Index

Introduction to the 1961 Edition

MAGNUS R. HESTENES

PROFESSOR OF MATHEMATICS

UNIVERSITY OF CALIFORNIA, LOS ANGELES

During the last decade, there has been a remarkable expansion in the demand for advanced mathematics by the engineer. This demand has arisen in part because of the increasing complexities created by technological progress.

In advanced design it is frequently necessary to study a carefully constructed mathematical model before creating the physical model. The modern rockets and satellites, for example, could not have been built and successfully launched without careful mathematical analysis of the physical problem at hand.

Not only does mathematics enter into initial planning and design; it enters into testing programs as well. Data are collected and interpreted in accordance with a statistical theory. Once a product has been designed and tested, a mathematical theory of quality control is frequently used in the manufacture of the product.

More recently, mathematics has been found to be a useful tool in the field of production planning.

Thus mathematics enters into all phases of engineering and production.

The modern high-speed computing machine is playing an ever-increasing role in physical, biological, and social sciences, in engineering, and in business. The effective use of computers requires the aid of persons with a high degree of mathematical training and proficiency. Almost every branch of mathematics has been used on problems that have been successfully attacked with the help of computing machines.

These machines can be used for experimentation as well as for solving intricate mathematical problems. For example, a traffic problem has been simulated on a computing machine, and experiments have suggested means of traffic control that have significantly increased the flow of traffic in a congested area.

It is clear that persons responsible for the operations of computing machines must be proficient in mathematics and must have on hand source material necessary for the solution of their problems.

The role of mathematics in engineering has been most aptly presented by Dr. Royal Weller in the Introduction to Modern Mathematics for the Engineer, First Series.

The topics selected for the present volume have been chosen to complement those found in the first volume. As is customary in mathematics, the mathematical theory is presented largely without regard to applications, except as illustrations of the theory. This is done in order to bring out the mathematical structure and in order to facilitate applications to problems that actually are of similar basic mathematical structure but that bear little superficial similarity to the applications being described. An attempt is made also to call attention to various mathematical fields that undoubtedly will play an important role in the science and engineering of the future.

Though it is not possible to cover all the phases of mathematics that are important for applications, it is hoped that this volume and its predecessor have exposed many of the most important and promising ones, have laid a firm foundation in modern mathematical thinking, and will stimulate the reader to further study of this most interesting and useful field of human endeavor.

PART 1

Mathematical Methods

1

From Delta Functions to Distributions

ARTHUR ERDÉLYI

PROFESSOR OF MATHEMATICS

CALIFORNIA INSTITUTE OF TECHNOLOGY

1.1Introduction

In mathematical physics, one often encounters impulsive forces acting for a short time only. A unit impulse would be described by a function p(t) that vanishes outside a short interval and is such that

It is convenient to idealize such forces as instantaneous and to attempt to describe them by a function δ(t) that vanishes except for a single value of t which we take to be t = 0, is undefined for t = 0, and for which

Such a function, one convinces oneself, should possess the sifting property

for every continuous function ϕ, and the corresponding property (obtained by integration by parts)

for every k times continuously differentiable function ϕ.

Unfortunately, it can be proved that no function, in the sense of the mathematical definition of this term, possesses the sifting property. Nevertheless, impulse functions postulated to have these or other similar properties are being used with great success in applied mathematics and mathematical physics.

The use of such improper functions can be defended as a kind of short-hand, or else as a heuristic means; it can also be justified by an appropriate mathematical theory. In Sec. 1.2 we shall indicate briefly some theories that can be employed to justify the use of the delta function. In order to provide a theoretical framework accommodating the great variety of improper functions occurring in contemporary investigations of partial differential equations, it seems necessary to widen the traditional concept of a mathematical function. The new concept, that of a generalized function, is abstract and cannot reproduce all aspects of the older concept of a function. In particular, it is not possible to ascribe a definite value to a generalized function at a point. Nevertheless, we shall see that in some sense such generalized functions can be described. In particular, it makes perfectly good sense to say that δ(t), which is a generalized function, vanishes on any open interval not containing t = 0.

