THE ORIGIN AND HISTORY OF CONVOLUTION I:

CONTINUOUS AND DISCRETE

CONVOLUTION OPERATIONS*

ALEJANDRO DOMINGUEZ-TORRES

This work was written while the author was at Applied Mathematics and Computing Group,

Cranfield Institute of Technology, Cranfield, Bedford MK43 OAL, UK. Now the author is at

Academic Division, Fundación Arturo Rosenblueth, México, D.F.1

*This work was sponsored by CONACYT, México. Act Number BA90074.

Chapter 1

Introduction

Nowadays there is not doubt of the uses and applications of the continuous and discrete

convolution operations in many branches of science. Moreover, the number of applications is so

large that trying to name and count them would take a long time. Some of these applications are

in signal and image processing, electric circuits, telecommunications, probability, statistics, etc.

Although the aforementioned applications, all of the modern books dealing with convolution,

which mainly deal with series and integral transforms rather than convolution, do not give any

information about its origin and history. This lack of information has also been present in the

earlier literature. It is worth to mention that the only two books given some remarks about this

history and origin are those by Doetsch [26] and by Gardner and Barnes [37]. The first author

gave some historical remarks in a series of notes marked throughout the text of his book and

commentd at the end of it. In the second book the brief notes of history appear in two subsections

of Appendix C. Both subsections are given in about one page.

From the above comments, it may be seen that the origin and history of convolution have not

been properly traced back. The present paper is an attempt to fill this gap.

Two important remarks must be done about the origin and history of convolution before going on

reading this paper. Firstly, they are not claimed to be complete. The reader may find that the

work of some authors is not fully commented or even mentioned. Secondly, the history given

herein is far away from being a critical one.

This paper has been divided in five chapters plus a list of references. Chapter 2 mainly concerns

with the real convolution integral, while Chapter 3 with the complex convolution integral.

Chapter 4 deals with the discrete convolution operation including some brief comments about

cardinal series. Finally, in Chapter 5 the names and notations given to the different convolution

operations are given.

1 On November 2010, the author is the Corporate Director of Postgraduate Studies at Universidad Tecnológica de

México. alexdfar@yahoo.com

2

In a second paper the author is at present doing an attempt to trace back the origin and history of

the so-called convolution theorems [28].

Chapter2

The Real Convolution Integral

2.1 Definition and Introductory Comments.

Let f,g be two real or complex valued functions of a real variable x . Assume u is a real variable

and construct the integral

b

a

duuxguf )()( , (2.1)

where the limits of integration a and b may finite or infinite or depend on the variable x . These

limits of integration as well as the sign to be taken in the argument of g x u( ) will be apparent in

what follows.

Since both variables x and u entering in (2.1) are real, that integral (2.1) will be called the real

convolution integral (RCI). Obviously these variables are considered as dummy variables and

may be changed through the present chapter.

As in the present, integral (2.1) occurred in the past in several areas of mathematics. Due to this

fact the present chapter has been divided in several sections, in each one of them is traced back

the occurrence of (2.1) in a particular area of mathematics.

It is difficult to establish the exact date when the RCI occurred by the very first time. As it is

described in the following sections of this chapter, the earliest occurrences of (2.1) the present

writer has found are in connection with the theory of series (general, Taylor and trigonometrical

series) and in connection with the theory of Beta function given by Euler.

2.2 Series, Taylor Series and the RCI.

Probably one of the first occurrences of the RCI in an explicit but particular form took place in

the year 1754 when the mathematician Jean-le-Rond D'Alembert (1717-1783) derived Taylor's

expansion theorem on page 50 of Volume I of his book Recherches sur differents points

importants du systeme du monde [73, p.111, footnote 1, and 65, pp.17-18]. Following Reiff

[op.cit., p.111] the derivation of Taylor's expansion theorem by D'Alembert is as follows. For a

function ( )z D'Alembert firstly wrote

( ) ( )z z u ,

Where

3

d

zddu

)(. (2.2)

Secondly he wrote

d z

d

d z

dzv

( ) ( ),

where now

2

2 )(

d

zddv (2.3)

D'Alembert continued with this process and found finally the expression

( ) ( ) ...z zd

dz

d

dz

1 2

2

2 (2.4)

In this derivation of Taylor's expansion theorem the limits of the integrals (2.2) and (2.3) were

taken from 0 to . Obviously these integrals are special cases of the RCI where one of the

functions entering into them is a constant unit function and the other ones are d dz d dz / , /2 2,

and so on, respectively.

Reiff [op.cit., p.111] and Nielsen [op.cit., p.18] point out that the series (2.4) was given by

D'Alembert without naming the work of Taylor. Because of this fact, Nielsen [op.cit., p.18] also

points out that Jean-Antoine-Nicolas Caritat de Condorcet (1743-1794) used to designate series

(2.4) as 'théoreme de D'Alembert'.

The above method for deriving Taylor's expansion theorem was known to Pierre Simon Laplace

(1749-1827) since he stated it in 1812 in a more modern form in his book dealing with the theory

of probability [54, pp.179-180]. As in the case of D'Alembert, Laplace did not mention Taylor's

work. Moreover, he did not mention D'Alembert work, the reason of this fact is difficult to guess

since Laplace met D'Alembert when the latter was then at the height of his fame [14, p.259].

In order to see the importance of the RCI in Laplace's method and to distinguish the differences

of it with respect to D'Alembert method, a translation to English language of Laplace's derivation

is given next [op.cit., pp.179-180].

"Consider the integral from z 0 ,

),()()( zxxzxdz

' ( )x standing for the differential of ( )x divided by dx . If analogously one designate by

"( )x the differential of ' ( )x divided by dx ., by ' ' ' ( )x the differential of ' ' ( )x

divided by dx and so on, one will obtain

4

),('')(')(' zxzdzxzzxdz

),('''2

1)(''

2

1)('' 22 zxdzzxzzxdz

Continuing in this way, one will obtain generally

)(321

)(''21

)(')(' )(2

zxn

zzx

zzxzzxdz n

n

)(321

1 )1( zxdzzn

nn

A comparison of this expression with the precedent one, one will have

( ) ( ) ' ( ) "( ) ( )( )x x z z x zz

x zz

nx z

nn

2

1 2 1 2 3

z)(xdzfzn321

1 1)(nn.

Setting x z t , the precedent equation will have the following form

( ) ( ) ' ( ) "( ) ( )( )t z t z tz

tz

nt

nn

2

1 2 1 2 3

)z'z(tdzfzn321

1 1)(nn

,

the integral being taken from z' 0 to z' z ".

Finally it is mentioned that according to Burkhardt [12, p.400, footnote 2065] an expression of

the type

,)()( duuxguf

where the limits of integration are from u 0 to u x ,was used by Sylvestre Françoise Lacroix

(1765-1843) on page 505 of Volume I of his book entitled Traité des différences et des séries.

From the title of Lacroix's book and from the Burkhardt's reference it may be inferred that the

above integral occurred in connection with some work related to series, however no description

of Lacroix's work is given herein since the present writer was not able to obtain Lacroix's book.

