒ – they are both given by φ ∫h(x)f(x)φ(x) dx. In other words, the definition of the product of a generalised function and an infinitely differentiable function is consistent with the ordinary rule for multiplying functions.

, that is, f =g if and only if 〈ƒ,φ〉 = 〈g,φ〉 for all φ. For ordinary functions we can talk about two functions being equal at a point. For generalised functions this does not make sense since there is no such thing as the ‘value’ of a generalised function at a point x. However, we can form an idea of the behaviour of a generalised function over an interval (a,b) by considering its action on test functions which vanish outside (a,b) and therefore give no information about the distribution outside (a,b).

Definition 1.19 Two distributions ƒ and g are said to be equal on (a,b) if 〈ƒ,φ〉 g,φ〉 for all φsuch that supp(φ) ⊂(a,b).

Thus, for example,δ = 0 on (O,a) for any a > 0, or indeed on (0,∞). Here 0 denotes the distribution φ 0 for all φ, which is generated by the function z(x) ≡ 0. In this case it is natural to say that the delta function has the value zero for all x > 0, and the next definition legalises that usage.

Definition 1.20 If ƒ is a distribution and g is a locally integrable function, ƒ is said to take values g(x) on an interval (a,b) if ƒ = g on (a,b) (in the sense of Definition 1.19). In this context we sometimes use ƒ(x) as an alternative notation for ƒ, and write ƒ(x) = g(x) for a < x < b.

Example 1.21 δ takes the value 0 for x > 0 and x < 0. In the notation explained above, this statement can be written δ(x) = 0 for x ≠ 0. But it should be remembered that δ(x) does not stand for the value of δ at a point x ; statements containing the notation ƒ(x) should be interpreted strictly according to Definitions 1.19 and 1.20. (At the same time, there is little harm in thinking of ƒ(x) as if it were the value of a function at a point, provided that you realise that it is only a convention and liable to be misleading. For example, δ (0) really is meaningless, since there is no ordinary function g such that δ takes values g(x) on an interval including 0.)

We now define operations analogous to simple changes of variable for ordinary functions.

Definition 1.22 (Change of Variable) For any distribution ƒ and any real a, a new distribution ƒ+a is defined by 〈ƒ+a,φ〉 = 〈ƒ,φ(x − a)〉. We call ƒ+a the translation of ƒ by a. When the notation ƒ(x) is used for ƒ (as explained in Definition 1.20), ƒ(x + a) is a convenient notation for the translation of ƒ.

If ƒ is a regular distribution generated by a function ƒ, then it is easy to see that ƒ+a is generated by the function ƒ(x + a). Thus ƒ (x + a) is a consistent and sensible notation. Of course, when we write f(x + a), we mean the function x ƒ(x + a); similarly, in Definition 1.22, strictly speaking we should not write 〈ƒ,φ(x + a)〉 since φ(x + a) is not a function but a number, but should write 〈ƒ,ψ〉 where ψ: x φ(x + a). It is pedantic and clumsy to insist on keeping to these rules all the time, however, and we shall assume that the reader can understand from the context whether f(x) stands for a number or a function.

Definition 1.23 (Change of Variable) For any distribution ƒ and any real a ≠ 0, a new distribution ƒ.a is defined by 〈ƒ.a,φ〉 = 〈ƒ,φ(x/aa . When the notation ƒ(x) is used for ƒ, ƒ(ax) is a convenient notation for ƒ.a.

If ƒ is a regular distribution generated by a function ƒ, then it is easy to see that ƒ.a is generated by the function ƒ(ax). Thus ƒ(ax) is a sensible and consistent notation. If ƒ is a distribution which takes values g(x) on (a,b) (in the sense of Definition 1.20), then ƒ(x + c) = g(x + c) on (a − c, b — c) and ƒ(cx) = g(cx) on (a/c,b/c) if c > 0 or on (b/c,a/c) if c < 0.

Example 1.24 〈δ(x − a),φ〉 = 〈δ(x),φ(x + a)〉 = φ(a). Thus δ (x − a) is a distribution which picks out the value at a of a test function to which it is applied. This is sometimes called the ‘sifting property’ of the delta function. We also have δ(x − a) = 0 for x ≠ a. Similarly, 〈δ (ax),φ〉 = 〈δ (x),φ(x/aa = φ a . Hence

(1.3)

We now proceed to the differential calculus of distributions. Our general plan for defining operations on distributions is to begin with locally integrable functions, find an expression for the operation applied to the corresponding regular distribution, and then generalise to all distributions. If ƒ is a differentiable function, ƒ the corresponding regular distribution, and df/dx the distribution generated by f′, we have

on integration by parts (there are no boundary terms because φ vanishes at infinity). For regular distributions corresponding to differentiable functions, we thus have

(1.4)

We shall use (1.4) as a definition of differentiation for any distribution. But first we must show that the right hand side of (1.4) is always a distribution.