In this chapter we shall outline two theories of generalized functions. One, essentially algebraic in nature, is restricted to generalized functions on a half line; the other, more closely related to functional analysis, places less restrictions on the independent variable. We shall also mention briefly other theories of generalized functions.

DELTA FUNCTIONS AND OTHER GENERALIZED FUNCTIONS

1.2The Delta Function

Since the delta function is the idealization of functions that vanish outside a short interval, it is plausible to try to approximate the delta function by such functions. Let s(t) be a function on (− ∞, ∞) satisfying the following conditions:

Then the function

sn (t) = ns(nt)

satisfies the conditions a and c and vanishes outside (−1/n, 1/n), and it may be regarded as approaching the delta function as n → ∞. Indeed, it can easily be proved that

for any continuous function ϕ, or even for any function ϕ that is integrable over some interval containing 0 and is continuous at 0. Furthermore,

uniformly in t over any finite interval, provided ϕ(t) is continuous over some larger interval. If s(t) is k times continuously differentiable, we also have

As a matter of fact, it is not necessary that s have the property b. If s has the properties a and c, then the condition (1.3) will hold for all functions ϕ that are bounded and continuous for − ∞ < t < ∞. Some examples of such approximations to the delta function that have been used by the great analysts of the last century are the following:

For a history of the delta function, see Ref. 16, Chap. V.

It may be remarked that we clearly have

showing that in some sense δ(t) is the derivative of the unit function

and indicating some connection between the theory of the delta function and that of generalized differentiation of discontinuous functions.

An entirely different kind of theory of the delta function was adumbrated by Heaviside (see Chaps. 2 and 3 and also Ref. 16, page 65) and more clearly pinpointed by Dirac (Ref. 2, pages 71 to 77) ; it was not carried out in detail, however, until more recently. According to this theory, the delta function is defined by its action on continuous functions, this action being given by the sifting property (1.1) or (1.3); any analytical operation that, acting on a continuous function ϕ, produces ϕ(0) is then a representation of the delta function.

We have seen that it is impossible to construct such an analytical operation in the form of a Riemann (or Lebesgue) integral; but it is possible to express it as a Stieltjes integral. Indeed,

for all continuous functions ϕ. If U were differentiable, we should have

so that here too the delta function appears as a generalized derivative of the unit function.

The two theories are not as far from each other as they might at first appear to be. Although the delta function cannot be expressed as an integral operation, it can be approximated by such operations, namely, the integral operations defined by means of the sn. Indeed, this is exactly the burden of Eq. (1.3).

1.3Other Generalized Functions

We have indicated theories of the delta function, the basic impulse function appropriate to functions on the line − ∞ < t < ∞. Clearly there are corresponding basic impulse functions on an arbitrary finite or infinite interval; functions of two variables in a plane, where an impulse function may be concentrated at a point, along a curve, or on a more general set of points; functions of several variables; functions of a point on a curved surface or, more generally, on a manifold; and so on. While it should be possible to devise an appropriate theory for each of these impulse functions, it is clearly preferable to seek a general theory embracing all of them.

Other generalized functions occur in connection with Fourier analysis, the modern theory of partial differential equations, etc., and one should like these subjects to be included in any useful theory of generalized functions. We shall give a simple example to indicate the application of generalized functions to partial differential equations.

This example concerns the hyperbolic partial differential wave equation

uxx − uyy = 0

Clearly f(x − y) + g(x + y) is a solution of this equation if f and g are twice continuously differentiable functions. Now, in many problems—e.g., problems of discontinuous wave motion—one should like to regard f(x − y) + g(x + y) as a solution of the wave equation even if f or g fails to be twice continuously differentiable. There are several ways of considering such weak or generalized solutions. From the point of view adopted here, the most natural approach is that of a generalized theory of differentiation, according to which every function has generalized derivatives that are generalized functions and satisfy the partial differential equation.

