I have a curious question I stumbled upon that may be interesting to some.


Consider **real**-valued continuous functions on the circle $f\_1(x),f\_2(x),f\_3(x)$ (so they are periodic in $x \mapsto x+2\pi$).


They have the following properties:


1. $\int\_0^{2\pi} \frac{dx}{2\pi} f\_i(x) = 0$
2. $ \int\_0^{2\pi} \frac{dx}{2\pi} f\_i(x) f\_j(x) = \frac{1}{3} \delta\_{ij}$
3. $f\_1(x)^2 +f\_2(x)^2 + f\_3(x)^2 = 1$ for all $x$.


Considered as square-summable vectors on the circle, the first two conditions mean that $f\_i(x)$ are orthogonal to the constant function $1$, and they are mutually orthogonal, each with norm $\frac{1}{\sqrt{3}}$. There are infinitely many functions that satisfy properties 1 and 2.


But is there a solution now also requiring property 3 to hold?


If yes, what is an example of such a set of functions? (Or more generically, how to construct them in general?)
If no, how do we rule them out?


Thanks.


\*\*Edit: if continuity is replaced by smoothness (at least 1-differentiable), does anything change?



Orthogonal functions on circle with constraints
I think we can do it smoothly. Think of $f=(f\_1,f\_2,f\_3)$ as a map from the circle into $\mathbb R^3$. I need it to take values in the unit sphere. Mark the three obvious great circles on the sphere, and note the twelve quarter-circles in this picture.


Here is a continuous solution. As $x$ goes from $0$ to $\pi/6$, $f(x)$ goes from $(1,0,0)$ to $(0,1,0)$ following a great circle path at constant speed. As $x$ goes from $\pi/6$ to $\pi/3$, $f(x)$ follows another such path, from $(0,1,0)$ to, say, $(0,0,1)$. You can keep going like this, in a twelve-part path that eventually traverses each of those quarter-circles once. The average of $f\_i$ is $0$ because four of the twelve terms are $0$ and the remaining eight cancel in pairs. For $i\neq j$ the average of $f\_if\_j$ is $0$ because eight of the twelve terms are $0$ and the remaining four cancel in pairs. The average of $f\_i^2$ is independent of $i$.


This can be modified to a smooth solution by arranging for each of the twelve paths to be not a constant-speed path but rather one that travels along a quarter-circle beginning and ending with a brief interval of staying still, in a symmetrical way.


