As we mentioned in the first chapter, general questions concerning whole numbers, rather than merely mechanical computations, can be extremely difficult to answer. Indeed this higher (as opposed to basic) arithmetic is one of the most difficult and fascinating branches of mathematics, and has been extensively studied by many of the greatest mathematicians. These mathematicians often spent an enormous amount of time and energy checking their conjectures with specific numbers and sifting through mountains of experimental data. (Mathematics is an experimental science after all. It differs from other experimental sciences, however, because often the amount of experimental evidence prior to the proof of a result might be very slim, and after the proof experimental evidence is strictly irrelevant, although psychologically reassuring.) Anyway, the computational feats of mathematicians of old are often extremely impressive, bearing in mind that they only had paper and pencil (or papyrus and stylus or whatever) to hand. What an advantage we have! Indeed what might Gauss have achieved with his own micro? Let us start our investigations. The emphasis throughout this chapter will be on the building blocks of arithmetic: the prime numbers.

Prime numbers

Recall that a natural number x is prime if it is only divisible by 1 and itself (in other words cannot be written as a product of natural numbers a and b unless a or b is 1). It is fairly clear that any number can be factored as a product of prime numbers and the resulting factorization is unique up to the order of the factors.

Introduction

This chapter is the core of the book. We have seen that if the function k is sufficiently well-behaved then the operator K defined by is a compact operator from L2(a, b) to itself, so results about compact operators will produce corresponding information about integral equations. Just as for general matrices, the amount that can be said about general compact operators is limited, and the most substantial conclusions follow if the operator has some symmetry, symmetry in this case being self-adjointness. It turns out that there is a powerful classification theorem for compact self-adjoint operators which describes the action of the operators in terms of their eigenvalues and eigenvectors.

The benefit of the theory of this chapter is that it expresses compact self-adjoint operators in a standard form, which is known to exist for each operator. This reduces many problems to one of evaluation of information about parameters already known to exist, and the techniques can be used to provide specific information about integral operators. We shall, in fact, return frequently to the structure theorem for compact self-adjoint operators to find the basis for a variety of techniques in subsequent chapters.

Many of the results given here are true in greater generality than we shall state them. In particular, the theory can be extended to compact operators on Banach spaces, although the additional complication is considerable and the results on self-adjointness do not extend satisfactorily.

Measure theory and (Lebesgue) integration are both very detailed and technical topics, to an extent that can be off-putting to anyone seeking results to use elsewhere. In this appendix we have attempted to suppress detail that is not required for the purposes in hand - for example, we do not need to know the definition of a measurable function, only enough to show that the functions we shall encounter are measurable. We have not stated the results in their greatest generality.

We shall need to use a vector space of functions which forms a Hilbert space and the natural choice of norm associated with this is given by ∥ φ ∥ = {∫|φ(t)|2dt}½. This definition does indeed define a norm on the set of continuous functions on [a, b] but the resulting space is not complete and we need to enlarge our class of functions, which in turn requires a suitable integral, the Lebesgue integral.

A necessary condition for a function φ: S → ℝ (where S is, for the moment, considered to be an interval, possibly of infinite length, in ℝ) to be Lebesgue integrable is that it be measurable. The definition of measurability need not concern us here; what we do need to know is that all continuous functions, monotonic functions and all step functions are measurable and that the modulus of a measurable function, the sum and product of two measurable functions and a real-valued function which is the pointwise limit of a sequence of measurable functions are all measurable.

Introduction

We have seen that the solution of an equation of the form φ = f + λKφ can, in some circumstances, be expressed as the sum of a series involving all the eigenvalues µn and eigenvectors φn of the operator K. From a theoretical point of view this solution is of undoubted importance, but its practical use is limited by the fact that there are comparatively few operators for which the sequences (µn) and (φn) are known. If these sequences are not known exactly, there remains the prospect of estimating the first N eigenvalues and eigenvectors of K and approximating the solution of φ = f + λKφ by truncating the associated series.