We shall outline in this chapter two theories of generalized functions. The first of these is algebraic in nature; indeed, it closely imitates the widening of the concept of number from integers to rational numbers. It is most successful with functions of a single nonnegative variable, although it has been extended to functions of several such variables and to functions of a single variable on a finite interval. It has the further advantage of providing a very natural approach to operational calculus as well as to generalized functions and generalized differentiation. Its greatest drawback at present seems to be its inability to cope with functions of unrestricted real variables or with functions of several variables ranging over an arbitrary region.

The second theory belongs more to the domain of functional analysis. In a sense, it might be compared with the extension of the concept of number from rational to real numbers, but the comparison is somewhat farfetched. The principal advantage of this theory is its ability to cope with all generalized functions needed at present. There are several approaches to this theory; we shall outline one of them in the simplest case of functions of a single real variable and briefly mention some others. The considerable number of different approaches to this theory is partly due to an endeavor to remove a basic difficulty remaining in it—the difficulty in defining the product of two generalized functions—and largely due to a desire to make this concept of generalized functions more easily accessible to applied mathematicians and engineers.

An entirely different attempt to cope with the problem of the delta function may be mentioned here. Schmieden and Laugwitz²⁰ have enlarged the concept of real numbers. Their system of numbers contains infinitesimally small and infinitely large numbers, and the analysis based on this system leads to functions, in the mathematical sense of this word, that behave like the delta function. Moreover, in this system the multiplication of two functions presents no difficulties.

MIKUSIŃSKI’S THEORY OF OPERATIONAL CALCULUS AND GENERALIZED FUNCTIONS

1.4The Definition of Operators

In Secs. 1.4 to 1.10, which are based largely on Ref. 12, t is a nonnegative variable, f = {f(t)} denotes a function of this variable, f(t) is the value of f at t the set of all continuous functions of twill indicate that a with similar notation for other sets, Θ will tentatively denote the function vanishing identically (later we shall see that we may replace this notation by 0), and l the function having a value equal to unity for every t ≥ 0 [except for the value 1 at t = 0, this is the restriction of U(t) to t ≥ 0]. The sifting property of the delta function appropriate to the interval 0 ≤ t ≤ ∞ may be expressed as

addition and multiplication by a scalar are defined in the obvious way, namely, for a and α , αa + βb is the function whose value at t is αa(t) + βb(t). These operations have the familiar properties. The convolution a * b, or simply ab, of two functions is defined by Duhamel’s integral

This operation has all the properties of multiplication, and it commutes with multiplication by a scalar; that is, ab = ba, a(bc) = (ab)c and hence may be written as abc, (a + b)c = ac + bc, (αa)b = α(ab), etc.

with addition and multiplication by scalars forms a vector space. The same set with addition and convolution forms a commutative ring, which will be called the convolution ring.

Integral operations can be expressed in terms of convolutions. In fact, la is the function having value at t equal to

We also have

ll = l² = {t}

and by induction,

for n = 1, 2, 3, … , so that convolution with the latter function expresses the effect of n denotes the real part of α, we may set

and call convolution with lα fractional integration of order α.

The convolution ring has no unit element; such that au = a for all a ∈ C. To see this, it is sufficient to note that lu = l means

for all t ≥ 0, which is clearly impossible. This means that the delta function appropriate to this case is certainly not a continuous function; actually, it is not any function.

A very important property of the convolution ring is contained in Titchmarsh’s theorem: For we have ab = Θ if and only if a = Θ or b = Θ (or both these equations hold). An elementary proof of this theorem was given by Mikusiński in Ref. 12, Chap. 2, and is reproduced in , division . The equations bu = a, bv = a imply b(u − v) = Θ, and if b ≠ Θ, then it follows that u = v; that is, the convolution equation bu = a has, with b ≠ Θ, at most one solution u. This solution may then be regarded as a/b. But, of course, bu = a . Clearly, bu(0) = 0, and hence bu = a will certainly have no solution if a(0) ≠ 0; the equation may fail to have solutions in other cases as well.

