Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 75 / 1 / c / Solution

Let $d$ be the dimension of a proposed real Chebyshev system on the circle, and regard its functions as continuous $2\pi$-periodic functions on the real line. Choose lifted points ${\theta }_0<{\theta }_1<\cdots <{\theta }_{d-1}<{\theta }_0+2\pi$ and set ${\theta }_d={\theta }_0+2\pi$. Move the points byFor every $s$ the points remain distinct on the circle. Indeed, each consecutive gap is a convex combination of positive gaps; the closing gap is also a convex combination of the original closing gap and ${\theta }_1-{\theta }_0$. The evaluation determinantis therefore nonzero for every $s$ by the determinant criterion for a Chebyshev system. It is continuous, so the intermediate value theorem forbids a sign change. At $s=1$, however, periodicity makes its rows the cyclic permutation of the initial rows, giving $D(1)=(-1{)}^{d-1}D(0)$. For even $d$ this reverses the sign, a contradiction. Hence every such periodic Chebyshev system has odd dimension; none has even dimension. This determinant argument avoids assuming differentiability or simple zeros of the functions.