5

2.3. Fourier Series and the RCI.

In dealing with the representation of a function by Fourier series several authors used expressions

of the type given by Eq.(2.1). Jean Baptiste Joseph Fourier (1768-1830) himself was the first one

in using such expressions as early as the year 1807 when he made the first announcement of his

investigations about the propagation of heat in solids before the French Academy. In order to

quote the RCI used by Fourier an English translation of Fourier 1822 book is used herein as main

source of reference [36].

In Art.235 of Fourier's book is found the following representation for a function F x( ) defined in

an interval from to (the notation is that used in the translation of Fourier's book)

)(2cos)cos(2

1)(

1)(

xxdFxF ,

which was reduced by him to one of the form

)(cos2

1)(

1)(

xidFxF , (2.5)

where the summation was taken from i 1 to i and where the limits of integration were set

to be from to . There is not doubt that (2.5) is a particular case of the RCI.

In Arts.415-416 of his book Fourier generalized the series to the case of integrals and from the

expression

0

px)dpcos(paf(a) dap

1f(x)

he derived the following RCI representation for f x( )

xa

px)sin(paf(a) da

p

1f(x)

when p .

In 1815 Siméon Denis Poisson in a paper submitted to the French Academy at the end of 1815

and in order to participate for the 'Prix d'analyse mathématique' derived, independently of

Fourier, the representation of a function by Fourier series [68]. In pages 85-86 of Poisson's work

it is found the following key RCI-expression for the derivation of such representation (the

notation differs slightly from that used by Poisson)

6

f(a)da

2l

1

l

da

l

a)ip(xki)cosexp(

i

l

pa)(x2cosexp(k)2l

f(a)dak)exp(exp(k)

where the integrals were taken from 0 to 1. Some variations of the above RCI were used

by Poisson in future works [69,70].

Peter Gustav Lejeune Dirichlet (1805-1859) in dealing with the same problem of Fourier series

wrote in 1829 [22] that that representation could be written in the (RCI) form (nowadays known

as Dirichlet integral)

dxn

x

)(sin2

))((sin)(

1

2

21

.

Georg Friedrich Bernhard Riemann (1826-1866) in 1866 in his work related to Fourier series also

used RCIs to prove some properties of the Fourier coefficients [74]. Among these properties was

included the so-called Riemann-Lebesgue Lemma. A description of his work can be found on

pages 244-247 in the book by Umberto Bottazinni [8].

Finally it is mentioned a result closely related to Fourier series obtained in 1885 by Karl

Weierstrass (1815-1897). Weierstrass proved that [98] if f x( ) is a single-valued function which

is continuous in an interval ( , )c c , then at any point x inside the interval

,,)(lim)( dtnc

txtfxf

c

cn

where ( , )v n denotes

n

r

rrvn

1

cos212

and n stands for

( n 1) ( m/ ( n 1) )2

with m1.

2.4 Differential Equations and the RCI.

In the field of differential equations (DE) some authors; v.g. [49, pp.191-192], [71, pp.104-109];

without given their source of reference attribute to Leonhard Euler (1707-1783) the use of a RCI

of the type

7

duuvuxxy v )()()( 1 (2.6)

for solving "any linear differential equation in which the coefficient of y r( ) is a polynomial in x

of degree r " [71, p.104]. A more inaccurate quotation is given in the book by Gardner and

Barnes [37, p.364]; these authors did not mention neither the exact RCI used by Euler, nor the

type of linear DE that is solved by such expression and nor their source of reference.

On the other hand, the present writer found that the problem of solution of DE by definite

integrals was studied by Euler in 1768 in Volume II of his Institutionum Calculi Integralis [32,

Vol.III], and to be more specific in Chapter X (Caput X: Constructione Aequationum Differentio-

Differentialium per Quadraturas Curvarum).

In problem 131 (Arts. 1049-1052) of that chapter it is found an study of the solution of the DE

L ud y u

duM u

dy u

duN u y u( )

( )( )

( )( ) ( )

2

20

by means of the integral

b

a

ndxxQuKxPy .)()()( (2.7)

Obviously (2.7) reduces to (2.6) in the very especial case in which K u u( ) and Q x x( ) .

Euler did not studied explicitly this special case, however he did study only the case Q x x( )

[op.cit., Arts.1050-1052].

At this point the following question arises, Is the preceding citation the source of reference of the

aforementioned authors? If the answer to this question is affirmative, then the facts have been

distorted in time. If the answer is negative, then that source of reference remains as a mystery to

be solved.

In the XIX century the solution of some DE, mainly partial DE describing physical phenomena

were expressed by means for RCIs. One of the earliest author in doing this was Fourier. In fact,

in Chapter IV of his book [36] it is found that the solution of the heat equation

tk

xx f x

2

20; ( , ) ( ) (2.8)

for a ring was expressed as

2

0

2 ).(exp)(cos)(2

1),(

i

i

ktixifdtx

8

Fourier went a step further and changing ( , )x t by v x t( , ) and f x( ) by ( )x in (2.8), he solve

the resulting equation for "the free movement of heat in an infinite line" and found for v the

expression

).2()(exp 1 2 ktqxqdqv

(2.9)

It is of historical interest to point out that according to Grattan-Guinness [40, pp.41-42] the

solution of (2.9) was first stated by Laplace in 1809 [53, pp.235-244] and then considered by

Fourier in his future research.

Fourier expressed also (2.9) in a slightly different form for the case in which k 1. He considered

the change of variable q x t ( ) / 2 and expressed (2.9) as

.p2

/4td)-(x-(a)expdav

2

(2.10)

The 3D version of (2.10) was also given by Fourier in connection with the solution of the "free

movement of heat in and infinite solid". Thus in Art.376 of his book it is found the expression.

4t

g)(zb)(ya)(xg)expb,f(a, dgdbdatv

2223/2

In Art.384 of the same book the corresponding expression for the case k 1 was also given.

These are probably the first time that a 3D RCI was stated explicitly.

At this point it is worth to mention that Weierstrass in 1885 [98] proved his famous theorem of

approximation of continuous functions by polynomials using an integral of the type given by

(2.10).

One of the first treatises for solving DE by means of RCIs is due to H. Mellin. Indeed, in 1896

Mellin published a paper [63] dealing with the relation of the RCI integral

)(

;)()(l

dtttx

where ( l ) denoted either one of the intervals ( , ),( , ), ( , )a b a b x a x b x or being a b, constants;

to the DE

( ) ( ) ( ) .a b xd y

dxa b x

d y

dxa b x yn n

n

n n n

n

n

1 1

1

1 0 0 0

In Arts.4-5 of his paper Mellin stated the formulae

9

b

a at

b

a

yx

f,dt

dFt)f(t)dty(x

x

dFf(t)

dt

dFt)dty(x

,

xb

a

xb

a at

yx

fdt

dFt)f(t)dty(x

dx

dFf(t)

t

dFt)dty(x

,

xb

xa

xb

xa

t)f(t)dty(xdx

dFf(t)

dt

dFt)dty(x ,

b

a

b

a

t)f(t)dty(xdx

dFt)y(x

xFdtf(t)

,

xbt

xb

a

xb

a

yx

f,dt

dFt)f(t)dty(x

dx

dFt)y(x

xFdtf(t)

,

xbt

xat

xb

xa

xb

xa

yx

f,dt

dFt)f(t)dty(x

dx

dFt)y(x

xFdtf(t)

,

where

F ck

k

k

n

( )

0

,

FF F

ck

k

nk v

v

kv

,

( ) ( )

0

1

0

1

,

being the n.,,0,1,k ,ck constants.

it is worth to mention that in Art.8 Mellin made an study of the RCI attributed to Euler:

(l)

a f(t)dt;t)(x

and in Arts. 13-14 he used the above formulae in conjunction with Convolution Theorem for

Laplace integrals to state existence theorems for the solution of the aforementioned DE.