The situation encountered here is very similar to that met upon the introduction of division of integers. There the feasibility of division (with the exception of division by zero) is ensured by the extension of the number concept from integers to rational numbers, and similarly here we shall ensure the existence of a unique solution of bu = a with b ≠ Θ by enlarging the convolution ring to a field of convolution quotients. Almost any construction of rational numbers from integers can be imitated; we shall follow the construction in terms of classes of equivalent ordered pairs of integers.

We shall consider ordered pairs (a,b, always assuming that the second element ≠ Θ. We call (a,b) and (c,d) equivalent if and only if ad = bcor a/b the class of all ordered pairs equivalent to (a,b), call a/b a convolution quotientthe set of all convolution quotients. Clearly the cancellation law (ac)/(bc) = a/b .

contains numbers, functions, and also the delta function and its derivatives. Thus, convolution quotients may be regarded as generalized functions, and they include the more common impulse functions on the half line t ≥ 0. We embed with the convolution quotient (ab)/b for any b ≠ Θ. Since

this embedding is independent of b. We shall call a function f integrable if it is absolutely integrable over every finite interval 0 ≤ t ≤ t, the convolution fb is defined and is a continuous function. It is thus natural to identify f with the convolution quotient (fb)/b for any b ≠ Θ. As in the previous case, the embedding is independent of bwith the convolution quotient (αb)/b, an embedding that is independent of b. Now b/b. This already shows that it is impossible to ascribe definite values to convolution quotients at a point t.

by the equations

It is necessary to verify that these definitions are independent of the ordered pair used in the representation of the convolution quotients involved. Now, if

preserves all these operations. For this reason, we may write f in place of (fb)/b and α in place of (αb)/b; in particular, we may write 1 in place of b/b. We also see that multiplication by 1 (which may be interpreted either as multiplication by the scalar 1 or as convolution multiplication by the convolution quotient corresponding to this number) reproduces f.

of convolution quotients is an algebra; i.e., it is a vector space under addition and multiplication by scalars and a field .

.

will primarily be considered here as generalized functions, but we shall see in the next section that they may act as operators, thus providing a convenient approach to Heaviside’s operational calculus.

1.5Differential and Integral Operators

We have seen that convolution multiplication of a function by l , that is to say by

, b ≠ 0, means differentiation. We shall see that this is not quite the case.

Let

a = {a(t)}

be a differentiable function and

an integrable function. Then

. On multiplying by s, we obtain

and thus see that, even in the case of a differentiable function a, the product sa represents the derivative function only if a vanishes at t = 0 (Sec. 2.1). This may be explained to some extent by interpreting all our functions as vanishing for t < 0, so that there is a jump of a(0) at t = 0 that contributes a(0)δ(t) to the derivative. This also explains why, even for a differentiable function a, the product sa is in general not a function; further, it shows some of the difficulties encountered in the early applications of Heaviside’s operational calculus. By applying Eq. (1.4) several times, we obtain by induction

for a function a that is n times differentiable and has an integrable nth derivative a(n). For such a function, sna is in general a generalized function. On the other hand, sna exists as a convolution quotient for any (not necessarily differentiable) function a, or any convolution quotient. The generalized function sna may be called the extended or generalized nth derivative of a.

Since 1 corresponds to δ(t), sn is the extended nth derivative of the delta function, and a polynomial in s with constant, i.e., scalar, coefficients is an impulse function.

Next we investigate simple rational functions of s. From Eq. (1.4) we have

From this, it can be proved by induction that

We are now ready to interpret any rational function of s. Such a function can be decomposed into a sum of a polynomial and partial fractions of the form (1.5), and every term of this decomposition can then be interpreted.

EXAMPLE 1.1.To decompose the rational function s³/(s² + 1).

Since

where j² = −1, we have

The operational calculus so developed may be used to solve ordinary linear differential equations with constant coefficients, and also to solve systems of such equations. It will be sufficient to illustrate the process by a simple example.

