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Dear David,

This is just a reflection on your question:

Since you assume that the curve is parametrized by arc-length, applying Plancharel's formula to $f'$ yields $$\sum n^2|a_n|^2 = 1.$$ Moreover, you also assume that the map $f' : S^1 \rightarrow S^1$ has degree 1 and Brezis's formula for the degree of a $C^1$ map from the circle to the circle (Google Kahane's paper Winding number and Fourier series for the formula and the amusing story behind it) yields $$\sum n^3|a_n|^2 = 1.$$

Averaging these two equations and assuming $a_1 = 0$ one gets that $$\sum_{|n|> 1} {n^2(1 + n) \over 2}|a_n|^2 = 1 \ {\rm and } \ \sum_{|n|> 1} n^2|a_n|^2 = 1 ,$$ but I don't see right now if this and could lead to a contradiction with $\sum n^2|a_n|^2 = 1$.

I don't know if this helps with your precise question, but I think Brezis's formula for the degree in terms of Fourier coefficients could come in useful.

3 added 9 characters in body

Dear David,

This is just a reflection on your question:

Since you assume that the curve is parametrized by arc-length, applying Plancharel's formula to $f'$ yields $$\sum n^2|a_n|^2 = 1.$$ Moreover, you also assume that the map $f' : S^1 \rightarrow S^1$ has degree 1 and Brezis's formula for the degree of a $C^1$ map from the circle to the circle (Google Kahane's paper Winding number and Fourier series for the formula and the amusing story behind it) yields $$\sum n^3|a_n|^2 = 1.$$

Averaging these two equations and assuming $a_1 = 0$ one gets
$$\sum_{|n|> 1} {n^2(1 + n) \over 2}|a_n|^2 = 1 \ {\rm and } \ \sum sum_{|n|> 1} n^2|a_n|^2 = 1 ,$$ but I don't see right now if this could lead to a contradiction.

I don't know if this helps with your precise question, but I think Brezis's formula for the degree in terms of Fourier coefficients could come in useful.

2 added 30 characters in body; deleted 12 characters in body

Dear David,

This is just a reflection on your question:

Since you assume that the curve is parametrized by arc-length, applying Plancharel's formula to $f'$ yields $$\sum n^2|a_n|^2 = 1.$$ Moreover, you also assume that the map $f' : S^1 \rightarrow S^1$ has degree 1 and Brezis's formula for the degree of a $C^1$ map from the circle to the circle (see Brezis's Google Kahane's paper Winding number and Fourier series for the formula and the amusing story behind it) yields $$\sum n^3|a_n|^2 = 1.$$

Averaging these two equations and assuming $a_1 = 0$ one gets
$$\sum_{|n|> 1} {n^2(1 + n) \over 2}|a_n|^2 = 1 \ {\rm and } \ \sum n^2|a_n|^2 = 1 ,$$ but I don't see right now if this could lead to a contradiction.

I don't know if this helps with your precise question, but I think Brezis's formula for the degree in terms of Fourier coefficients could come in useful.

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