Spaces of distributions

1. A Historical Note on Schwartz Space and
Test or Bump Functions
Alejandro Domínguez
December 2013
Some preliminary definitions
The definition of rapidly decreasing functions is as follows.
1
Definition (Rapidly Decreasing Functions)
A function  :

is said to be a rapidly decreasing function

A.   C   ,
B. xi
d j  x 
dx j
 (infinitely differentiable);
 M ij , for all integers i, j  0 .
Notice that property B in the Definition 1 gives the name to these functions. In fact, this property means
that the function and all of its derivatives decrease faster than any polynomial. Also notice that the set of
rapidly decreasing functions forms a linear space, called the Schwartz space, denoted by S  ,
other words: S  ,

 . In
is the linear space of all complex valued infinitely differentiable functions having
all their derivatives decreasing faster than any polynomial.
An example of a function belonging to S  ,
Figure 1. Graph of   x   x e
i
x
2
 is (see Figure 1)   x   xi e x
2
, for all i  0 .
for i  0,1, 2, 3, 4 (in black, red, green, pale green, and blue respectively).
1

2. Of course, any complex valued infinitely differentiable function  having a compact support [i.e.,
  C0  ,
 ] belongs to S 
a maximum in
,
 . This is so since any derivative of 
is continuous and x i j  x  has
. Particularly, these last functions are called test functions or bump functions. The
linear space of test functions is denoted as D  ,
 . Of course this last space is a subspace of S 
,
.
A function of this type is (see Figure 2).
 1 2
 1 x
  x   e ,
0,

x  1;
x  1.
Figure 2. Example of a test or bump function.
This is probably one of the simplest examples of a test function; however, it has an awful property. In fact,
following an argument similar to that given on p. 16 of (Griffel, 2002), since support of  is the interval
 1,1 , then   x   0
about
for x  1 and x  1 , so all derivatives vanish at 1 . Hence the Taylor series of 
1 is identically zero. But   x   0 for 1  x  1 , so  does not equal its Taylor series. Test
functions are thus peculiar functions; they are smooth, yet Taylor expansions are not valid. The above
example clearly shows singular behavior at x  1 .
Notice that in the theory of distributions, it is not necessary to use explicit formulas for rapidly decreasing
functions or test functions: They are used for theoretical purposes only.
Another related set of functions are the so-called functions of slow growth.
2
Definition (Functions of Slow Growth)
f:

is said to be a function of slow growth

A.
f C  ,
;
B. There exists a B  0 such that
d j f  x
dx
j
  as
O x
2
B
x .

3. The set of functions of slow growth is denoted as N  ,
polynomial is an element of N  ,
 . Moreover, if
 . From this definition it is obvious that any
f  N  ,  and   S  ,  , then f   S  ,  .
It should also be observed that (Lighthill, 1958, p. 15) calls the elements of S  ,
while the elements of N  ,

good functions,
 fairly good functions.
The historical note
The Schwartz space is called so after the French mathematician Laurent Moise
Schwartz (March 5, 1915 – July 4, 2002). This space was actually defined by
Schwartz in his paper (Schwartz, 1947-1948). In fact, on p. 10 of this paper it
can be read the definition of the Schwartz apace and its relation to the space of
test functions:
Main Scientists on
Schwartz Space and
Test or Bump
Functions
Soit  S  l’espace des fonctions   x1 , x2 , xn  indéfiniment
dérivables (au sens usuel), et tendant vers 0 à l’infini plus vite que
2
2
toute puissance de 1 r ( r 2  x12  x2   xn ) ainsi que chacune
de leurs dérivées.
On peut encore dire que, si   S  , tout produit d’un polynôme
par une dérivée de  (ou toute dérivée du produit de  par un
polynôme) est une fonction bornée, et réciproquement ; nous
dirons pour abréger que  est «à décroissance rapide à l’  ainsi
que ses dérivées». L’espace  S  admet évidemment, comme
sous-espace particulier, l’espace  D  , des fonctions 
indéfiniment dérivables à support compact.
Nous introduirons dans  S
Portrait 1. Laurent Moise
Schwartz (March 5, 1915 –
July 4, 2002) (http://wwwhistory.mcs.standrews.ac.uk/Biographies/
Schwartz.html).
 une notion de convergence. Des
 j   S  convergeront vers o si le produit par tout polynôme de
toute dérivées des  j (ou toute dérivée du produit des  j par tout
polynôme) converge uniformément vers o dans tout l’espace. On
voit que des  j   D  , convergeant vers o dans  D  , convergent
aussi vers o dans  S  , mais la réciproque est inexacte. On montre
aisément que  D  , considéré comme sous-espace vectoriel de
 S  , avec la topologie induite par celle de  S  , est dense dans
S  .
The method of multiplication of a suitable function by a test function and then
integrate the result (as it is the case of the Theory of Distributions) is as old as
the beginning of mathematical analysis. On years 1759-1760, the Italian born
mathematician Joseph-Louis Lagrange (January 26, 1736 – April 10, 1813)
published a paper where this method is used in relation to the integration of the
sound wave equation (Lagrange, 1759-1760). On §6 Lagrange established the
next problem and a method for finding its solutions:
6. Problème I.  Étant donné un système d’un nombre infini de
points mobiles, dont chacun dans l’état d’équilibre soit déterminé
3
Portrait 2. Joseph-Louis
Lagrange (January 26, 1736
– April 10, 1813)
(http://www-history.mcs.standrews.ac.uk/Biographies/
Lagrange.html).