EXAMPLE 1.2.To solve the differential equation

By applying Eq. (1.4) twice, we have

and hence

Now,

Hence we have the solution

Such equations can be solved even if the right-hand sides are generalized functions, for instance, delta functions, as with the differential equation satisfied by the Green’s function. For instance, the solution of

obtained in a manner similar to that for the above Example 1.2, is

We note that Eq. (1.5) is in agreement with the Laplace transform of eαttn−1/(n − 1)! in case s denotes a complex variable. We shall see in Example 1.8 that this is not a coincidence. Thus, tables of Laplace transforms may be used in interpreting rational functions of s.

1.6Limits of Convolution Quotients

that has all the desirable properties is known. We shall follow Mikusiński in introducing a notion of convergence that is at any rate simple, has many of the desirable features of convergence, and appears to be adequate for the applications of convolution quotients to operational calculus and to partial differential equations. According to this concept of convergence, a sequence of convolution quotients is regarded as convergent if it has a common denominator and if the numerators, which are continuous functions, are convergent in the sense outlined above. Thus, we shall say that a sequence of convolution quotients an converges to a, in symbols

an → aorlim an = a

if there is a q and if furthermore the sequence of continuous functions qan converges to qa uniformly over every finite interval 0 ≤ t ≤ t0. Clearly a itself is then a convolution quotient.

It is fairly easy to prove that the limit, if it exists, is unique and has most of the usual properties. In particular, the sequence a, a, a, … converges to a; the sum (product) of convergent sequences is convergent and tends to the sum (product) of the limits; and a sequence of scalars is convergent in the ordinary sense if and only if the corresponding sequence of convolution quotients converges in the sense outlined here. However,

does not necessarily hold even if it is assumed that bn ≠ 0 and lim bn ≠ 0.

We shall now give some examples and comment on them.

EXAMPLE 1.3.To prove that {sin nt} → 0.

We have

, even for ordinary functions, demands much less than ordinary convergence. It thus allows us to ascribe limits to sequences of functions that would ordinarily be regarded as divergent, and it also opens the way to a representation of some convolution quotients as limits, in this sense, of ordinary functions. (See also Example 1.5.)

EXAMPLE 1.4.To prove that for , cn → 0 as n → ∞.

For a fixed t0, there exists an M > 0 such that |c(t)| ≤ M for 0 ≤ t ≤ t0.

We shall prove by induction that, for n = 1, 2, …,

This relationship clearly holds when n = 1. If it holds for n = 1, then

and this completes the proof by induction of the inequality (1.6). Since the right-hand side of (1.6) converges to 0 uniformly for 0 ≤ t ≤ t0, we have cn .

EXAMPLE 1.5.To prove that if f(t) is absolutely integrable over 0 ≤ t ≤ ∞ and

then

We set

Now,

and we shall prove that this function approaches {t} = l² uniformly over every finite interval. Set

First assume that 0 ≤ t ≤ δ. Then

For δ ≤ t ≤ t0,

In either case, for any δ between 0 and t0,

Given t> 0, we first choose δ so that 0 < δ /(2A) and then choose N so that

We then have |gn(tfor 0 ≤ t ≤ t0 and n ≥ N, showing that gn(t) converges to 0 uniformly for 0 ≤ t ≤ t0. Thus, l²{nf(nt)} converges to l² uniformly on every finite interval.

We have accordingly found a family of approximations to the delta function. This should be compared with the approximations discussed in Sec. 1.2. The result suggests that many other convolution quotients might be represented as limits of functions.

1.7Operator Functions

We shall now consider convolution quotients that depend on parameters. For the sake of simplicity, we shall consider a single parameter x varying over a closed and bounded interval I: α ≤ x ≤ β, and we shall denote the domain α ≤ x ≤ β, t ≥ 0 of the xt plane by D. An operator function a(x) assigns to each x ∈ I a convolution quotient a(x). Mikusiński calls such a function a parametric function , so that a(x) = {a(x,t)}, and considers a(x) as a continuous operator function if there exists a q such that b(x) = qa(x) is a parametric function and b(x,t) is continuous in D; he says a(x) is k times continuously differentiable with respect to x if there exists a q such that b(x) = qa(x) is a parametric function that is k times continuously differentiable with respect to x; and he sets

Continuous and differentiable operator functions have many of the usual properties, and differentiation obeys the familiar rules. It is unnecessary for us to go into further details here. Instead of this, let us consider some examples.