In pass it is mentioned that in this same paper Mellin introduced the so-called Mellin convolution

and developed similar results than those given for the RCI.

2.5 Integral Equations and the RCI.

In 1823 and 1826 Niels Henrik Abel (1802-1829) published two papers [1, 2] which are rightly

believed to be the earliest publications to contain what is now called an integral equation. The

second of these papers is a revised an improved version of the first. one. Abel solved in these

papers the famous problem of tautochrone curves by reducing it to an integral equation which

now bears his name. To be specific, Abel in the second paper considered the following situations:

10

"Let BDMA be a curve whatever. Let BC be horizontal straight line and CA be a vertical

straight line. Suppone that a particle urged by gravity moves on the curve, a point D whatever

being its point of departure. Let be the time which passed when the particle is at the given

point A, and let a be the height EA [ DE and MP are horizontal; E and P are on CA ].

The quantity shall be a certain function of a which shall depend on the form of the curve.

Reciprocally, the form of the curve shall depend on this function. We proceed to examine how

with the help of a definite integral, one can find the equation of the curve for which is a

given continuous function of a ".

Letting AM s and AP x and t be the time taken by a particle in running through the arc

DM , then if ( )a it follows that

a

xa

dsa

0

)( .

Abel solved this equation for the variable s and obtained the expression

x

ax

daas

0

)(1

. (2.11)

Moreover Abel solved for s the more general expression

a

nxa

dsa

0)(

)( (2.12)

and obtained

0

1)(

)(sinnax

daans . (2.13)

Abel's equation and several others analogous to it were solved by Joseph Liouville (1809-1882)

using the notion of fractional derivatives and integrals. Liouville's procedure was purely formal

and he seemed to be unaware of Abel's work. The solution of (2.12) published by Liouville in

1832 [57, 58] is easily derived from (2.13). Indeed, if it set

x

dvxs0

,)()(

where v( ) is a function such that v( )0 0 , then a substitution in (2.13) gives

11

x

n

x

ax

daandv

0

1

0

,)(

)(sin)(

or equivalently

x

nax

daa

dx

dnxv

0

1)(

)(sin)(

. (2.14)

At this point it is worth to mention that at the end of the XIX century integral equations of the

RCI type were considered in detail by Volterra. an brief account of Volterra's work is given

below in Section 2.6

Returningto the works by Abel and Liouville, the equations derived by them were the source of

inspiration of several authors to derive and define fractional derivatives and integrals. Complete

accounts of this subject are given in Chapter I of the book by H.T. Davis [20] an in two papers by

B. Ross [75, 76]. Herein only the fractional operations of the RCI type are briefly quoted next.

In the field of fractional calculus, Riemann in 1847 was the first author after Abel and Liouville

in using an integral of the type (2.12). He sought a generalization on Taylor's series expansion

and derived the following expression for fractional integration [75, p.6 and 76, p.4]

x

c

r

r

r

dkkurxrdx

xud.)()(

)(

1)( 1 (2.15)

In a paper written in 1863 and published in 1864, H. Holmgren [20, p.20 and 76, p.7] considered

Riemann's expression (2.15) as his point of departure for a long memoire on the subject, and later

on in 1867 applied his theory to the integration of a differential equation of the type [20, p.20 and

76, p.7]

( ) ( ) .a b x c xd y

dxa b x

dy

dxa y2 2 2

22

2 1 1 0 0

It is worth to notice that the method used by Holmgren resembles the supposed (and up to now

not confirmed) method used by Euler in the solution of DE by definite integrals (see Sec. 2.2. of

this chapter).

Similar integral to those of (2.15) were studied later by A.D. Grünwald in 1867 and A. V.

Letnikoff in 1872, both in connection with fractional operations and the solution of particular

integrals [76, pp.7-8].

2.6 Volterra's Work and the RCI.

The modern theory of integral equations was initiated almost simultaneously by Eric Ivar

Fredholm (1866-1927) and by Vito Volterra (1860-1940) in the last decade of the XIX century.

12

Volterra's work is based in what he called functions of lines. The theory of these functions of

lines began in 1887 with a series of papers published by Volterra in the Rendiconti de la Real

Accademia dei Lincei and which were condensed and resumed by Volterra himself and J. Pérès in

two books [95, 96], from which the following quotations are taken.

In order to understand how the RCI emerged from Volterra's work, some preliminary definitions

are firstly given [96, pp.5-6].

Given two functions f x y( , ) and g x y( , ) of two variables, the integral (Volterra's notation)

y

x

y)(x,gfy)dxx)g(x,f(x,

was called by Volterra composition of the first kind, while the integral

b

a

y),(x,gfy)dxx)g(x,f(x,

where a b and are constants, was called by him composition of the second kind.

Volterra also mentioned that [95, p.100]:

"the operations of composition are an extension to the case of an infinite number of variables

of the notion of the product of two square matrices a bir rs and ; or the composition of the

corresponding linear substitutions ( , , , , ,..., ).i r s n 1 2 Composition of the second kind

correspond to the case of general square matrices, while that of the first kind correspond to the

case of matrices a a r iir ir in which for 0 "

Corresponding to these type of composition, Volterra defined two different types of

permutability. The permutability of the first kind was defined as

y

x

x)dxx)g(x,f(x,y)(x,gf

y)(x,fgx)dxx)f(x,g(x,

y

x

while the permutability of the second kind was defined a similar way where the integrals ran from

a b to .

A name for f g

was also given by Volterra [96, p.6]:

13

"Nous dirons que cette fonction f g

est la resultante de la composition de f g et ".

The name resultante was later used by several authors to designate the RCI (see Sec. 5.3. herein).

Volterra also considered the very special case in which the function f x y( , ) is permutable with a

constant, say the unity [96, p.9 and 95, pp.109-110]; i.e.,

y

x

y

x

dyfdxf .),(),(

From this equality was where Volterra derived the RCI. His idea is a follows, let ( x, y )be the

common value of these two quantities. By differentiating it is obtained

f ( x, y )y x

or equivalently

x y0 .

This partial differential equation implies that ( x, y )and therefore f x y( , ) functions only of the

difference y x . Further, it can be shown that all functions of the difference y x are permutable

(of the first kind) with another and with the unity. In fact, let x y , then

y

x

y

x

;)dx)f(yg()dx)g(yf(

i.e;

f g( y x ) g f ( y x ).

The set of functions permutable (of the first kind) with the unity was called by Volterra group of

the closed cycle in connection with its applications to the theory of heredity.

