3.3 THEORY OF FUNCTIONS
221
By what we have proved, f (x) depends only on I1 x I1 or equivalently, same, on IIx112; in other words, there exists a function a : JR -> R such that f(x) = a (11XI12) ,
xEJ3.
It remains to verify the additiveness of the function a. To this end, fix ), p E R+ arbitrarily, and choose vectors x, y E JR3 so that A = IIx112' P = IIy112, and xly. Then, in view of (1) and the Pythagorean theorem, 2a(.\) + 2a(p) = 2a (11x112) + 2a (IIy112) = 2f (x) + 2f(y)
=f(x+y)+f(x-y)=a(Ilx+y112)+a(IIx-yl12) = a (11x112 + I1y112)
+a
(11x112 + I1y112) = 2a(A
+ p).
Thus a is additive on R+ and, as we may choose its value on JR_ arbitrarily,
setting a(-a) = -a(A), A E R+, we get the additive function desired. Conversely, simple substitution shows that the functions of the form (3) are solutions of the problem (1)-(2). Remarks. 1. We obtain the same solution if we assume (1) only for y satisfying Ilyll = 1 and, in a more general way, consider vectors of a real inner product space of dimension at least three. Of course, in this case we encounter further, essential difficulties.
2. The two-dimensional case has proved to be still harder. It is an open question whether in this case the problem (1)-(2) admits solutions different from (3).
Problem F.48. For any fixed positive integer n, find all infinitely differentiable functions f : I[8n -+ JR satisfying the following system of partial differential equations: n
>aikf
0,
k=1,2,....
i=1
Solution. Let n E N, and consider the system of partial differential equations n a2kf
= 0,
k = 1, 2, ...
(1/n)
i=1
for the infinitely differentiable unknown function f : Rn -+ R. Obviously, linear combinations of any partial derivatives of solutions are solutions
again. We show that there is a universal solution in the sense that all solutions are linear combinations of partial derivatives of this universal solution. We begin with an important observation.
Lemma 1. If f is a solution of system (1/n), then 3f = 0 (i = 1, 2, ... , n), so f is a polynomial of degree not greater than 2n - 1 in each variable.