I am working on an adaptation of the Remez algorithm for best uniform approximation of an continuous functions by sums of cosines, i.e. we want $f(x) \approx \sum_{j=1}^n \gamma_j\, \cos (\omega_j\, x)$. In order to get some theoretical results, I need an upper bound for the zeros of a sum of cosines on a specified interval.
Let $I=[0,R]$ be some interval. We may say that the family of functions $\mathcal F$ defines a Haar space (Chebyshev space) of (Haar) dimension $d$, if every $h$ in the span of $\mathcal F$ has at most $d-1$ roots in $I$. Note that this definition is a bit unusual, since $d\neq n$ is allowed. We choose now $\mathcal F=\{\cos(\omega_j \cdot)\mid j=1,\dots,n\}$ for an increasing sequence of frequencies $0\leq\omega_1<\dots<\omega_n$. I am interested in an upper bound of the roots of some non zero function $h\in \text{span }\mathcal F$ in dependency of the highest frequency $\omega_n$ and the interval length $R$. I am sure that the number of roots is bounded, but I can not give it explicitly.
Clearly, while choosing $g(x)=\cos(\omega_nx)$ we have a function with $\lfloor \frac {\omega_n \pi} R\rfloor $. So, for the haar dimension we have $\lfloor \frac {\omega_n \pi} R\rfloor \leq d$.
In oder to get some more concrete results, I considered similar problems like proving that $\{\cos (k \cdot)\mid k=0,\dots ,n\}$ is a Chebyshev system of dimension $n$. Sadly, the proofs related to the fact that we can represent $\cos(k\cdot)$ as a linear combination of $\cos(\cdot)^j$ and then we can use the fact that the standard polynomials form a Chebyshev system. Sadly, this aproach does not work here. Moreover, I looked on the proof that $\{x^{\lambda _j}\mid 0\leq \lambda_1<\dots<\lambda_n\}$ forms a Chebyshev system. I was hoping to use this result for my problem, sadly without success.
I am grateful for any help about showing that $\mathcal F$ forms a Chebyshev system with explicit dimension $d$ or any literature about this problem.