I am trying to connect things I learned and combine the ideas in Functional Analysis with other areas that I know so I can understand both of them better. So here's a question I got in my mind:
Suppose $f:\mathbb{R} \longrightarrow \mathbb{R} $ is a continuous function in $L^2(\mathbb{R})$. One knows that $f + i \cdot0$ belongs to the space $L^2(\mathbb{C})$. So we could naturally ask the question that what information could we get from the Fourier series of $f$ about the dynamics of the group $\langle f \rangle$ acting on $\mathbb{R}$? Do other possible estimates of $f$ with another family of functions help us in that regard? What properties of the dynamics of $\sin$ and $\cos$ waves/functions could be inherited by $f,$? Can we choose some specific $g$ for an initial $f$ so that $ f+ i.g \in L^2(\mathbb{C}) $ and study the dynamics of $f$ by understanding the dynamics of the group $ \langle f,g \rangle$ using the Fourier series of $ f + i \cdot g $? Could we restrict ourselves with studying the dynamics of only the members of this family (trigonometric polynomials) or a finite linear combination of them? What about other families that estimate $f$? Could we generalize these ideas on other spaces like manifolds or Hilbert Spaces?