Proposition 1.25 For any distribution ƒ, the functional φ −〈ƒ,φ′〉 is a distribution.

Proof If φ, then φ′, so φ −〈ƒ,φ′. It is clearly linear. If φn , then φ′(by the standard theorem on differentiating uniformly convergent sequences), hence −〈ƒ,φ′n〉 → −〈ƒ,Φ′〉 which proves that the functional φ −〈ƒ,φ′〉 is continuous.

We can now define differentiation for generalised functions.

Definition 1.26 The derivative of a generalised function f is the generalised function ƒ′ defined by 〈ƒ’,ø’〉 = −〈ƒ,φ′〉 for all φ .

Equation (1.4) shows that the distribution generated by the derivative ƒ’ of a function f is the same as the derivative of the distribution ƒ; these two possible ways of interpreting the symbol ƒ′, for a differentiable function ƒ, are identical. Our new definition of differentiation is consistent with ordinary calculus.

The advantage of our theory over ordinary calculus is that every generalised function is differentiable; this follows from Proposition 1.25. If ƒ is a locally integrable function which is not differentiable, the distribution ƒ′ is called the generalised derivative of ƒ. For example, a continuous but not differentiable function has a generalised derivative with a discontinuity; a function with a simple step discontinuity has a generalised derivative involving a delta function.

Example 1.27 x is a locally integrable function, differentiable for all x ≠0, but certainly not differentiable at 0. The generalised derivative is calculated as follows. For any test function φ,

integrating by parts and using the fact that φ vanishes at infinity. We define a function sgn(x) (read as ‘sign of x’) by

(1.5)

It is unnecessary for our purposes to specify the value of sgn(0), since it generates the same distribution sgn for any choice of sgn(0). We now have, from the above,

, so | x |′ = sgn. Note that since the generalised derivative is a distribution, we cannot speak of its value at a point, only over an interval. Thus the awkward question ‘what is the value at x = 0 of the derivative of | x | ?’ has no meaning in our language.

Example 1.28 since φ vanishes at infinity. Hence

(1.6)

Example 1.29 The derivative of the delta function is defined by 〈δ’,φ〉 = −〈δ,φ′〉 = −φ’(0). Similarly, the n-th derivative of δ is given by 〈δ(n),φ〉 = (− 1)nφ(n)(0).

It is easy to show that the derivative of a sum of generalised functions equals the sum of the derivatives. The rule for differentiating a product needs a little more care. If h is a smooth function and f ,

showing that

(hf)′ = hf’ + h’f .

We have now constructed the basic algebra and differential calculus of distributions. In the next section we consider distributional convergence.

1.4 Convergence of Distributions

. However, it is useful also to define the idea of convergence for distributions. Just as for continuity, we cannot use an ‘ε − δ.

Definition 1.30 A sequence (fn) of distributions is said to be convergent if the sequence of numbers (〈fn,φ.

Notice that this definition does not involve the existence of a limiting distribution towards which the sequence tends. It is framed strictly in terms of the sequence itself (unlike the usual definition of convergence in elementary analysis, which requires one to find the limit of a sequence before the definition can be used). However, it can be proved that if a sequence of distributions converges, then there is in fact a limit distribution.

Theorem 1.31 If (fn) is a convergent sequence of distributions, then there is a distribution F such that 〈fn,φ〉 → 〈F,φ. We write fn → F as n → ∞.

Given a convergent sequence of distributions (fn), we can always define a linear functional φ lim(〈fn,φ〉). This is the required distribution F, but to prove that it is a distribution, that is, to prove that it is a continuous functional, is not easy. We shall not need the theorem in this book, and for its proof we refer the reader to the standard text-books (see section 1.7).

The next theorem relates distributional convergence to ordinary convergence for regular distributions.

Theorem 1.32 (Distributional Convergence) If F, f1, f2, . . . are locally integrable functions such that fn → F uniformly in each bounded interval, then fn → F distributionally.

Proof Let φ be a test function with supp(φ) ⊂ (a,b), then

, 〈fn,φ〉 → 〈F,φ〉.

It is not in general true that if fn → F where fn and F are locally integrable functions, then fn → F; the convergence must be uniform, as in Theorem 1.32 (or some other extra condition must be satisfied). This will be illustrated by Example 1.34 below. But first we must consider a simpler example.