EXAMPLE 1.6.To discuss the function a(x) = {cos (x − t)}.

This is a continuous parametric function for any interval I. By virtue of the results in Example 1.2, this operator function can be expressed as

The function is indefinitely differentiable with respect to x, and the reader may easily verify that its derivatives can be computed by differentiating the explicit form. The function a(x) satisfies the operator differential equation

a″(x) + a(x) = 0

and the initial conditions

a(0) = {cos t}a′(0) = {sin t}

EXAMPLE 1.7.To discuss the function hα(x), defined as follows: For x ≥ 0 let

h(x,t) = 0 if 0 ≤ t < xh(x,t) = 1 if x ≤ t

and set

h(x) = {h(x,t)}hα(x) = lαh(x)

, then hα(x) is a parametric function, and

Thus hα(x, and it is k ; further,

. Since

lβhα(x) = hα+β(x)

it follows that hα(x) is infinitely differentiable, in the sense of differentiation of operator functions, with respect to x; and

for all α. This differential equation, together with

hα(0) = lαh(0) = lα+1

suggests writing

hα(x) = lα+1e−sx

We shall justify this later.

Of particular importance is

h−1(x) = e−sx

, we set

h(x)f = {g(x,t)}

and find

Consequently, for

h−1(x)f = sg(x) = g1(x)

we have

g1(x,t) = 0for 0 ≤ t < xg1(x,t) = f(t − x)for x ≤ t

Thus, g1(x, t) is simply the function f(t) shifted by x, and e−sx is the shift operator.

By a direct computation, it can be verified that

h(x)h(y) = lh(x + y)

for x ≥ 0, y ≥ 0, and this relationship holds for all real x, y, provided that we define h(x) for negative values of x by the equation

h(x)h(−x) = l²

We have thus defined hα(x) for all complex α and for all real x. In particular,

h−1(x)h−1(−x) = 1

We now turn to integration of operator functions with respect to x. Let ϕ(x) be absolutely integrable over I = [α,β], and let a(x) be a continuous operator function and q ≠ 0 such that qa(x) = b(x) is a continuous parametric function. We then set

It can be proved that this definition is independent of q and that the integral has all the usual properties. Infinite integrals may then be defined as limits of finite integrals.

EXAMPLE 1.8.To prove that for any integrable function f, the integral exists and is equal to f.

This result is the background for the coincidence noted at the end of Sec. 1.5. It should be remarked, though, that here s is an operator, so that the integral is not a Laplace integral; also, it might be noted that the result to be proved holds without any restriction on the growth of f(x). Since

le−sx = h(x)

we have

As β → ∞, the last function approaches lf uniformly over every finite interval 0 ≤ t ≤ t0. Thus

exists and is equal to lf.

1.8Exponential Functions

, there exists an interval I containing x = 0 and a differentiable operator function e(x) on I that satisfies the differential equation

e′(x) = we(x)

and the initial condition

e(0) = 1

We then say that w is a logarithm and set

e(x) = exw

It is fairly easy to prove that in this case e(x) exists for all real x, is infinitely differentiable, is uniquely defined by the differential equation and the initial conditions, and has the properties

exw ≠ 0for all x(exw)−1 = e−xwexweyw = e(x+y)w

EXERCISE

Verify that s is a logarithm and that (see Example 1.7)

e−xs = h−1(x)

are logarithms. The element s is a logarithm, and so are real multiples of s, but it can be proved that js are logarithms, and

The series representation holds also for other w; thus it holds for integrable (rather than continuous) functions, or for w = 1. But it does not hold for all logarithms; for instance, the series fails to converge for w = s, which is a logarithm. If u and v are logarithms, then αu + βv is a logarithm for real, but not necessarily for complex, α and β.

