A Fourier cosine series of type $f(x)=\frac{a_0}{2}+\sum_{k=1}^\infty a_k \cos(2^k x)$.

Question

For a function $f(x)$ defined on $x \in [0,\pi]$, one can write $f(x)$ as $$f(x)=\frac{a_0}{2}+\sum_{k=1}^\infty a_k \cos(kx)$$ for some coefficients $a_k$. Fourier claimed that any function $f(x)$ can be expanded in terms of cosines in this way, where $$a_k=\frac{2}{\pi} \int_0^\pi f(x) \cos(kx) dx.$$ Are there similar results for a different cosine series, as $\sum_{k=1}^\infty a_k \cos(2^{k}x)$?