Will Fourier Series converge even if you only use Prime Integer frequencies?

Question

So there is a Fourier Series for a function $f(x)$ with period $P$:

$$ f(x) = \frac{A_0}{2} + \sum_{n=1}^{N} A_n \cdot \cos \left(\frac{n 2 \pi x}{P} + \phi_n \right) $$

Let $\frac{2 \pi x}{P} = t$ then

$$ f(x) = \frac{A_0}{2} + \sum_{n=1}^{N} A_n \cdot \cos \left(n t(x) + \phi_n \right) $$

Suppose that for this particular function $f(x)$ the series converges, and my question is the following:

If I only use prime numbers for my frequencies $n$, will the series still converge? Even if it needs to be longer than the original one?

Answer 1

As TonyK said, the series of prime-numbered terms will converge to a different function, because a Fourier series of $f$ is unique. Unless, of course, we had $A_n=0$ for all composite $n$.

Indeed, the new sum, call it $g$, will be orthogonal to every trigonometric polynomial $\cos \left(n t(x) + \phi_n \right)$ with composite $n$. If the original series had $A_n\ne 0$ for some composite $n$, then we have a lower bound on the modulus of $\int (f-g) \cos \left(n t(x) + \phi_n \right)$, which gives us a quantitative way of saying that $g$ will not be very close to $f$.