Exponential functions arise in the solution of partial differential equations, in which they often correspond to fundamental solutions.

EXAMPLE 1.9.To prove that s½ is a logarithm.

Let us set

Q(x) = {Q(x,t)}R(x) = {R(x,t)}

where

Clearly, Q(x) = −R′(x). Now the function

although a parametric function, fails to be continuous; but the function

is continuously differentiable with respect to x when x > 0, and

On the other hand, we have

and upon introducing a new variable of integration v by

we obtain

so that

Q′(x) = −s½Q(x)

Moreover, l²Q(x) approaches {t} = l² uniformly in every interval 0 ≤ t ≤ t0, and hence

In the course of this work we have also seen that

and

where erf denotes the error function. For fixed x > 0, Q(x,t) as a function of t increases for 0 < t < x²/6 and decreases thereafter. Thus,

Since this expression approaches zero, uniformly in t, as x → ∞, we see that exp (−xs½) → 0 as x , a exp (xs½) is a bounded continuous parametric function for x ≥ 0, then a , the function

a exp (xs½) + b exp (−xs½)

is a bounded continuous parametric function for all real x, then a = b = 0.

1.9The Diffusion Equation

Let us briefly indicate the application of the technique developed here to the diffusion equation

uxx(x,t) = ut(x,t)

in the half plane − ∞ < x < ∞, 0 ≤ t < ∞ (subscripts indicate partial derivatives). If

u(x) = {u(x,t)}

is a parametric function possessing continuous partial derivatives with respect to x and t, and a continuous second partial derivative with respect to x, our partial differential equation may be replaced by the operator differential equation

where ϕ(x) = u(x,0).

If ϕ(x) is an integrable function, this differential equation may be solved by the method of variation of parameters. Two solutions differ by an operator function of the form

a exp (xs½) + b exp (−xs½)

; and, by the remark at the end of the preceding section, a solution that is a bounded continuous parametric function is unique within the class of such functions. It may be verified that

is a bounded continuous parametric solution of Eq. (1.7) if ϕ is a bounded measurable function. In this sense, the function

may be regarded as a (unique) generalized solution of our boundary-value problem if ϕ(x) = u(x,0) is a bounded measurable function. Actually, this solution is differentiable, indeed analytic, for t > 0 and for all x; and although it is not a continuous function for t ≥ 0, it satisfies the initial condition in the generalized sense that

u(x,t) → ϕ(x)as t → 0t > 0

at least for all those x at which ϕ is continuous. By a more refined analysis, this result can be extended to measurable functions that, instead of being bounded, are assumed to satisfy an inequality

|ϕ(x)| ≤ A exp (B|x|α)

where A, B, and α are constants and 0 ≤ α < 2.

Other problems involving parabolic equations, and problems involving the wave equation and other hyperbolic equations, can be solved by means of this operational calculus, but so far no significant and successful applications to elliptic partial differential equations, such as Laplace’s equation, are known.

1.10Extensions and Other Theories

Mikusiński¹⁴ has extended this theory to functions of several variables, t1, …, tn, ranging over the cone

t1 ≥ 0, …, tn ≥ 0

He¹³ has also developed the corresponding theory for convolution quotients of functions on a finite interval, α ≤ t ≤ β.

An alternative theory has been proposed by J. D. Weston,²⁶,²⁷ whose generalized functions are operators acting on certain perfect functions rather than convolution quotients of functions.

DISTRIBUTIONS

1.11Testing Functions

We now turn to generalized functions of an entirely different type, to distributions.²¹ There are several different approaches to this theory, most of them resembling the theories of the delta function indicated in Sec. 1.2 in that distributions appear either as generalized limits of functions or else as characterized by their action on certain classes of functions. We shall present here the second point of view for generalized functions of a single real variable t ranging over the entire real line (−∞, ∞). Alternative approaches and extensions will be mentioned in Sec. 1.18.

