How to build smooth families of functions satisfying a power unity?

Question

The equation $$f_1(t)^2+f_2(t)^2 = 1$$

Has the very famous solution $\cases{f_1(t) = \sin(kt)\\f_2(t) = \cos(kt)}$

Sometimes called the trigonometric unity or the triangle union. Sin and cos are also functions which are famous for being very well-behaved.

Can we systematically find other families of functions:

$$\sum_{k=0}^N f_k(t)^n = 1$$

while still enforcing / encouraging well-behavedness in some sense? Bonus points if you can manage to do it using only linear algebra and nothing non-linear at all.

Answer 1

Easily.

Choose arbitrary functions (as well-behaved as one likes) $g_i$ with $|g_i| \le 1$ for $i < k$. Define $$f_1 = g_1\\f_2 = g_2\sqrt{1 - f_1^2}\\f_3 = g_3\sqrt{1 - f_1^2 - f_2^2}\\\vdots$$

Finally, choose $f_k$ so that $f_k^2 = 1 - f_1^2 - f_2^2 - ... - f_{k-1}^2$. Obviously if the functions are to be continuous, this is just a matter of choosing whether or not to change signs each time it comes to $0$.