5. LIE'S CONCEPT OF A CONTINUOUS GROUP OF TRANSFORMATIONS WITHOUT THE ASSUMPTION OF THE DIFFERENTIABILITY OF THE FUNCTIONS DEFINING THE GROUP.
It is well known that Lie, with the aid of the concept of continuous groups of transformations, has set up a system of geometrical axioms and, from the standpoint of his theory of groups, has proved that this system of axioms suffices for geometry. But since Lie assumes, in the very foundation of his theory, that the functions defining his group can be differentiated, it remains undecided in Lie's development, whether the assumption of the differentiability in connection with the question as to the axioms of geometry is actually unavoidable, or whether it may not appear rather as a consequence of the group concept and the other geometrical axioms. This consideration, as well as certain other problems in connection with the arithmetical axioms, brings before us the more general question: How far Lie's concept of continuous groups of transformations is approachable in our investigations without the assumption of the differentiability of the functions.
Lie defines a finite continuous group of transformations as a system of transformations
having the property that any two arbitrarily chosen transformations of the system, as
applied successively result in a transformation which also belongs to the system, and which is therefore expressible in the form
where
are certain functions of
and
. The group property thus finds its full expression in a system of functional equations and of itself imposes no additional restrictions upon the functions
. Yet Lie's further treatment of these functional equations, viz., the derivation of the well-known fundamental differential equations, assumes necessarily the continuity and differentiability of the functions defining the group.
As regards continuity: this postulate will certainly be retained for the present—if only with a view to the geometrical and arithmetical applications, in which the continuity of the functions in question appears as a consequence of the axiom of continuity. On the other hand the differentiability of the functions defining the group contains a postulate which, in the geometrical axioms, can be expressed only in a rather forced and complicated manner. Hence there arises the question whether, through the introduction of suitable new variables and parameters, the group can always be transformed into one whose defining functions are differentiable; or whether, at least with the help of certain simple assumptions, a transformation is possible into groups admitting Lie's methods. A reduction to analytic groups is, according to a theorem announced by Lie[10] but first proved by Schur,[11] always possible when the group is transitive and the existence of the first and certain second derivatives of the functions defining the group is assumed.
For infinite groups the investigation of the corresponding question is, I believe, also of interest. Moreover we are thus led to the wide and interesting field of functional equations which have been heretofore investigated usually only under the assumption of the differentiability of the functions involved. In particular the functional equations treated by Abel[12] with so much ingenuity, the difference equations, and other equations occurring in the literature of mathematics, do not directly involve anything which necessitates the requirement of the differentiability of the accompanying functions. In the search for certain existence proofs in the calculus of variations I came directly upon the problem: To prove the differentiability of the function under consideration from the existence of a difference equation. In all these cases, then, the problem arises: In how far are the assertions which we can make in the case of differentiable functions true under proper modifications without this assumption?
It may be further remarked that H. Minkowski in his above-mentioned Geometrieder Zahlen starts with the functional equation
and from this actually succeeds in proving the existence of certain differential quotients for the function in question.
On the other hand I wish to emphasize the fact that there certainly exist analytical functional equations whose sole solutions are non-differentiable functions. For example a uniform continuous non-differentiable function
can be constructed which represents the only solution of the two functional equations
where
and
are two real numbers, and
denotes, for all the real values of
, a regular analytic uniform function. Such functions are obtained in the simplest manner by means of trigonometrical series by a process similar to that used by Borel (according to a recent announcement of Picard)[13] for the construction of a doubly periodic, non-analytic solution of a certain analytic partial differential equation.
[10] Lie-Engel, Theorie der Transformationsgruppen, vol. 3, Leipzig, 1893, §§ 82, 144.
[11] "Ueber den analytischen Charakter der eine endliche Kontinuierliche Transformationsgruppen darstellenden Funktionen," Math. Annalen, vol. 41.
[12] Werke, vol. 1, pp. 1, 61, 389.
[13] "Quelques théories fondamentales dans l'analyse mathématique," Conférences faites à Clark University, Revue générale des Sciences, 1900, p. 22.