Let $f$ be an $L^2$ function on the unit circle $f \in L^2(S^1, d \theta)$. This is equivalent to giving a Fourier series $\sum_{n \in \mathbb{Z}} a_n e^{i n \theta}$ with $\sum_{n \in \mathbb{Z}} | a_n|^2 < \infty$. Then for every subset $S$ of $\mathbb{Z}$ the series $\sum_{n \in S} a_n e^{i n \theta}$ corresponds to an $L^2$ function $f_S$ on the circle.
Assume now that $f$ is continuous (or $L^p$ for some $p>2$ ($\,p\ge 1$ ?), or some Holder class, etc). Is the function $f_S$ again of the same class?
Observation: For smooth functions ( analytic functions) it's true, the Fourier coefficients are characterized as decreasing faster than any power of $|n|$ ( some exponential $q^{|n|}$, $|q|<1$) .
This is a broad question with no definitive answer. You are asking for a description of characteristic functions $\chi_S$ that are Fourier multipliers bounded on $L^p$, or $C^\alpha$, etc.
Certainly, this is so when $S$ finite or $\mathbb{Z}\setminus S$ is finite. The best known nontrivial case is $S=\{n\in \mathbb{Z}:n\ge 0\}$ which is directly related to the Hilbert transform. Thanks to M. Riesz, we know that $\chi_S$ is a multiplier on $L^p$ for $1<p<\infty$. It's not a multiplier on $L^1$ or $L^\infty$, or on $C(\mathbb{T})$. It is a multiplier on $C^\alpha$ for $\alpha\in(0,1)$, however. None of these results are trivial.
Suggested reading: An Introduction to the Theory of Multipliers by Ronald Larsen, and the references to the Wikipedia article on Fourier multipliers.