Given a continuous map $f:S^1\to C$ \mathbb{C}$from the unit circle to the complex numbers, one can form its Fourier series$\sum_{n=-\infty}^\infty a_n\exp(in\theta)$. I want to stick with those$f$that give simple closed curves, bounding a closed topological disk, going round the disk in a counter-clockwise direction, and parametrized proportional to arclength. I am happy to add the hypothesis that$f'(t)$is a continuous function of$t$and that, for$t\in S^1$,$|f'(t)| = 1$. Is it then true that$a_1\neq 0$? If this is true, is$|a_1|$bounded away from zero as$f$varies? It may be that some other normalization might make the second question more tractable: for example, instead of normalizing the length to be$2\pi$by a change of scale, as I have done above, one could require that a disk of unit radius be contained in the disk bounded by$f$. Any such normalization of$f$would be highly acceptable. I'm motivated by trying to describe the space of shapes'' in the plane, by using Fourier descriptors, a topic of interest both in machine vision and in microscopy in biology. 3 Put in a greater than sign to emphasize my question Given a continuous map$f:S^1\to C$from the unit circle to the complex numbers, one can form its Fourier series$\sum_{n=-\infty}^\infty a_n\exp(in\theta)$. I want to stick with those$f$that give simple closed curves, bounding a closed topological disk, going round the disk in a counter-clockwise direction, and parametrized proportional to arclength. I am happy to add the hypothesis that$f'(t)$is a continuous function of$t$and that, for$t\in S^1$,$|f'(t)| = 1$. Is it then true that$a_1\neq 0$? If this is true, is$|a_1|$bounded away from zero as$f$varies? It may be that some other normalization might make the second question more tractable: for example, instead of normalizing the length to be$2\pi$by a change of scale, as I have done above, one could require that a disk of unit radius be contained in the disk bounded by$f$. Any such normalization of$f$would be highly acceptable. I'm motivated by trying to describe the space of shapes'' in the plane, by using Fourier descriptors, a topic of interest both in machine vision and in microscopy in biology. 2 made title more explicit, added motivation for question # Simple closed curves and thecoefficentof$\exp(i\theta)$intheassociated Fourier series Given a continuous map$f:S^1\to C$from the unit circle to the complex numbers, one can form its Fourier series$\sum_{n=-\infty}^\infty a_n\exp(in\theta)$. I want to stick with those$f$that give simple closed curves, bounding a closed topological disk, going round the disk in a counter-clockwise direction, and parametrized proportional to arclength. I am happy to add the hypothesis that$f'(t)$is a continuous function of$t$and that, for$t\in S^1$,$|f'(t)| = 1$. Is it then true that$a_1\neq 0$? If this is true, is$|a_1|$bounded away from zero as$f$varies? It may be that some other normalization might make the second question more tractable: for example, instead of normalizing the length to be$2\pi$by a change of scale, as I have done above, one could require that a disk of unit radius be contained in the disk bounded by$f$. Any such normalization of$f\$ would be highly acceptable.
I'm motivated by trying to describe the space of shapes'' in the plane, by using Fourier descriptors, a topic of interest both in machine vision and in microscopy in biology.