Q:

Fourier series and transform related to Epicycles

Let $\gamma:\mathbb{R}\to\mathbb{C}$ be a continuous periodic curve having a bounded variation.
1) Is it true that one can find a sequence of numbers $(r_n)_{n\in\mathbb{N}}\subset (0,\infty)$ and some function $\varphi:\mathbb{R}\times \mathbb{R}\to\mathbb{R},\ \varphi\in C^{1}$ such that the sequence of curves:
$$\gamma_n:\mathbb{R}\to\mathbb{C},\ \gamma_{n}(t)=\sum_{k=0}^n r_k e^{i\varphi(k,t)}$$
converges uniformly to $\gamma$ on $\mathbb{R}$? Moreover can we choose $(r_n)_{n\in\mathbb{N}}$ to be decreasing or strictly decreasing? Can we choose $\varphi (x,t)=2x\pi t+f(x),\forall (x,t)\in\mathbb{R}\times \mathbb{R}$, where $f$ is a $C^1$ function?
2) Is it true that there is a function $r:\mathbb{R}\to \mathbb{C},\ r\in L^1 (\mathbb{R})$ such that $r(k)=r_k e^{if(k)},\ \forall k\in\mathbb{N}$ and 
$$\gamma(t)=\int_{-\infty}^{\infty}r(x)e^{i2\pi x t} dx\ ?$$
I'm trying to prove this theorem: http://www.u.arizona.edu/~aversa/scholastic/Mathematical%20Power%20of%20Epicyclical%20Astronomy%20%28Hanson%29.pdf
See the begining of the last page. You can take a look also, at:https://www.youtube.com/watch?v=QVuU2YCwHjw

A:

The answer to (1) is yes, due to the Jordan criterion for convergence of the Fourier series. 
The answer to (2) is no, since by the Riemann-Lebesgue lemma $\gamma$ cannot be periodic (or even almost periodic) and be the (inverse) Fourier transform of an $L^1$ function. 

