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Given a nondegenerate smooth simple closed convex curve $f: [0,2\pi]\to \mathbb C \setminus \{0\}$ with winding number (around origin) $1$, and $f$ have zero mean. 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 $(f_n)$.

  1. Is $(f_n)$ linearly independent?
  2. Is $(f_n)$ total in $L^p([0,2\pi])$ for some $p$?
  3. 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 total 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?

Update: I have put a bounty. While the problems are not particularly interesting, I think solving them can be a chance to see new techniques. In addition, I do want to see an answer for the problems. I have been thinking about them since the first time I learned Fourier series. I think the first two questions should be true; otherwise, I would be surprised.

To make the problems more promising, I have added the conditions that the curve is convex and $f$ has zero mean, though I think they might not be very important.

I also found some posts regarding Fourier series of closed curves: Simple closed curves and the coefficent of $\exp(i\theta)$ in the associated Fourier series and What geometric information is carried by the Fourier coefficients of the components of a closed curve?

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    $\begingroup$ The function $f$ should be a $2\pi$-periodic function defined on $(-\infty,\infty)$. $\endgroup$ Commented Mar 17 at 13:49
  • $\begingroup$ @IosifPinelis Thank you! I have edited the question. $\endgroup$ Commented Mar 17 at 13:57
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    $\begingroup$ TeX note: set difference should be typeset as $\mathbb C \setminus \{0\}$ \mathbb C \setminus \{0\}, not $\mathbb C \backslash \{0\}$ \mathbb C \backslash \{0\} (note the difference in spacing). I have edited accordingly. $\endgroup$
    – LSpice
    Commented Mar 17 at 14:50
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    $\begingroup$ dense' should be replaced twice with total', isn't it? $\endgroup$ Commented Mar 17 at 18:45
  • $\begingroup$ @ChristopheLeuridan True. Have edited the question. $\endgroup$ Commented Mar 18 at 9:40

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Partial answer: About linear independence, it is true that if $f$ is non constant then the dilations $f_n(x)=f(nx), n\in \mathbb{N}$ are linearly independent. In fact suppose for a finite sum we have \begin{equation} \sum_{n=1}^{N} c_n f_n(x) = 0, \,\, a.e.-x, \end{equation} then the Fourier coefficients must vanish so we arrive at the equation \begin{equation} \sum_{n=1}^N \sum_{{ d \in \mathbb{N} \, : \, dn=m}} c_n \hat{f}(d) = 0, \forall m =1,2,3... \end{equation} and analogously for the negative Fourier coefficients. Let $$ d_0=\min\{d \in \mathbb{N}: \hat{f}(d)\neq 0 \} \quad n_0=\min\{ n\in\mathbb{N} : n\leq N, c_n \neq 0 \}. $$ Then we have that $$ \sum_{n=1}^N \sum_{{ d \in \mathbb{N} \, : \, dn=d_0 n_0}} c_n \hat{f}(d) = c_{n_0}\hat{f}(d_0) = 0 $$ which is a contradiction.

EDIT: If you want to consider dilations and reflections, i.e. $f_n(x)=f(nx), n\in \mathbb{Z}$, just write the function as a sum of an even and an odd part $f=g+h$, then $$ \sum_{n=-N}^N c_nf(nx)=0 \iff c_0g(0)+\sum_{n=1}^N(c_n+c_{-n})g(nx) + \sum_{n=1}^N(c_n-c_{-n})h(nx) =0. $$ If $g,h$ are non trivial then you get two equations for "positive dilations" and so the problem is reduced in the previous case.

The problem about total sets or a Schauder basis I expect to be much more complicated. In the article "A Hilbert space of Dirichlet series and systems of dilated functions in $L^2(0,1)$" of Hedenmalm, Lindqvist and Seip the authors give a characterization of the functions for which $f(nx)$ is a Riesz system (an isomorphic image of an orthonormal basis) for $p=2$. In particular it is not always true that the system of dilations is total in $L^2(0,2\pi)$.

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  • $\begingroup$ I also found some posts regarding Fourier series of closed curves and added them to the question. $\endgroup$ Commented Mar 20 at 6:46
  • $\begingroup$ Thank you for the answser and the reference. You have proven that the positive dilations are linearly independent provided $f$ is non constant. However, if $f$ is an even function, then $f_{−1}-f_1=0$, and $(f_n)$ is linearly dependent. $\endgroup$ Commented Mar 20 at 6:50
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    $\begingroup$ @ZhangYuhan This case can be reduced to the previous one, see the edit $\endgroup$ Commented Mar 20 at 8:01

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