The group of the closed cycle coincide with the set of functions of one variable y x t . Indeed,

the composition of two functions f g and belonging to the group can be written as

14

t

0

y

x

t

0

(t)gf)d)g(f(t

)d)g(tf()dxx)g(yf(

He also noticed that f g( t )

is a new function of the group. This is the way Volterra arrived to the

RCI.

It is not difficult to see at this point that the so-called group of the closed cycle coincides with

what nowadays is known as linear-shift(-time) invariant (LSI) systems.

In order to illustrate the theory developed by Volterra the following example given by him is

considered [95, Chap.VI, Sec.4]. If represents the angle of torsion and P the torsion couple,

the relation between them is to a first approximation

kP,

where k is a constant determined from physical considerations. but actually the relationship is

more complicated than this since depends not only upon P but also upon the history of the

elastic body, the torsion of which is being studied. This second approximation is the form of an

integral equation

t

dssPstKtkP ,)()()( (2.16)

where K t s( ) is the coefficient of heredity.

Equations of the type (2.16), where the RCI is a characteristic feature of it, are instances of a

general theorem characterizing the so called LSI systems. The formulation and some history of it

are the purposes of the next section of this chapter.

2.7. LSI Systems and the RCI.

Consider a LSI system whose physical action is completely described mathematically by a linear

operator A . Let u x( ) be the function defined by the expression

0 if 0

0 if 1)(

x

xxu

and let c x( ) be the response of A to function u x( ); i.e., c x A u x( ) ( ) . For an arbitrary function

f x x h x A f x( ), , ( ) ( ). 0 let Then, under the above conditions, it is possible to express h x( ) in

terms of f x c x( ) ( ) and . This expression whose derivation can be found in [37] is given as

15

x

0

0.x ,)d(x)c(xf'f(0)c(x)h(x) (2.17)

Equation (2.17) was derived and used by Jean Marie Constant Duhamel (1797-1872) in at least

two of his papers [30, 31]. Some authors; e.g., John R. Carson attaches the name of Duhamel to

this equation [17, p.16, footnote 1], however this does not seem to be justified since Liouvill's

solution of Abel's equation [see Eq.(2.14)] is of this form and occurred one year before of

Duhamel's papers.. Indeed, (2.17) can be expressed as

x

0

,.)d)c(xf(dx

dh(x)

which is exactly of the form of Eq.(2.14).

Independently of the above derivations, L. Boltzman in 1874 and J. Hopkinson in 1877 obtained

similar expressions to that of (2.17) in the solution of some problems of physics [5 and 48]. Some

authors also attached the name of these authors to that equation [13, p.56].

In the present century, Eq.(2.17) was independently derived and published by Carson [17, p.16,

footnote 1]. The exact reference to his publication is not given in his book (it is the guess of the

present writer that the publication was around the years 1917-1919). As it was pointed out before,

Carson credited Duhamel the derivation of (2.17). He also mentioned that (2.17) was

independently communicated to him by H.W. Nichols and by Stuart Ballantine. However the

exact references were not given.

2.8 Special Functions and the RCI.

There is a great amount of occurrences of the RCI in the theory of special functions. These

occurrences appeared mainly in the formulation of some of their properties and it seems that they

were not formulated by the respective authors having the RCI in mind. In this final section of this

chapter some of those occurrences are quoted chronologically without discussing their derivation.

Euler was probably the first author in using the RCI in the theory of special functions when he

used this integral in his discussion of the Beta function (the Greek letter B was first introduced by

Jaques P.M. Binet in 1839 [15, Vol.II, p.272] ). Indeed in 1768 Euler wrote the expression [32,

Vol,I, Chap.XII, pp.269-270]

,z)(u

dzzu

1

1

3

3

2

2

1

1v

1vm

(2.18)

where the limits of the integral are to be understool from z z 0 to . Note that the

transformation x z u z / ( ) gives

16

0

1

0

111

;)1(1

)(dxxx

uzu

dzz v

v

v

an expression which was also proved by Euler.

In the year 1848 Oskar Schlömilch (1823-1901) applied the Gamma function to solve some

definite integrals and stated the following expression [79, Erste Abtheilung, pp.110-11]

x

0

x

0

x

0

131m1l )d()(x...d)(d)(x

x

0

13ml )dr.F()(xs)m(l

(s)(m)(l)

Schlömilch also made reference to the formulas found by Abel [see Eqs.(2.11) and (2.13)].

Concerning Legendre polynomials, in 1848 F. Neumann found that the Legendre polynomials of

the second kind Q zn ( ) could be expressed in terms of the Legendre polynomials of the first kind

P zn ( ) . The expression he found was [106, p.320]

1

1

,)(

2

1)(

yz

dyyPzQ n

n

where n is a positive integer and z is a real number between -1 and 1. A second property of the

RCI type for the Legendre polynomials is that given by H.V. Lowry in 1932. This property is

[71, pp.31-32]

x nn

nn

xtdttxP

0

212

2 .2

sinsin)cos(

Finally some properties of the RCI type concerning Bessel and associated functions are quoted.

These are

0

00 ;)()(sin2

)( dxxJkx

kxkJ

0

00 ;)()cos(2

)( dxxJkx

kxkY

x

nmnm dt

t

tJtxJnxJ

0

.)()(

)(

17

The first two formulae were given by N.J. Sonine in 1880 in his study of cylindric functions [89

and 97, p.433]. The last formula was derived in 1905 by H. Bateman using convolution theorem

for Laplace trnasform [97, p.380]. Some special cases of this last formula were derived

independently of Bateman by W. Kapteyn in three papers published in the period 1905-1907.

Bateman in 1812 also used the following integral for some developments of the potential function

[97, p.389]

.)()()(exp 22

0 dttfytxkJkz

Chapter 3

The Complex Convolution Integral

3.1. Definition and Introductory Comments

Let f g and be two complex valued functions of the complex variable z . If is a complex

variable, the integral

C

dgzfi

)()(2

1 (3.1)

where i2 1 and C is a suitable curve in the complex plane, will be called the complex

convolution integral (CCI).

The history of the CCI started when the theory of complex integration started to be developed. As

it is well know, the main author who contributed to this development was Augustin-Louis

Cauchy (1789-1857): The brief historical quotations of the CCI given in this chapter start with

the work of this author.

3.2. Cauchy's Integral Formula and the CCI.

Following Bottazinni [8], a particular form of (3.1) was formulated by Cauchy in 1830 in a long

article presented to the Academy of Science of Turin in 1831. Cauchy also gave several

reformulations of this particular form in two papers appeared in 1834 and 1841. These three

papers have an interesting history which is also given in Bottazinni's book [op.cit., pp.157-158].

The aforementioned formulation of Cauchy is not other than the formulation of the so-called

Cauchy integral formula in the theory of functions of complex variable. If f is a continuous and

finite function for x X together with its derivative and where x X ip exp( ) for p ,

then according to Bottazinni [op.cit. p.158], Cauchy stated the formula

.)(

2

1)( dp

xx

xfxxf (3.2)

18

This formula seems to be a RCI at first sight. However after performing some algebraic

manipulations into it, it can be seen that it is a CCI. Indeed, the right hand side of (3.2) can be

written as [8, pp.176-177]

C

p

p

p

p

.xdxx

)xf(

2pi

1

exp(ip)dp iXxx

)xf(

2pi

1dp

xx

)xf(x

2p

1

(3.3)

Obviously (3.3) is a CCI and is the way Cauchy integral formula is known nowadays.