Example 1.33 dn → δ as n → ∞, where dn(x) = n/π (1 + n²x²) is the sequence considered in section 1.1. This statement is usually written more concisely as

(1.7)

To prove it, we must show that ∫dn(x)φ(x) dx → φ(0) as n ,

where (a, b) is an interval containing supp(φ), and a < 0 < b. By means of the substitution nx = y, the first and third integrals here are easily shown to tend to zero as n → ∞. In the second, use the mean value theorem to write |φ(x) − φ(0) |≤M |x| , where M = max I φ′(x, which is easily evaluated exactly, and shown to tend to zero as n → ∞. This completes the proof of (1.7).

Example 1.34

(1.8)

(integrating by parts), → − φ′(0) = 〈δ′,φpointwise as n → ∞ for all xis a true statement, it does not represent the true state of affairs as clearly as (1.8) does.

Equation (1.8) is a special case of the following theorem.

Theorem 1.35 (Termwise Differentiation) If F, f1, f2, . . . are distributions such that fn → F as n .

Proof

.

Using Theorem 1.35, (1.8) follows immediately from (1.7). Notice how much simpler this theorem is than the corresponding result for ordinary functions: in that case, (fn.

Our outline of the basic framework of distribution theory is now complete. Essentially all the usual results of calculus hold in the generalised theory; for instance, one can prove that the limit of a product or a sum is the product or sum of the limits, and so on. Similarly one can define convergence of a distribution which depends on a continuously varying parameter t as well as for distributions depending on an integer parameter n. All this will be found in the systematic text-books listed in section 1.7.

1.5 Further Developments

In this section we shall introduce another piece of notation for simplifying work with generalised functions, and then consider some important examples of singular distributions.

Notation 1.36 For any distribution f and test function φstands for the number 〈f,φ〉.

One can tell whether the integration sign stands for a genuine integral or the action of a functional on a test function by whether the integrand contains ordinary functions or generalised functions f. Many authors make no distinction between the notations for ordinary and generalised functions. This is a reasonable policy because whenever confusion is possible, it is harmless: that is, the two alternative interpretations of a formula lead to the same result in the end. Thus, if f is a regular distribution generated by the function f, then ∫f(x)φ(x) dx = ∫f(x)φ(x) dx by definition of f. If you look back at Definitions 1.22, 1.23, and 1.26, you will see that they are constructed so that if the integral notation for distributions is used, they correspond to the usual rules for simple changes of variable and integration by parts (the boundary terms being zero because φ has bounded support). Therefore, using the integral notation one can manipulate the integrals just as if they stood for genuine integration, while feeling assured that the rules of distribution theory are being strictly obeyed. Thus, for instance, we can write the first result in Example 1.24 as

(1.9)

and prove it by a change of variable in the integral, x − a = y, so that the left hand side of (1.9) becomes ∫δ(y)φ(y + a) dy = φ(a). Equation (1.9) is the key property of the delta function in the informal treatment usual in physics or mathematical-methods books.

We shall now consider some examples of singular distributions. So far we have met the delta function and its derivatives, which are singular according to Definition 1.15. That definition does not quite correspond to the use of the word ‘singular’ in ordinary analysis. The function log | x | has what is generally called a singularity at x = 0. But it is locally integrable (its indefinite integral is x log | x | −x, which is a continuous function), and generates a regular distribution. Its derivative 1/x, however, is not locally integrable, and does not generate a regular distribution. We must therefore consider more carefully the derivative of the generalised function log| x |.

The difficulty is that the integrand has a singularity so strong that it must be excised from the domain and the integral defined by a limiting process. The result of such a process is called an improper integral. If I,δ) tends to a definite limit as ε and δ independently tend to zero (through positive values), then the improper integral ∫φ(x) dx/x is said to converge to that limit. In general this will not be the case, unless φ(0) = 0. But if we take δ , and then let ε → 0, we find that I, ) tends to a limit. Taking δ → 0 gives a different limit, and thus many different values can be assigned to ∫φ(x) dx/x = δ. The value obtained in this way (if it exists) is called the principal value of the integral. We must now prove that I, → 0.

Notation 1.37 ↓ 0 means e tends to zero taking positive values only.

Lemma 1.38 For any test function φ↓0.

Proof Integrating by parts gives

The integrals containing loglxl are convergent as ε ↓ 0. And [φ) − φ| = −φ′(θ | for some θ ↓ 0 because φ.

Definition 1.39 If f is a function defined for x ≠ 0, we define the principal value whenever the limit exists.

Lemma 1.38 shows that the limit exists when f(x) = φ(x)/x, and that

(1.10)

We can now define a distribution corresponding to 1/x, in the same way as for locally integrable functions, but with ∫x−1 φ(x) dx given a definite value by the principal-value rule.

Definition 1.40 P/x