Given a nondegenerate smooth simple closed curve $f: [0,2\pi]\to \mathbb C \backslash \{0\}$ with winding number (around origin) $1$. Let $f_n: [0,2\pi]\to \mathbb C \setminus \{0\}$ be such that $f_n(t) := f(nt)$ for $n\in \mathbb Z$ (viewing $f$ as a $2\pi$-periodic function on $\mathbb R$). I have several questions concerning the sequence $(f_n)$.
- Is $(f_n)$ linearly independent?
- Is $(f_n)$ dense in $L^p([0,2\pi])$ for some $p$?
- Does $(f_n)$ form a Schauder basis of $L^p([0,2\pi])$ for some $p$?
This sequence is exactly the basis of Fourier series if $f(t)=\exp(it)$. The general case can be seen as an extension of Fourier series.
A well-known intuition for Fourier series is that it represents the transition from time-domain to frequency-domain. This intuition naturally leads to the question: Does the particular waveform of $\exp(it)$ matter so much? Can we decompose a periodic function into waves of other shapes?
Indeed the complex exponential function is so special, that the resulting Fourier series has many good properties such as othogonality and convolution theorem, which obviously can't be extended to the general case. However, I still think the questions above are meaningful. I think answering them can bring us insights of Fourier series and its "frequency-domain" intuition.
The winding number condition is necessary, sense if we choose $f(t) = \exp(2it)$, then $(f_n)$ can't be dense in $L^p$.
Here is a relevant post (Generalize Fourier transform to other basis than trigonometric function) which asked a similar question. Another relevant post: Why sin and cos in the Fourier Series?