Eq.(3.2) was used by several authors as soon as Cauchy papers were published. The next

important application of this formula was given by P.A. Laurent (1813-1854) when he

established his famous series expansion theorem in 1843 [56]. The CCI appeared in Laurent's

paper when he stated that the coefficients of the series expansion.

f x a a x s a x s a x sn

n( ) ( ) ( ) ( ) 0 1 2

2

b

x s

b

x s

b

x s

1 2

2

2

3( ) ( ) ( ),

where f is an analytic function regular in the open annulus R x s R ', are given as

' 1;

)(

)(

2

1

C nn dtst

tf

ia

C

1n

n dt;s)f(t)(t2pi

1b (3.4)

with C ' and Care the circles whose common center is the point s and whose radii are R' and R ,

respectively.

Concerning the theory of Legendre polynomials P xn ( ) , L. Schläfli in 1881 [78] used the results

of Cauchy and Laurent to show that P xn ( ) admitted the following CCI representation [47, p.30]

C nn

n

n dxi

xP ,)(2

)1(

2

1)(

1

C being a large circle whose center is the point x .

On the other hand, it is not difficult to show that the an 's in (3.4) are equivalent to the expressions

' 1

)(

!

1

)(

)(

2

1

C

n

nnds

sfd

ndt

st

tf

ia

.

19

Thus if r is allowed to be a non-negative number, then the above expression suggests to define

fractional derivatives of f x( ) by means of the formula

C n

n

xdtxt

tf

i

nxfD

,)(

)(

2

!)(

1 (3.5)

where C is a closed curve in the complex plane about the point t x . Eq (3.5) is also a CCI and

by means of that H. Laurent (not to be confused with P.A. Laurent) in 1884 defined fractional

derivatives [20, p.66].

3.3. Pincherele's Work and the CCI.

Although the aforementioned occurrences and uses of the CCI, none of the above authors made a

complete study of (3.1). The earliest study of that equation is perhaps that made by S. Pincherele

in 1908 [67]. Pincherele's study was made in connection with the solution of the complex integral

equation

Pzzgdzzfzsk

i),()()(

2

1

(3.6)

where P 0and k z( ) and g z( ) are given functions while f z( ) is unknown. Pincherele

succeeded in the solution of (3.6) using as tool the unilateral Laplace transform. His results and

the results of other authors are resumed in Chap. 17 of Gustav Doetsch's book [26].

Chapter 4

The Discrete Convolution

4.1. Definition and Introductory Comments.

Let ix , and iy be two real or complex sequences such that i . The discrete

convolution (DC) of these sequences is a new sequence defined by the expression

x y in i n

n

(4.1)

Notice that if a sequence, 1;N,0,1,i ,x 1i has a number of terms N1 and the sequence

.,1,,1,0, 2 Niyi has a number of terms N2 , then the DC of these sequences may be written

in the form

x y , i 0,1, ,N 1; N N N 1n i n 1 1 2

n 0

i

. (4.2)

20

The DC of two finite sequences having N1 and N2 terms, respectively, therefore is a new

sequence having N N1 2 1 terms.

4.2. Cauchy's Work and the DC

The earliest study of the DC is perhaps that performed by Cauchy in his famous book entitled

Cours D'Analyse de L'École Royale Polytechnique which appeared in 1821 [18]. It is of

importance to point out that Cauchy did not give in any part of his book the source of references

he used to establish his results concerning DC and the other topics studied in it, therefore it is

difficult to say if a previous author studied or made use of the DC.

In what follows in this section the results stated by Cauchy are quoted. These quotations are

taken from the aforementioned book by Cauchy.

The first result concerning DC is stated in Chap.IV in connection with the multiplication of

series. On page 141 is established and proved that if

u u u un

n

0 1 2

0 1 2

, , , , , ;

, , , , ,

(4.3)

are two [absolutely] convergent sequences composed only of positive terms and having sum s

and s', respectively, then

u v

u v u v

u v u v u v

u v u v u v u vn n n n

0 0

0 1 1 0

0 2 1 1 2 0

0 1 1 1 1 0

,

,

,

will be a new convergent sequence having sum ss'

The condition of positiveness of the terms in the sequences given in (4.3) was removed by

Cuachy on page 147 and then he proved the corresponding results for real arbitrary [absolutely]

convergent sequences.

The preceding two results were then used by Cauchy to establish a theorem and three corollaries

concerning the multiplication of power series. The theorem is given on page 157 and estates that

if the two sequences

a a x a x a x

b b x b x b x

n

n

n

n

0 1 2

2

0 1 2

2

, , , , ,

, , , , ,

(4.4)

21

are convergent for certain value of the variable x and such that they have sums s and s',

respectively, then

a b

a b a b x

a b a b a b x

a b a b a b a b xn n n n

n

0 0

0 1 1 0

0 2 1 1 2 0

2

0 1 1 1 1 0

,

( ) ,

( ) ,

( )

will be a new convergent sequence which have sum ss' .

The first corollary of this theorem is given on pages 157-158 and in it Cauchy stated that under

the conditions given to the sequences (4.4) the product of series in given as

( )( )

( ) ( ) .

a a x a x b b x b x

a b a b a b x a b a b a b x

0 1 2 0 1 2

2

0 0 0 1 1 0 0 2 1 1 2 2

2

and then he concluded saying that the product of the sums of two sequences is a new sequence of

the same form.

In the second corollary [op.cit., p.158] he extended the result of the first corollary to the case

when an arbitrary number of series is taken into account. In the third corollary [op.cit., p.158] he

considered the very special case in which in (4.4) a b0 0 , a b a b1 1 2 2 , , , and he wrote the

expression

( ) ( ( ) ).a a x a x a a a x a a a x0 1 2

2 2

0

2

0 1 0 1 1

2 22 2

Following the analysis given on page 159, he replaced in (4.4) the sequence b b x b x0 1 2

2, , , by a

polynomial composed of a finite number of terms and:

"on obtient une formule qui ne cesse jamais d'etre exacte, tant que la série

a a x a x0 1 2

2, , ,

demeure convergente".

Under the above discussion he then established that if the sequence a a x a x a xn

n

0 1 2

2, , , ,

converges, the product of the sum of this sequence by the polynomial

kx lx px qm m 1 ,

where m is an integer number, is a new convergent series of the same type where the general

term will be

22

( ) ,qa pa la ka xn n n m n m

m 1 1

"pourva que l'on considère comme nulles dans les premieres termes celle des quantités

a a a an n n m n m 1 2 1, , , ,

qui se trouveront affectées d'indices négatifs: en d'autres termes, on aura

( )( )km lx px q a a x a xn m 1

0 1 2

2

qa qa pa x0 1 0( ) ( )qa pa la ka xm m

m

1 1 0

( )qa pa la ka xn n n m n m

m

1 1"

On the other hand, Cauchy also considered the case in which the sequences involved in (4.3) take

complex values. The theorem concerning this case is established on page 283 of his book.

