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| ISBN-13: | 9780486141329 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 04/26/2012 |
| Series: | Dover Books on Mathematics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 390 |
| File size: | 11 MB |
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Applied Functional Analysis
By D.H. Griffel
Dover Publications, Inc.
Copyright © 1985 D.H. GriffelAll rights reserved.
ISBN: 978-0-486-14132-9
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 (distances measured from the centre of the rod, say) is ∫baρ(x)dx. 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:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (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,
dn(x) = n/[π(1 + n2x2)]. (1.2)
Then dn(x) -> 0 as n -> ∞ for any x ≠ 0, and dn(0) -> ∞ as n -> ∞ (see Fig. 1). Also [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and if a< 0 < b, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 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 replacesδ 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.
1.2 Basic Distribution Theory
There are several ways of generalising the idea of a function in order to include Dirac'sδ. 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 f: R -> C is {x: f(x) ≠ 0}, written supp(f). A function has bounded support if there are numbers a,b such that sup(f) [subset] [a,b].
Definition 1.2 A function f: R -> C is said to be ntimes continuously differentiable if its first n derivatives exist and are continuous. f is said to be smooth or infinitely differentiable if its derivatives of all orders exist and are continuous.
Definition 1.3 A test function is a smooth R -> C function with bounded support. The set of all test functions is called P.
Example 1.4 is a test function with support (a,b).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
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 on the space P is a map f: P -> C such that f(aφ + bψ) = af(φ) + bf(ψ) for all a,b [member of] C and φ, ψ [member of] P.
A map P -> C means a rule which, given any φ [member of] P, produces a corresponding z [member of] C; we write z = f(φ). We also speak of the 'action' of the functional f on φ producing the number f (φ). The notation φ [??] f(φ) stands for the phrase 'φ is mapped into the number f(φ)'; see Appendix A.
Examples 1.7 (a) φ [??] φ(0) is a functional, easily seen to be linear. (b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] dx is a functional; the integral converges because φ has bounded support. It is not linear. (c) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] dx is a linear functional for any function f sufficiently well behaved to ensure convergence of the integral.
We must now define convergence in the space P. 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 -> Φ in P 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 [member of] [a,b]; and (iii) for each k, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as n -> ∞, uniformly for x [member of] [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 f on P is continuous if it maps every convergent sequence in P into a convergent sequence in C, that is, if f(φn) ->f(Φ) whenever φn -> Φ in P. A continuous linear functional on P is called a distribution, or generalised function.
This definition of continuity is modelled on one of two alternative definitions of continuity of an ordinary R -> R function. The other common definition for R -< R functions is: f is continuous if for any ε > 0 there is a δ > 0 such that |f(x) - f(y) | < ε whenever |x - y| < δ. This can be shown to be equivalent to the condition that f map every convergent sequence of numbers into a convergent sequence. We adopt the latter as our definition in P, because there is no analogue in P 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 <f, φ> to denote the 'action' of the distribution f on the test function φ; in other words, <f, φ> is the number into which f maps φ. The reason for using this odd-looking notation, rather than f(φ), will appear shortly.
(Continues...)
Excerpted from Applied Functional Analysis by D.H. Griffel. Copyright © 1985 D.H. Griffel. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
PrefacePart I. Distribution Theory and Green's Functions
Chapter 1. Generalised Functions
1.1 The Delta function
1.2 Basic distribution theory
1.3 Operations on distributions
1.4 Convergence of distributions
1.5 Further developments
1.6 Fourier Series and the Poisson Sum formula
1.7 Summary and References
Problems
Chapter 2. Differential Equations and Green's Functions
2.1 The Integral of a distribution
2.2 Linear differential equations
2.3 Fundamental solutions of differential equations
2.4 Green's functions
2.5 Applications of Green's functions
2.6 Summary and References
Problems
Chapter 3. Fourier Transforms and Partial differential Equations
3.1 The classical Fourier transform
3.2 Distributions of slow growth
3.3 Generalised Fourier transforms
3.4 Generalised functions of several variables
3.5 Green's function for the Laplacian
3.6 Green's function for the Three-dimensional wave equation
3.7 Summary and References
Problems
Part II. Banach spaces and fixed point theorems
Chapter 4. Normed spaces
4.1 Vector spaces
4.2 Normed spaces
4.3 Convergence
4.4 Open and closed sets
4.5 Completeness
4.6 Equivalent norms
4.7 Summary and References
Problems
Chapter 5. The contraction mapping theorem
5.1 Operators on Vector spaces
5.2 The contraction mapping theorem
5.3 Application to differential and integral equations
5.4 Nonlinear diffusive equilibrium
5.5 Nonlinear diffusive equilibrium in three dimensions
5.6 Summary and References
Problems
Chapter 6. Compactness and Schauder's theorem
6.1 Continuous operators
6.2 Brouwer's theorem
6.3 Compactness
6.4 Relative compactness
6.5 Arzelà's theorem
6.6 Schauder's theorems
6.7 Forced nonlinear oscillations
6.8 Swirling flow
6.9 Summary and References
Problems
Part III. Operators in Hilbert Space
Chapter 7. Hilbert space
7.1 Inner product spaces
7.2 Orthogonal bases
7.3 Orthogonal expansions
7.4 The Bessel, Parseval, and Riesz-Fischer theorems
7.5 Orthogonal decomposition
7.6 Functionals on normed spaces
7.7 Functionals in Hilbert space
7.8 Weak convergence
7.9 Summary and References
Problems
Chapter 8. The Theory of operators
8.1 Bounded operators on normed spaces
8.2 The algebra of bounded operators
8.3 Self-adjoint operators
8.4 Eigenvalue problems for self-adjoint operators
8.5 Compact operators
8.6 Summary and References
Problems
Chapter 9. The Spectral theorem
9.1 The spectral theorem
9.2 Sturm-Liouville systems
9.3 Partial differential equations
9.4 The Fredholm alternative
9.5 Projection operators
9.6 Summary and References
Problems
Chapter 10. Variational methods
10.1 Positive operators
10.2 Approximation to the first eigenvalue
10.3 The Rayleigh-Ritz method for eigenvalues
10.4 The theory of the Rayleigh-Ritz method
10.5 Inhomogeneous Equations
10.6 Complementary bounds
10.7 Summary and References
Problems
Part IV. Further developments
Chapter 11. The differential calculus of operators and its applications
11.1 The Fréchet derivative
11.2 Higher derivatives
11.3 Maxima and Minima
11.4 Linear stability theory
11.5. Nonlinear stability
11.6 Bifurcation theory
11.7 Bifurcation and stability
11.8 Summary and References
Chapter 12. Distributional Hilbert spaces
12.1 The space of square-integrable distributions
12.2 Sobolev spaces
12.3 Application to partial differential equations
12.4 Summary and References
Appendix A. Sets and mappings
Appendix B. Sequences, series, and uniform convergence
Appendix C. Sup and inf
Appendix D. Countability
Appendix E. Equivalence relations
Appendix F. Completion
Appendix G. Sturm-Liouville systems
Appendix H. Fourier's theorem
Appendix I. Proofs of 9.24 and 9.25
Notes on the Problems; Supplementary Problems; Symbol index; References and name index; Subject index