Since distributions will be defined in terms of their action on certain classes of functions, the resulting notion of generalized functions will depend on the class of testing functions on which distributions act. We shall use two spaces of testing functions: One has proved useful in applications to Fourier analysis, and the other has been employed in connection with partial differential equations. Other classes of testing functions have also been used.⁵

be the set of infinitely differentiable functions decreasing rapidly as t → ± ∞. More precisely, ϕ if all derivatives ϕ(k) exist and if for any integer k and any polynomial P(t), P(t)ϕ(k)(t) → 0 as t is a vector space in the sense that, for any two elements ϕ1 and ϕand any two real or complex numbers c1 and c2, the function c1ϕ1 + c2ϕ. We shall use 0 indiscriminately to denote the number 0 and the function identically equal to zero for all values of t. We now introduce a notion of convergence in S. A sequence of functions ϕn in S is said to converge to 0 if, for any fixed k and fixed polynomial P(t), P(t)ϕn(k)(t) → 0 uniformly for all real t as n → ∞. A sequence of functions ϕn is said to converge to ϕ if ϕn − ϕ converges to 0. We shall indicate this by writing ϕn → ϕ as n → ∞.

Let cn → c (in the sense of convergence of numbers) and let ϕn → ϕ and θn → θ, as n → ∞; then it is easy to see that also cnϕn → cϕ and ϕn + θn → ϕ + θ, as n → ∞. Thus multiplication by a number and addition of functions are continuous operations. From now on, we shall usually omit the qualifying phrases appearing in the parentheses above, since the nature of the entities involved will indicate which space we are in and which notion of convergence should be used.

be the set of infinitely differentiable functions ϕ vanishing outside some finite interval, the interval depending on ϕ . There are such functions. As an example, let us define

where a and b are real numbers, a < b, and c, d, α, and β are positive numbers. Clearly, ϕ vanishes outside a finite interval and is infinitely differentiable except possibly at a and b, and it is easy enough to show that ϕ . A sequence of functions ϕn is said to converge to 0 if there is a finite interval I such that each ϕn vanishes outside I, and if, for each fixed nonnegative integer k, ϕn(k)(t) → 0 uniformly for all real t (or all t in I) as n → ∞. A sequence of functions ϕn is said to converge to ϕ if ϕn − ϕ converges to 0, and we shall write ϕn − ϕ → 0 and ϕn → ϕ as n .

will be called testing functions.

In studying the action of other functions on testing functions, we shall start with continuous functions of t and proceed to functions that are merely locally integrable in the sense that they are integrable over every finite interval. In this context a function of t will be said to be of slowgrowth if its growth is dominated by that of some polynomial; in other words, a function f is of slow growth if (1 + t²)−Nf(t) is bounded for some N.

1.12The Definition of Distributions

For continuous functions f of slow growth,

converges for each ϕ and defines an evaluation of f . In classical analysis, we think of a function f as characterized by its values f(t) for all real t. We now claim that, alternatively, we can characterize such a function by its evaluations f〈ϕalso possess the same values for all t and hence are identical. It will be sufficient to show that f〈ϕ〉 = 0 for all ϕ entails f(t) = 0 for all t. Indeed, suppose f(t0) ≠ 0 for some t0, say f(t0) > 0. Since f is a continuous function, there is some interval I around t0 on which f is positive. Take any interval (a,b) in the interior of I and define

Then clearly f〈ϕab〉 > 0, and accordingly the assumption f(t0) ≠ 0 for some t0 is inconsistent with f〈ϕ〉 = 0 for all ϕ in S. Incidentally, we see that a continuous function of slow growth is completely characterized by its evaluations on the ϕab for rational a and b, but we shall continue to think of it in terms of its evaluations on all ϕ .

, and the proof given above shows that at points of continuity the values of such a function are completely determined by the evaluations of the function. On the other hand, the values at points of discontinuity are not at all determined. For instance, the