Finally, it is of importance to mention that Cauchy also approached the DC from the point of

view of double sequences, which are studied in NOTE VII of his book. On pages 542-543 he

stated that if

u u u u0 1 2 3, , , , ;

0 1 2 3, , , ,...

are two convergent sequences having sums s and s' and if the following table is constructed

u u u u0 0 1 0 2 0 3 0 , , , ,

u u u0 1 1 1 2 1 , , ,

u u0 2 1 2 , ,

u0 3 ,

then the vertical sums

u0 0 ,

u u0 1 1 0 ,

u u u0 2 1 1 2 0

,

u u u un n n n0 1 1 1 1 0

will be a new convergent sequence, and the sum of this new sequence will be equal to ss' .

As it can be seen at this point, the discussion made by Cauchy concern both (4.1) and (4.2).

Improvements of his results concerning convergence were given by several authors in the second

half of the las century. These results can be found, for example, in the book by G.H. Hardy [44].

4.3. Statistics and the DC.

23

An application of discrete convolution, probably the first one and which seems no to be in

connection with Cauchy's work, was given by actuaries and vital staticians in the XIX century.

This application was mainly in dealing with the problem of graduation of statistical data by

linear compounding. The main idea of this method consists of the replacement of a sequence of

observed values ru (of a sequence of true values rU ) by a sequence r , where each vr is

obtained by a linear compound given by the expression

v b u b u b ur r r r r r r ( )1 1 1 1

( )b u b ur r r r2 2 2 2

(4.5) ( )b u b ur n r n r n r n

for a range of 2 1n terms and on the assumptions that the finite differences of rU beyond

certain order j may be neglected.

Assume now that the b s' in (4.5) are such that b b ar k r k k , then (4.5) becomes

r r r r r r n r n r na u a u u a u u a u u 0 1 1 1 2 2 2( ) ( ) ( ).

It is not difficult to see that this expression is a DC formula. Indeed, it may be written as

r k r n

k n

n

a u

(4.6)

The fundamental conditions under which such replacements of ur by linear compounding of the

u s' are legitimate were well set out by W.F. Sheppard in three papers published in the period

1912-1915 [86, 87 and 88]. See Hugh H. Wolfenden's paper for an account of these papers [104].

The determination of the b s' or a s' may be affected by interpolation, fitting by least squares, or

simply reduction of error processes. According to Wolfenden [op.cit., p.83], the earliest

application of (4.6) in interpolation was that of Griffith Davies in 1834 in connection with the

mortality table of the Equitable Society. The application of (4.6) to the problem of fitting by least

squares were indicated briefly by C.L. Landré in 1901 [52], and were fully worked out by

Sheppard in his papers. The formulae for reduction of error were indicated by G.F. Hardy (not to

be confused with G.H. Hardy) in 1909 [43] and fully examined by Sheppard in his papers, and

some of them afterward were rediscovered independently by R. Henderson in 1916 [45] and J.

Larus in 1918 [55]. An account of these papers may be found in Wolfenden's paper and book

[104, 105].

The paper published by Wolfenden in 1925 was based on the work by, until then unknown,

Erastus L. De Forest and then it became known that the determination of the a s' in (4.6) in the

case of interpolation, fitting by least squares and those of reduction of error, which make the

mean square error in 4 a minimum, had previously been discussed very fully by De Forest.

These discussions appeared in a series of papers published in the Smith sonian Reports of 1871

24

and 1873, in a pamphlet in 1876 and in 1877-1880 in a Journal of Des Moines, Iowa, called The

Analyst (A monthly Journal of Pure and Applied Mathematics). The complete list of the papers

written by De Forest is given at the end of Wolfenden's paper.

According to Wolfenden, De Forest also made an extensive investigation of the effects of

applying some of his linear compounding formulae repeatedly, and in this instance also reached

some important conclusions on a matter which was suggested, but not closely examined, by other

authors in later years. De Forest noted clearly the manner in which when a linear compounding

formula is repeated in a large number of times, the curve of the coefficients ultimately tends to a

central bell-shaped potion with an infinite number of small oscillations at each end. To be

specific, De Forest observed that the limiting form of the curve of coefficients of (4.6) becomes

the normal probability curve when these coefficients are symmetric, while in the unsymmetrical

case [see Eq.(4.5)] he reached an unsymmetrical probability curve. It is important to point out

that these observations in the symmetrical case were proved many years later, to be specific, in

1948, by Isaac J. Schoenberg [81].

The next importance occurrence of (4.6) in statistics was in the year 1946 in a paper written by

Schoenberg [80]. This occurrence of (4.6) is closely related with the work of De Forest just

described. Indeed, in that paper Schoenberg approached the problem of smoothing the sequence

of equidistant data ny by a series of the type

F y L , L L ,n n n n n n

n

(4.7)

from the point of view of Fourier series of the functions T u u( ) ( ) and whose Fourier

coefficients are nn Ly and , respectively [80, pp.50-56]. In his paper, he characterized the

smoothing properties of (4.7) in terms of ( )u and established the conditions oin ( )u such that

(4.7) reproduces the values of yn of a polynomial of degree not exceeding a given integral

number m. Two years later, Schoenberg returned to the problem of smoothing and, as it was

pointed out before, he proved the De Forest's observation about the bell shaped form of the

iterates of (4.7). In the following years, Schoenberg and his students made several investigations

of the subject of smoothing data by (4.7). These investigations were resumed by him in a paper

published in 1953 [82].

In pass it is worth to mention that in Schoenberg's paper of 1946 the B-splines were introduced

by the very first time in the modern mathematical literature.

4.4. Special Functions and the DC.

In the theory of special functions, and mainly in the theory of Bessel functions, there exist several

properties of them which are given by means of DC formulae. Some of these formulae are briefly

quoted in this section.

As usual let J xn ( ) denote the Bessel function of the first kind and of order n. The earliest

property of J xn ( ) given by a DC formula seems to be given by P.A. Hansen in a paper originally

25

published in German in 1843 and which its translation to French appeared in 1845. The

expression given by Hansen was [42 and 97, pp.30-31]

J x J x J x J x J xr r r r

rr

1 1 1

10

1

2 2( ) ( ) ( ) ( ) ( ).

This expression is a particular case of a more general one derived independently by C.G.

Neumann in 1867 [64, p.40, and 97, p.30] and E.C.J. von Lommel in 1868 [59, pp.26-27, and 97,

p.30]. The general expression is

J y z J y J zn m n m

m

( ) ( ) ( )

(4.8)

This expression for z y and for all n was also derived by L. Schläfli in 1871 [78, pp.135-137,

and 97, p.30].

In 1867 Neumann also derived the following formulae for the case 0 [64, and 97, p.30],

where Y x ( ) denotes the Bessel function of the second kind and order ,

Y Y Z J z mm m

m

( ) cos( ) ( ) ( ) cos( ).

Y Y Z J z mm m

m

( ) ( ) ( ) ( ) ( )sin sin

,

with Z z Zz2 2 2 cos ; while Lommel derived the expression

( ) ( ) ( ) ( ) ( ) ,

1 2 02

1

2

0

2r

r n r r

r

n r

r

n

J z J z J z J z

and the espression

Y z t Y t J zm m

m

( ) ( ) ( ),

(4.9)

for z t .