We shall in fact consider approximation methods for the equation φ = f + λKφ in the following chapter. There it will emerge that a knowledge of the dominant eigenvalues of K (that is, its largest positive eigenvalue and its negative eigenvalue of largest magnitude) is of particular value in implementing several of these methods.

In an integral equation arising from a practical problem, λ itself is a quantity of some significance. A simple illustration of this fact was given in Example 1.3, where λ = 1/µ is proportional to the square of the frequency of vibration of a thin rod. The eigenvalues in this case give the natural frequencies of oscillation of the rod, that is, the frequencies at which resonance will occur in certain forced motions of the rod.

Introduction

We have already defined a non-negative operator K to be one which is self-adjoint and such that (Kφ, φ), ≥ 0 for all φ. A positive operator is self-adjoint and satisfies the stronger condition (Kφ, φ), > 0 for all φ ≠ 0. In Mercer's Theorem we saw that if the operator generated by a continuous kernel k(x, t) (0 ≤ x, t ≤ 1) is non-negative, then k(x, t) can be expanded in a uniformly convergent series and the eigenvalues of the operator are such that. If we know that all of these eigenvalues are positive then each of them does not exceed the value of the integral, which is therefore an upper bound for the largest eigenvalue.

Given an arbitary operator T in a Hilbert space, we can construct the operator T*T which is non-negative since (T*Tφ, φ) = ∥Tφ∥∥2 ≥ 0. We are therefore well supplied with non-negative operators. In practice, however, we usually need to determine whether a given operator is non-negative and this is a difficult issue to resolve. We give it some attention in this chapter, because there are many techniques, especially in approximation theory, which apply only for non-negative or positive operators. The eigenvalue bound referred to above is a simple example.

As we intend to apply our results to integral equations, we shall restrict attention to compact operators and those closely related to them.

Introduction

Chapter 4 was devoted to the spectral theorem in which we characterised a compact self-adjoint operator in terms of its eigenvalues and eigenvectors, and from this we were able to deduce substantial qualitative results about integral equations. Far reaching though these may be, they have the drawback that they involve the eigenvalues and eigenvectors of the particular integral operator being investigated. In many cases the exact determination of eigenvalues and eigenvectors will itself be a difficult problem, so in this chapter we shall derive a body of results giving characterisations of the various eigenvalues of a particular operator, and relations between the eigenvalues of the sum and product of two operators and those of the summand operators. The techniques used here yield some results immediately on the approximation of one operator by another, and these can be used to estimate the eigenvalues of otherwise recalcitrant operators. The main emphasis of this chapter, however, is not on approximation techniques, which will be dealt with in Chapters 7 and 8, but on suitable characterisations of the quantities involved which will form the foundations of these approximation techniques.

The Rayleigh quotient

If H is a Hilbert space and T a bounded linear map from H to itself, then associated with each vector φ ∈ H there is a scalar quantity (T φ, φ), which is real if T is self-adjoint, for then (Tφ, φ) = (φ, T*φ) = (φ, Tφ) = (Tφ, φ).

It often happens that the most concise and illuminating method of solving even the most practical problem in mathematics involves the use of abstract ideas and techniques. This is particularly true of integral equations, where much progress can be made by using both direct and abstract techniques side by side.

The advantage of reformulating an equation, such as an integral equation, as an ‘abstract’ problem in a Hilbert space is that many of the important issues become clearer. In the abstract setting, a function is regarded as a ‘point’ in some suitable space and an integral operator as a transformation of one ‘point’ into another. Since a point is conceptually simpler than a function this view has the merit of removing some of the mathematical clutter from the problem, making it possible to see the salient issues more clearly. It is thus easy to visualise elegant general structures which can be translated into results about the original concrete problem. To obtain these results in a useful form, however, a second step is needed, for elegant general results tend to produce only elegant generalities and a further process is required to recover hard specific facts about the solutions sought. We use the abstract framework of functional analysis to derive the general structures and more ad hoc techniques for the recovery.

There is all too often a gap between the approaches of a pure and an applied mathematician to the same problem, to the extent that they may have little in common.