4. par la variable x , et dont le premier et le dernier, qui répondent á
x  0 et à x  a soient supposés fixes, trouver les mouvements de
tous les points intermédiaires, dont la loi est contenue dans la
d2z
d 2z
formule 2  c 2 , z étant l’espace décrit par chacun d’eux
dt
dx
durant un temps quelconque t .
Qu’on multiplie cette équation par Mdx , M étant une fonction
quelconque de x , et qu’on l’intègre en ne faisant varier que x ; il
est clair que si dans cette intégrale, prise en sorte qu’elle
évanouisse lorsque x  0 , on fait x  a , on aura la somme de
toutes les valeurs particulières de la formule
d2z
d 2z
Mdx  c 2 Mdx , qui répondent à chaque point mobile du
2
dt
dx
système donné. Cette somme sera donc
Portrait 3. Norbert Wiener
(November 26, 1894 –
March 18, 1964)
(http://www-history.mcs.standrews.ac.uk/Biographies/
Wiener_Norbert.html).
d 2z
d 2z
Mdx  c  2 Mdx .
 dt 2
dx
This method was also used by Norbert Wiener on 1926 for solving linear
partial differential equations of second order (Wiener, 1926). On pp. 582
Wiener wrote:
8. Operational Solution of Partial Differential Equations.
Before we enter in this topic in detail, it is important to consider
the nature of the solution of a partial differential equation. Let us
consider the linear equation
A
 2u
 2u
 2u
u
u
B
C 2  D  E
 Fu  0 ,
x 2
xy
y
x
y
Portrait 4. Kurt Otto
Friedrichs (September 28,
1901 – December 31, 1982)
(http://www-history.mcs.standrews.ac.uk/Biographies/
Friedrichs.html).
where for simplicity’s sake, we shall suppose that the coefficients
have as many derivatives as we shall need in the work which
follows. If u satisfies this equation, it must manifestly possess the
various derivatives indicated in the equation. As is familiar,
however, in the case of the equation of the vibrating string, there
are cases where u must be regarded as a solution of our
differential equation in a general sense without possessing all the
orders of derivatives indicated in the equation, and indeed without
being differentiable at all. It is a matter of some interest, therefore,
to render precise the manner in which a nondifferentiable function
may satisfy in a generalized sense a differential equation.
Let G  x, y  be a function positive and infinitely differentiable
within a certain bounded polygonal region R on the XY plane,
vanishing with its derivatives of all orders on the periphery of R .
Then there is a function G1  x, y  such that
4
Portrait 5. Jean Leray
(November 7, 1906 –
November 10, 1998)
(http://www-history.mcs.standrews.ac.uk/Biographies/
Leray.html).

5.   Au
xx
 Bu xy  Cu yy  Du x  Eu y  Fu  G  x, y  dxdy
R
  u  x, y  G1  x, y  dxdy
R
for all u with bounded summable derivatives of the first two
orders, as we may show by integration by parts. Thus the
necessary and sufficient condition for u to satisfy our differential
equation almost everywhere is that
 u  x, y  G  x, y  dxdy  0
1
R
For every possible G (as the G s for a complete set over any
region), and that u possesses the requisite derivatives.
Portrait 6. Sergei Lvovich
Sobolev (October 6, 1908 –
January 3, 1989)
(http://www-history.mcs.standrews.ac.uk/Biographies/
Sobolev.html).
Other famous mathematicians have also used the method of multiplication of a
suitable function by a test function and then integrate the result; e.g., (Leray,
1934), (Sobolev, 1936), (Courant & Hilbert, 1937), (Friedrichs, 1939), (Weyl,
1940), (Schwartz, 1945), (Bochner & Martin, 1948).
The test or bump functions   D  ,
properties have an interesting history:

holding the following two additional

   x  dx  1 ;