Concerning Schaläfli, he also gave the following expression in his paper of 1871 [op.cit., pp.139-

141, and 97, p.289]

S t z S J z z tn n m m

m

( ) ( ); ;

where S tn ( ) are the nowadays so-called Schläfli polynomials defined by the expression

26

)2/(

0

2

0

.1;2!

)!1()(

,0)(

n

m

mn

n nt

m

mntS

tS

On the other hand, in 1872 L. Gegenbauer in his studies dealing with Bessel functions defined the

so called Neumann's polynomials of order n the formula [38, and 97, p.273 and p.290]

O tn

n

n m m n

n m t mm

n

( )( ) cos ( )

( )( ),

1

4 1

2

2

2

2 2

10

In a paper dated August 1879 and published in 1880 N.J. Sonine proved that if C z ( ) is defined

by the expression [89]

C z J z Y z ( ) ( ) ( ) ( ) ( ), 1 2

where 1 2( ) ( ) and are arbitrary periodic functions of with period unity, then

C z t C t J zm m

m

( ) ( ) ( ).

(4.10)

Formulae (4.8), (4.9) and (4.10) were also proved by J.H. Graf in a paper dated March 1893 and

published the same year [39, pp.141-142]. In this same paper Graf also proved that [op,cit.,

pp.142-144]

m

m imzJizZ

izZJ ),(exp)(

)(exp

)(exp)(

2/

where Z z 2Zzcos and zexp( i ) Z .2 2

Finally, outside of the theory of Bessel functions, the following expression was derived by C.

Runge in 1914 [77]

n

r

rnrn

n yHxHr

nyxH

0

/2 ),()(2

2

where H xn ( ) denotes the Hermite polynomials of degree n 0 1 2, , , .

4.5. Cardinal Series.

A convolution formula closely related to the DC is that given by the expression

27

a f z nn

n

( ) (4.11)

Obviously this formula reduces to the DC, depending of the values of n , if z is allowed to take

integral values only. In the literature (4.11) is better known as cardinal series. A major factor

affecting current interest in the cardinal series is its importance in the sampling theory of band-

limited functions or signals. Althouhg this application, its origin concerns with the problem of

interpolation as will be seen bellow. Herein only some historical remarks concerning (4.11) will

be given. For major accounts the reader is referred to the works of A.J. Jerri [50] and J.R.

Higgins [46].

The first explicit use of (4.11) occurs in a brief note by Félix-Edouard-Justin-Emile Borel (1871-

1956) in 1898 in a paper dealing with the problem of expansion of a function by Taylor series

[6]. On page 1002 of this paper it is found a expression of the type

( z )

sin( z ) a

z n

n

n 0

(4.12)

to get information about how the power series coefficients na of a function f x a zn

n( )

determine its singularities.

The next year, Borel in dealing with the problem of interpolation used the following formula [7]

sin( t ) c ( 1)

t n,n

n

n

(4.13)

which can be written as

ct n

t nn

n

sin

( )

( ).

On page 83 of this last paper Borel mentioned that he deduced the series from Lagrage's

interpolation formula.

Independently of Borel, in 1900 John Dougall expanded the solution P z ( ) of Legendre's

equation

(1 z )d y

dz2z

dy

dzn(n 1)y 0, z 1,2

2

2

as a series of Legendre polynomials [29]

).()1(1

11sin)(

0

zPnn

zP n

n

n

28

The corresponding expansion for the second solution Q z ( ) of Legendre's differential equation

was given by H.B.C. Darling in 1923 [21].

On the other hand, since P z P zn n 1( ) ( ), it is readily seen that if f P z( ) ( ) then Dougall's

formula may be regarded as a case of the following interpolation formula

fn

nf n

n

( )( )

( )( )

sin

(4.14)

which was established by Jaques Hadamard (1865-1963) in 1901 [41] after an extensive study of

Borel's paper of 1898.

Hadamard's formula (4.14) is analogous to the fundamental interpolation formula of Charles de la

Vallée Poussin (1866-1962) appeared in 1908. The interpolation scheme due to de la Vallée

Poussin considers the finite interpolation formula [72, p.327]

sinmt

m

f n m

t

n

nma

b ( ) ( / ),

1

where f x( ) is a given function defined on the finite interval a b, , and the summation is

understood to be over those n for which n m a b / , .

According to W.L. Ferrar [34, p.333], F.J.W. Whipple in an unpublished manuscript dated 1910

introduced the cardinal series and discovered several of its properties, including the band-limited

nature of its sum.

Later, in 1915, Edmund T. Whittaker rediscovered again the cardinal series in connection with

the problem of interpolation of equidistant data [99]. In his paper Whittaker did not make

references to previous work. The expression used by Whittaker was [op.cit., p.86]

f a nx a n

x a nn

( )( ) /

( ) /,

sin (4.15)

which reduces to (4.14) if m a n . E.T. Whittaker did not call (4.15) cardinal series, the name

seems to first appeared in the works of Ferrar [34] and J.M. Whittaker (second son of E.T.

Whittaker) [101, 102].

After the above rediscovering of the cardinal series, it came up a period where they were used

and extended (this extension considered more general expressions no necessarily of the

convolution type) to deduce properties of entire functions from their known behaviour at a

sequence of points. Among the authors who made extensions are J.F. Steffensen [90], T.A.

Brown [10, 11], M. Theis [91], K. Ogura [66], W.L. Ferrar [33, 34, 35], T.M. MacRobert [60,

61], I.M. Sheffer [85], E.T. Copson [19] and J.M. Whittaker [101,102]. A brief account of some

of these papers are given in Story One of Higgin's paper [46, pp.53-57].

29

The cardinal series (4.15), appeared also in the russian literature. It was firstly given by V.A.

Kotel'nikov in 1933 in dealing with certain problems of communication [51].

In the american literature, it was mainly C.E. Shannon in 1949 who introduced a series of the

type given in (4.15), this was also in dealing with problems in communication [84]. Although

Shannon results were published in 1949, his paper was apparently written in 1940, however its

contents seem to have been in circulation in the United States by 1948.

The general theory of interpolation formulae of the type (4.11) with f x f x( ) ( ) started in

1946 with I.J. Schoenberg. The results obtained by him, his students and other authors in the

period 1946-1973 were stated in his paper of 1946 and in his monograph of 1973, In pass, it is

pointed out that in his paper of 1946 the B-splines were given a name and were used by the very

first type to solve the problem of interpolation.

The most recent account concerning cardinal series from an introductory point of view is given in

the book by R.J. Marks [62].

Chapter 5

The Notations and Names of Convolution 5.1. Introduction.

Some of the convolution operations defined in the preceding chapters have been denoted and

named in several ways in the literature. As it wil be seen below, the notation used for these

operations has been almost uniform in the literature. Concerning the names, these were usually

given either after the work of some author or after the study of the properties of these operations.

5.2. Notations.

Probably the first expression used to denote the RCI is that given by Volterra. Indeed, as it was

seen in Sec. 2.6., Volterra denoted and defined the so-called composition of the first kind as

y

z

y).(x,gfy)dxx)g(x,f(x,

Although the above notation is quite general, in the particular case that f x y( , ) and g x y( , )

belong to the group of the closed cycle (see Sec. 2.6.) Volterra used the same notation and he

wrote:

t

0

(t).gf)d)g(f(t

This notation can be considered as the most primitive notation of the very well known star-

notation or asterisk-notation:

30

t

tgfdgtf0

).)(()()( (5.1)

It is the guess of the present writer that the star-notation was firstly used by Gustav Doetsch.

Thus for example, Doetsch used the above notation in a paper published in 1927 [25, p.23].

Other notation were also given by Doetsch. For example on page 161 of [26] is stated that the

right-hand-side of (5.1) may be denoted as

f gt

0

(5.2)

in order to be distinguished from the RCI

,)d)g(f(t

which he denoted as

f g

. (5.3)

This notation also appeared on page 127 et. seg. of a paper published in 1938 by F. Tricomi [93].

Notations (5.2) and (5.3) seem to be difficult to write and have disappeared from the literature.

Doetsch himself did not use them in his book of 1970 [27].

Concerning the notation for complex convolution, Doetsch gave no notation in his paper of 1927

nor in his book of 1943. However in his book of 1970 he used the notation

f go

to designate that operation. On the other hand, Gardner and Barnes on page 275 of their book

[37] used the notation

to designate the CCI of functions F s F s1 2( ) ( ) and .

5.3. Names.

As it was seen in Sec. 2.6., in the theory of integral equations developed by Volterra the notion of

composition of two functions f x y g x y( , ) ( , ) and played and important role. Considering only

31

the composition of the first kind, it was also seen that if these functions were also permutable

functions with the unity, then

f ( x,y ) f g( x,y ) g f ( x,y )

f g( y x ) g f ( y x )

f ( y x )

from which easily follows the RCI

t

dtgft0

.)()()( (5.4)

Thus it can be said that the composition reduces to the operation (5.4).

The name composition is one of the first names attadched to (5.4) and has been used and

preferred frequently, since the times of Volterra's work, in the French literature. In that literature

the name has been modified slightly and some times it appears as produit de composition

(product of composition). The name composition has also been used as an alternative name in

some German literature where the name faltung is preferred; e.g., [4, p.56., and p.285 Remark-

Quotation 40] and [26, p.157]. Mereover, the name composition has been extended to designate

the RCI

.)()( dtgf (5.5)

Another name derived from the work by Volterra and Pérès, although less common, to designate

(5.5), is that of resultant [96, p.6]. This is the name used by E.C. Titchmarsh [92, p.51] and by B.

Van der Pol and J. Bremmer [94]. The name resultant has been also used to designate (5.4).

Indeed, in his book Doetsch [26, p.157] suggested that this name may be used as an alternative

English name to designate that operation; while G.H. Hardy in his book stated [44, p.98,

footnote]:

"The German name equivalent of resultant is faltung".

The name faltung to designate (5.4) seems to be given by the very first time by Doetsch in two

papers appeared in 1923 [23, 24]. To be more specific, (5.4) was called by Doetsch

faltungsintegral [25, p.23].

In the period of time immediately after the occurrence of the aforementioned Doetsch's papers,

the name faltung was the most common to designate (5.4) and/or (5.5). In his book Doetsch [25,

p.157] pointed out that the name faltung was preferred by many American authors. This

affirmation is supported by the following quotation taken from the book by Norbert Wiener [103,

p.45]:

32

"The quantity

dxxygxf )()(21

is known as the Faltung of f x( ) and g x( ) (there is not good English word), and the

sequence

nmn ba

as the Faltung of the sequences nn ba and ".

Wiener also used the word faltung to designate (5.5) [103, p.71].

Concerning the CCI, on page 23 of his paper of 1927 Doetsch designated this integral as

Facherintegral. However in his book he designated it as komplexe Faltung [27, p.167].

The name faltung has also been used to designate the discrete convolution operation [103, p.45].

On the other hand, it is difficult to say when the name convolution occurred by the very first time

in the literature. Doetsch himself suggested the name as an English translation of the german

name Faltung [26, p.157.]. On page 228 of the book by Gardner and Barnes it is found the

following quotation [37]:

"The process expressed by the integral [(5.4)] will be called convolution in the real domain, or

real convolution, and the functions [entered into it] will be said to be convolved".

On pages 231-233 of the same book by Gardner an Barnes, it was given what was probably the

first graphical interpretation of the RCI. This interpretation was given by convolving the

functions exp( ) ( ). t t t and exp At the end of this example these authors also pointed out that:

"It can be seen from this example that 'convolution' denotes a mathematical process that can be

interpreted graphically by folding, translating, multiplying, and integrating".

The process of folding in this graphical interpretation agrees with the translation to English of the

German word faltung, which means folding. It also agrees with the definition given on pages

952-953 of the 1978 edition of the Oxford English Dictionary. Indeed, according to this

dictionary, the word convolution concerns with the action of folding, and according to its

etymological roots it concerns with the action of rolling up together.

Gardner and Barnes also designated the CCI as [op.cit., p.275]: "convolution in the complex

domain or more briefly complex convolution".

Nowadays the name convolution has become in common use in the literature to designate either

the RCI, or the CCI, or the DC. In the past other names have been used to designate these

operations or special instances and variations of them. For example, the integral

33

x

xdxcxfxcfxh0

;0;)()(')()0()( (5.6)

which is equivalent to the integral

x

xdxcxfdx

dxh

0

;0;)()()(

some times has been referred as superposition theorem or Duhamel's theorem [16, pp.30-31, and

p.301]. The first name is due to, in the derivation of (5.6), the superposition principle plays an

important role [37, p.234]. On the other hand, Duhamel used the superporsition principle to

derive (5.5), this is the reason of the second name. The names of Boltzman and Hopkinson has

been also attached to (5.5) [13, p.56] (see also Sec. 2.7.).

Although the above names were originally given to (5.6), some authors have used them to

designate in general the RCIs. For example, R.B. Blackman and J.W. Tukey on page 73 of their

book [3] pointed out that:

"Convolution is often called by a variety of names such as Superposition Theorem,

Faltungsintegral, Green's Theorem, Duhamel's Theorem, Borel's Theorem, and Boltzman-

Hip'kinson Theorem".

In this book no references were given to this quotation. The name Borel's Theorem in the above

quotation is not justified since in some early literature the name is given to Convolution Theorem

for Laplace integral [28]. The name Green's Theorem remains as a mystery for the present writer

since the aforementioned authors gave no references.

Ronald N. Bracewell on page 24 of his book [9] stated a similar paragraph to the above of

Blackman and Tukey. He wrote:

"The word 'convolution' is coming into more general use as awareness of its oneness spreads

into various branches of science. The German term Faltung is widely used, as is the term

'composition product', adapted from the French. Terms encountered in special fields include

superposition integral, Duhamel integral, Borel's theorem, (weighted) running mean,

crosscorrelation function, smoothing, blurring, scanning, and smearing".

The reader is referred to Bracewell's book for the justification of the last six names.

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