lim  x;    lim
 0
1
 x
      x
 0 
 
n
Until the year 1944 these functions does not have a specific name. It was in
this year when, in the study of differential operators, the German born and
American mathematician Kurt Otto Friedrichs (September 28, 1901 –
December 31, 1982) proposed the name “mollifier” (Friedrichs, 1944, pp. 136139).
Portrait 7. Richard Courant
(January 8, 1888 – January
27, 1972) (http://wwwhistory.mcs.standrews.ac.uk/Biographies/
Courant.html).
According to Wikipedia in its entry “Mollifier”, the paper by Friedrichs
(http://en.wikipedia.org/wiki/Mollifier - cite_note-Laxref-0)
[…] is a watershed in the modern theory of partial differential
equations. The name of the concept had a curious genesis: at that
time Friedrichs was a colleague of the mathematician Donald
Alexander Flanders, and since he liked to consult colleagues about
English usage, he asked Flanders how to name the smoothing
operator he was about to introduce. Flanders was a puritan so his
friends nicknamed him Moll after Moll Flanders in recognition of
his moral qualities, and he suggested to call the new mathematical
concept a “mollifier” as a pun incorporating both Flanders’
nickname and the verb ‘to mollify’, meaning ‘to smooth over’ in a
figurative sense.
[The Soviet mathematician ] Sergei [Lvovich] Sobolev [October
6, 1908 – January 3, 1989] had previously used mollifiers in his
epoch making 1938 paper containing the proof of the Sobolev
5
Portrait 8. David Hilbert
(January 23, 1862 –
February 14, 1943)
(http://www-history.mcs.standrews.ac.uk/Biographies/
Hilbert.html).

6. embedding theorem [Sobolev, S. L. (1938). Sur un théorème
d’analyse fonctionnelle. (in Russian, with French abstract),
Recueil Mathématique (Matematicheskii Sbornik), 4(46)(3), 471–
497], as Friedrichs himself later acknowledged [Friedrichs, K. O.
(1953). On the differentiability of the solutions of linear elliptic
differential equations. Communications on Pure and Applied
Mathematics VI (3), 299–326].
There is a little misunderstanding in the concept of mollifier:
Friedrichs defined as “mollifier” the integral operator whose
kernel is one of the functions nowadays called mollifiers.
However, since the properties of an integral operator are
completely determined by its kernel, the name mollifier was
inherited by the kernel itself as a result of common usage.
Portrait 9. Hermann Klaus
Hugo Weyl (November 9,
1885 – December 9, 1955)
(http://www-history.mcs.standrews.ac.uk/Biographies/
Weyl.html).
Portrait 10. Salomon
Bochner (August 20, 1899 –
May 2, 1982) (http://wwwhistory.mcs.standrews.ac.uk/Biographies/
Bochner.html).
References
Bochner, S. & Martin, W. T. (1948). Several complex variables. Princeton, USA: Princeton University
Press
Courant R. & Hilbert, D. (1937). Methoden der Mathematischen Physik, II. Berlin, Deutschland: Verlag
von Julius Springer (see pp. 469-470).
Griffel, D. H. (2002). Applied functional analysis. New York, USA: Dover.
Friedrichs, K. O. (1939). On differential operators in Hilbert spaces. American Journal of Mathematics,
61, 523-544.
Friedrichs, K. O. (1944). The identity of weak and strong extensions of differential operators.
Transactions of the American Mathematical Society, 55, 132–151.
Lagrange, J. L. (1759-1760). Nouvelles recherches sur la nature et la propagation du son. Miscellanea
Taurinensia, II (Œuvres de Lagrange, Tome Premier (1867), 150-316, Gauthier-Villars, France:
Paris.
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7. Leray, J. (1934). Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Mathematica, 61,
193-248.
Lighthill, M. J. (1958). Introduction to Fourier analysis and generalised functions. Cambridge, United
Kingdom: Cambridge University Press.
Schwartz, L. (1945). Généralisation de la notion de fonction, de dérivation, de transformation de Fourier,
et applications mathématiques et physiques. Annales de l’université de Grenoble, 21, 57-74.
Schwartz, L. M. (1947-1948). Théorie des distributions et transformation de Fourier. Annales de
l’université e Grenoble, 23, 7-24.
Sobolev, S. L. (1936). Méthode nouvelle à résoudre le problème de Cauchy pour les équations linéaires
hyperboliques normales. Matematicheskiui Sbornik, 1 (43), 39-71
Weyl, H. (1940). The method of orthogonal projection in potential theory. Duke Mathematical Journal, 7,
411-444.
Wiener, N. (1926). The operational calculus. Mathematische Annalen, 95, 557-584.
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